A view from nowhere - quantum reference frames & uncertainty

0:00 I'm not having said that she is from Pittsburgh and was a colleague of ROG for the last five, six years. Another colleague of ROG was Michael Dixon, who is from the University of Indiana, and he will talk to us about a view from nowhere, quantum mechanics, Welcome, Reference, Frames, and Uncertainty. Thanks, of course, to Jeff and the other organizations for the invitation. And that is, in fact, the title of my talk. Also, I should simultaneously thank, and to some people in the audience, apologize. I've given versions of this talk some quite radically different from what I'm about to say but nonetheless in my own mind versions of the same material at various places and many people who were at those places at those times are also here so I apologize for saying some of the same stuff and I also thank you and you can thank yourself because of some of the comments you made to me I won't say exactly the same thing and I want to particularly thank Harvey Brown who's been pretty helpful to me I'm actually an officer this year and I've had some conversations both over email and in person with Harvey about reference frames and some of that has informed what I'm going to say today I won't go through I won't explain these three parts of the talk until it's more appropriate to do so so I'll just start with motivation There are a few motivations, actually, for giving this talk, in spite of the fact that many of you have heard some version of it before. And one has to do with the nature of the conference being in memory of Rob. I guess in 1996 or 7, Rob and I decided to write a book. And then about 2001, we decided not to write a book, having written part of it. And basically, we had lots of reasons for ditching the project. The main one probably was that we were both getting more and more distracted by other projects.

2:30 And, in fact, we talked on the phone when we finally decided to set this one aside. Both of us said, yeah, you know, there are other things I'm interested in working on. I don't think I really can keep working on this book in all honesty. And we both discovered in the sort of careful way that you do that we both felt exactly the same way about it. and then it turned out only a few months later we discovered that what we were both distracted by was an attempt to understand Boer in what turned out to be somewhat similar terms and that is the origin of this particular project even though five minutes from now I won't say the name Boer again because it's sort of gotten beyond that a bit but just to explain with Bohr historically. One of the arguments that I've made in previous papers and talks is that for Bohr, and I should also mention that I was also motivated by some stuff that Arthur Pine and others have written, particularly Mara Beller. Arthur and I actually disagree about this, but since I'm not going to give a talk about Bohr, I'll spare myself. Arthur's correcting me in public. But what I've argued elsewhere understands the uncertainty principle as arising from a disturbance not of the system being measured in the way you might imagine in the Heisenberg microscope case, the uncertainty arises by, you actually disturb the momentum of the thing you're measuring by measuring its position, for example, in the way you're all familiar with, but rather by disturbing the reference frame that defines what you mean by momentum or position by making the corresponding measure. I think there are some pros and some cons of this view, of the view that Bohr, in my opinion, expressed. I think the con, or a con, of Bohr's own writing on this issue is that he implicitly ran up against the fact that at his time a satisfactory quantum theoretic count of what it would mean to be a reference string. And he also, I think, rather explicitly ran up against the fact that he didn't

5:00 either care to express or perhaps didn't understand some of the mathematics that, in my view, is actually essential to describing what you would mean by a quantum mechanical reference string. On the pro side, I think that Board's attitude illustrates acutely aware of the need to say how mathematical objects, such as whatever object you might use to represent position or momentum, how those objects actually do represent physical properties. That is to say, he was acutely aware of the need to clarify the empirical significance of bits of mathematics. And it's that that I want, it's from that point that I want to take off. And so one of the issues I want to keep in the front of our mind is how do these familiar bits of mathematics, like X's and P's, how do they get their empirical significance? What exactly is their empirical significance? And why I frame the issue in that way will become clear as I go. That's sort of the historical motivation for this talk. There's a conceptual motivation as well. Well, imagine the following standard 6th grade textbook problem. You say something like a train traveling from Baltimore to Chicago takes 10 hours, and from Chicago to Seattle takes 40 hours. What's the average speed of the train? Well, the diligent student will go to an app that's probably go to mapquest.com or something like that these cities, divide by the total number of hours, and they'll come up with an answer of something like 50 and a third miles per hour. I did it. I'm a diligent student. A clever but lazy student would say the average speed of the train is zero relative to itself. Why do you have to, in some technical sense, accept that answer? Well, you have to accept that answer because you never stated in the question what was the record trait relative to which you were asking the question. You never stated it explicitly. And so you might think that the clever student is annoying because they quite clearly circumvented the point of the question. Or you might think they're lacking in common sense, or that they're a smart owl.

7:30 technically speaking, you have to accept the answer. And say they're lacking common sense. Because, of course, we all know that implicitly when you ask the question, you meant the reference frame in which these cities are at rest. We all know that's what I meant when I stated the question. And that's why we can say to the clever student, that was very clever but annoying. So we all know when we do physics, when we ask questions pain questions, we all know most of the time what was the intended frame of reference and we don't state it explicitly. But what I want to say today is that not paying careful attention was slowing back up. So in other words, what we know is implicitly how to connect the spatio-temporal notions that appeared in the question with the physical world. We implicitly all know how to make this connection in the case of a train question. But what I want to say today is not paying careful attention to the manner in which those spatiotemporal notions, which you might represent by X's and V's and P's and whatever, not paying careful attention to how they get empirical content can, in fact, be problematic sometimes. I think it has a few consequences. of. One of them is it can lead to misunderstanding or misinterpretation or at least incomplete understanding of certain very familiar results. And my first example is going to be of that sort. So when I talk about the hydrogen atom, I'll talk about the sense in which not being careful when you write down the solution to the show in your grade of hydrogen atom, not being careful about what you mean by X and P in that solution can actually lead you to that anyone in this room has ever had that misunderstanding. And I mean that sincerely, not facetiously. But I want to pick the example because it's very simple and it illustrates how you could come to have that misunderstanding but not be careful. Another consequence that could arise from not being careful about how X's and P's are connected with empirical content, rather given

10:00 empirical content, it can lead to a lack of awareness of the potential for a stipulated, implicitly stipulated frame of reference, such as the frame in which Chicago, Baltimore, and Seattle are at rest. It can lead to a lack of awareness that this sort of implicitly specified reference frame may not actually be legitimate to play that The thing to which you're implicitly appealing as your frame of reference may not actually suitably play such a role. Let me give you an example to illustrate this point, and again, it's not an example where people have made this mistake, it's simply an example to illustrate the point. There is, I found out recently, reading literature in astrometrics, which is the initially boring sounding science of measuring where things are, as long as those things are very, very far away from us. And there is an astrometric, well, there's more than one, but to keep things simple, there is an international common reference frame. It's defined by the positions in the sky of 219 bodies that are specified. and basically so the idea is if you are moving with respect to the frame defined by these 219 bodies there's an earth mass then you are moving and if you're not then you're not, or to be more careful about it, if you're accelerating with respect to them, you're really accelerating so these things are split to define an inertial frame what's left? it's just, well about it. That just happens to be the number. In fact, it's going to be revised. And this is the point that I wanted to mention. I think when I say this, we'll see what it is not terribly important, what the number is. There are, apparently there are in the works, again, this is, I'm not an expert on any of this stuff, okay? I just started reading this picture a week ago. But apparently there's going to be some satellite visions or satellite observations, I guess I have to say, in the not too distant future, I think in the next three years or so, in which all these astrometricians believe that they'll improve on this reference frame. What do you mean improve on this reference frame? Well, the point is this. Why do they think those things define a reference frame? Why have those bodies been picked out as defining this reference frame?

12:30 Well, the answer is that it actually goes back to discussions that happened in the 19th century. So put the ICRF on hold for a minute. Let me back up the 19th century for just two minutes. One of the things that happened in the late 19th century was people became concerned about the fact, the apparent fact, I think it is the fact, that Newton's law of inertia, as stated, doesn't seem to have any empirical content. So what does Newton's law, what does the law of inertia say? It says if there are no forces acting on a body, then it moves in a straight line, essentially. I don't want to get bogged down if you don't think that's the right way of saying it, because it's not going to be terrible for you. Well, of course, the problem is that you don't have given to you antecedently a notion of straight line or force-free. In fact, very often you would say, I think the straight lines are the lines followed by force-free bodies. Or if you want to grant yourself the notion of the straight line, then you'll say, well, the things that are smoothing in those straight lines, those things are force-free. So you don't actually have any way of hooking this thing up with the world without making some assumptions. And so various people worried about, how can I restate the law of inertia? in such a way to make it an actual empirical principle. And I don't want to rehearse the discussion because I don't have time to. And again, I'm not really an expert on this, but I started reading this stuff about three weeks ago. the upshot is that you have to define you have to state the law of inertia in terms of more than as it happens, two bodies. So, for example, you can say, suppose I have, you can eliminate the need for an antecedent notion of a straight line, for example, by saying, suppose I have three bodies that are moving proportionally the same, sorry, they're moving in proportional velocity. So, that just means that if one of them moves three units of distance in a given amount of time, we'll be able to move some k times three, and that will be true into the indefinite future. So, they're moving in proportional velocities, and they're non-coplanar. Then you can say any fourth force-free body that is introduced into the system will also move proportionally to those three. So you've eliminated the need to mention straight lines.

15:00 But you haven't eliminated the notion of force-free. So then there's an additional discussion. Well, how can I characterize bodies as being force-free without simply having to assert that they are such? That's a harder discussion, of course. introduce, if you're familiar from discussions from the late century and early 20th, that you always introduce extra forces, if you like. one thing that people did was to try to characterize in a fairly generic way what they meant by force free. So they would say things like, well, something can be a force only if it obeys the so-called action-reaction principle, and so on and so forth. I don't want to get into that discussion because let's come back down to the ICRF, International the way it's done there is you simply appeal to the theories that happen to be around right now and you say well the forces that can be acting on the body are the forces we know about from gravitational theory, electromagnetic theory and so forth and so the way this ICRF gets updated or the way that people anticipate that it will get updated is that they will be able to make measurements of motions of bodies relative to these things and those motions will have in them anomalous accelerations And moreover, those bodies that have these anomalous accelerations, they'll be anomalous in the sense that after I've taken account of all the forces acting on the body, it still has some acceleration in it that I can't account for by any force. And so then I had to say, aha, my reference frame wasn't actually inertial in the first place. Because that anomalous acceleration is relevant to this supposedly inertial reference. So that's how these things get sort of adjusted. You can imagine how this would go. I mean, what you do, in fact, what happened historically, is you start off by saying the sun is inertial. And then you'd find motions of everything with respect to that. Then you'd find out that there are anomalous accelerations with respect to the sun. And so you'd say, well, the sun actually isn't completely inertial. And you can start just trying to do this constant updating process that goes on. Okay. The point of this discussion is simply to say that if you naively say to yourself, well, let's just let the four walls of this room define our reference range. And then you start doing physics with respect to that, find yourself in trouble. If you don't keep in mind the fact that those four walls were stipulated by you to be a reference frame, and they may not serve the purpose well enough for you. They may actually be accelerating. They may not be inertial, and that could be important, because your laws of motion are to be valid in that frame.

17:30 Whether it's important, of course, depends on a lot of things, notably the accuracy with which you care about your results and the amount of acceleration or rotation or what have you that's happened. That's why the sun is good enough for lots of purposes. ICRF so far is good for any purpose because no one's been able to make the measurement accurate enough to determine whether or not there are anomalous accelerations in the structure. Okay, so the next main point is not only could you misunderstand some of the results that you derive in the theory, but you could actually be misapplying the theory in the first place by working in a frame in which the theory is not actually valid. Or more generally, for the purpose that you're using. Finally, what I want to say is sort of the main point of this talk is that by being careful about these things, by being careful about how our mathematical representations of position and momentum, for example, actually connect up with the empirical world, by being careful about the status of those things that we stipulate to be called reference strengths, We can actually learn something about the theory of your questions, in this case, the theory of your questions on the camera. Okay. What about this business, the view from nowhere? It has nothing to do with Nagel or anything like that. But it's not completely irrelevant. I mean, Laura, I mean, I did actually hear the connection between my partner in the content of this talk. So, first of all, a couple of points about my own attitude, which I'm not going to argue for these points, okay? I'm just telling you that so you know where I'm coming from. First of all, my attitude is vaguely relationalist, despite the arguments of some people in this room that I shouldn't be a relationalist. And I don't have an argument. If you ask me afterwards, okay, you said you weren't going to argue over what your argument is a relationalist. I don't have an argument, okay? I should say, however, that nothing that I say should be translated into the context of general relativity. Absolutely nothing that I say should be attempted to be translated into that context. I'm pretty sure that serious problems arise, and I don't know how to do translation anyway. Actually, I know that serious problems arise, and I could say what some of them are,

20:00 but I don't want to think about general relativity, and therefore not about functionality. my attitude is also vaguely operationalist in the sense that physical bodies are going to be used to define reference frame it doesn't mean that there has to be a kind of one to one connection between physical bodies and reference frame so I don't mean that every reference frame is determined by the four walls of some room the ICRF is perfectly good none of those 219 bodies is the reference frame you don't measure things relative to that star there relative to a frame that is defined in a clever way in terms of the positions of those vibes. I think this is vaguely operationalist because, well, my attitude towards that fact is vaguely operationalist. I don't think you could do better than that. I think I can also appeal to those who don't have my vaguely operationalist tendencies by saying, look, as a matter of fact, this is how we do measurements like that. As a matter of fact, we measure positions and momenta and velocities and angles and what have you relative to some frame that is defined by actual physical bodies. Even if you believe that there's a frame out there that's the frame that's not defined by physical bodies, you can still acknowledge that as a matter of course, we don't make measurements with the fact that that thing. Since I'm vaguely relationalist and vaguely operationalist, I don't think that thing exists. and nonetheless I think it's important to ask the question what would it be like to be in that thing and describe physics from that point of view and that's why I call this the view from nowhere it's the view from this particular frame that doesn't exist my last proviso is that nothing that I say is quantitative so my discussion is going to be purely qualitative of the uncertainty relations. I'm not going to have anything to say about the particularly quantitative nature of the uncertainty relations. So probably uncertainty relations is a somewhat misleading term of phrase here, and I should rather just talk about non-cognitivity. So here's the problem that's implicitly raised by the discussion up to this point. How can we be careful about the legitimacy of a reference frame?

22:30 How can we manage this? clue about how that happens in the case of making sure that your frame is inertial. You look for a novelist acceleration. The first problem that arises is, obviously, that we don't have access to these frames. However, we can imagine, and we often do imagine, the existence of such a frame. And then we describe our physics from that point of view. So, for example, you often, when you do climate mechanics, for example, you write down a wave function, phi of x. What does that x come from? Well, that's the x in the L2, right, of R3 Hilbert's state. That x is R3. Where did that R3 come from? You gave it to yourself. So suppose that we allow ourselves the luxury of imagining, as we often do, I say, allow ourselves the luxury of imagining that we're working in this absolutely legitimate frame of reference for our current purposes. And we can certainly ask, well, what would we say about our actual situation, our actual relational measurements from that point of view? And then we can further, and this is what I actually am going to do in a moment, that's why I'm rehearsing this strategy, if you like, and then we can further say, well, can I then kick the ladder out from underneath and forget about the fact that I was imagining in this frame to which I don't have access? And is anything that I was able to say preserved after I do that? Or, I mean, so there's two possibilities. One is I imagine myself to be sitting in this perfect reference frame and I describe actual relational measurements in its terms. And then I say, well, but I don't really have access to that. What do I lose by realizing that I don't have access to it? It might be I lose everything. And so everything I was able to say from that frame just goes away when I acknowledge that I don't actually have access to that. but actually what happens is not everything goes away you actually learn something and what you learn I claim is the uncertainty relations or non-primitivity and so that's the basic basic idea and the way we're going to get there after this quick example is I'm going to talk about the primitivity result and the connection that they have the connection that they make between uncertainty relations

25:00 And then finally I will talk about a quantum theory of reference frames. quick with the hydrogen atom because as it says at the top of the slide there, it's a rather familiar story. And so here on this slide is what I at least did some years ago when I first did the hydrogen atom. I imagine many of you did as well. You say, and then if you look in textbooks, I mean the first sentence there is almost the first sentence of the chapter on the hydrogen atom in every textbook. The cooling potential depends only And this is between a proton and an electron, as we can see in its expression. So we immediately reduce the problem using this symmetry to a simpler problem, the problem with the hydrogen atom, of course. And then you perform a separation of variables, and you do a lot of messy, but as I say, straightforward mathematics, and you get a solution for the radial and angular parts of the wave function. Again, this should all be quite familiar. But somewhat more careful treatment, in my opinion, is given in some books. I think perhaps, from my tastes anyway, it is largely a matter of taste. Schiff is a good place to look. What Schiff does is he says, look, what do you do when you normally start to write down the wave function, try to solve the Schrodinger equation for a two-particle system, well, the relevant Hilbert space is L2 of R to the sixth. So you write down a wave function in that thing, you write down the Hamiltonian in those coordinates, and then you turn the crank. So that's what he does. And then he re-coordinitizes R6 into center of mass variables and relative position variables. And I'm not going to go through And then you change the spherical coordinate, separate the variables, and do the same mathematical stuff that you did before, and you get the answer. The point that I want to raise here is that when you look at things from this point of view, this step, the step from here to here, makes it blindingly clear that the coordinates you're working in are relational.

27:30 When you do this particular change, somewhere in this paragraph, you get rid of that. That's the motion in the center of mass of the entire hydrogen atom. And you just worry about this stuff. So you've gotten rid of three of the coordinates, and you solve for the other three. In fact, if you transform back to the absolute coordinates, so put the center of mass bit back in and transform the whole mess. You have to put it back in to transform the absolute coordinates because the coordinates get mixed up in the reorganization. So you've got to put that part back in. You assume it's free, of course. And you assume that the motion of the hydrogen atom at the whole is free. And then you perform the transformation, and I'm not going to show you the result. I wrote it down once, and it would actually, it's a mess. I suppose with some clever redefinitions of functions, I could write it down as some simple-looking function. But it ain't simple, no matter how you slice it. And you can see why, because the coordinates get mixed up. And so what you end up with is an entangled wave function for the compound system. In other words, by looking at things from Schiff's point of view, it becomes, I think, very clear that the standard energy levels of the hydrogen atom are energies associated with the relative position and motion of the two particles. And they're not something you can associate with one or the other of those two particles. You can't think to yourself, I've now told you what the energy of the electron is in its orbital motion around the proton. It's an energy literally associated with the relative position of the two. Now, again, I don't claim anyone in this room has ever made this mistake. The point is just that if you don't think clearly about this, you can easily find yourself thinking the wrong thing, namely that what I told you is what the orbital energy of the electron is. And that's not at all what I told you. And if you try to wipe that down, you end up with a mess, right? Because you're going to get an entangled state involving out the proton and know how to test it. Okay, end of example.

30:00 But I need my notes. Okay, fine. So let me change gears now. That was really just to illustrate my first lesson of their coordinates, for example, can lead to misunderstanding. Now, the real kind of meat of the talk starts At least the technical aspect of the talk. And none of these results, except perhaps on the very last transparency, are original to me. And nor are they particularly obscure or unknown. So I'm going to go fairly quickly through them, especially given audience. Just in case this material isn't terribly familiar to you, here's the big picture, which you should keep in mind as I race through the next three or four transparencies. First of all, observables can be defined in terms of their transformational properties. And I'll try to flesh that statement out in a few moments, but the point is that things like position and momentum can actually be fully defined, kinds of transformational properties that they should have. Okay, second point about impermintivity theorems. Such an observable, as an observable defined in terms of its transformational properties, forms part of a so-called system of impermintivity. And we can then apply impermintivity theorem to derive, for example, the impermintivity relations. That's a big picture point. I'll flesh that one out as well. and I'll say a lot more about this last one in a bit but just get it on the table with respect to reference frames the transformations that are used to define these observables also implicitly define the notion of a reference frame at least once you're given some sort of absolute background frame when I use the word absolute with respect to a frame I don't mean necessarily something that's absolute in say the Newtonian sense of an absolute frame I mean something that is absolutely precisely legitimate for the purpose to which you put it. So, for example, if you're going to write down equations of motion and solve them, it's an exactly inertial reference string.

32:30 Okay, well, I claim that the above facts, quite familiar facts, actually, suggest the connection between the notion of a reference string and the uncertainty relations. And that's the connection is what I want to discuss, All right, I am going to race through this stuff because if you've seen it before, you only need a reminder. And if you haven't seen it before, the big picture points are really all you're going to get anyway, I suspect, unless you're much faster than me. So first of all, how do you have to define observables in order to define them in terms of their transformational properties? Well, the most convenient way of doing this, maybe, in fact, the only sort of feasible way of doing it, I don't know, but it's certainly the way it's always done, is to define them as positive operator value measures. And I'll say as a side point that for many cases, especially cases involving angular momentum and spin, you actually have to make the positive operator value measures and not projection value measures. But I'm not going to go enough into the technical details to show you why that's true. But it is actually important that they be positive operating values in general. Fortunately for the sake of doing position momentum which will be a last primary example projection value measures are absolutely fine. So a bit of notation. H is complex separable Hilbert space. L is bounded operators on H. S is a locally intuitively S is the set of possible values of the observed that we're about to define. And B of S is the synology of we're all subsets of S. I will leave the definition up for a moment, but for those to whom it's not familiar, let me just say that a positive operator value measure is a map. I'm just going to define it in terms of projection operators. It's a map from of possible values of the observable to projection operators on Hilbert space. And this map exactly corresponds to the spectral decomposition. It's sort of the spectral decomposition in reverse, if you like. It's take the spectral decomposition of the operator in question, or more precisely, take the spectral family associated with the operator. You can't actually write it out as a sum, if you like, but they still have spectral families.

35:00 And it's a map from values to members of these structural families, or sets of values to members of these structural families. This is the exact definition here. So let's think about position and momentum for a moment. I will very quickly, depending on the time, do another example that's not just position and momentum. But let me start with that familiar case. In this case, the space S, this so-called compact problem of space, is just R3, and where else S can be thought of as representing propositions of the form of the particle located at some point is 4 and it's R in delta, where delta is a set of points from R3. And then you can define, in this case, a projection value, so PV measure. And I'll say more about how that measure is defined at a moment. And then you can define the position operator essentially by integrating with respect to this projection value measure. Momentum works the same way. And here's the point. The requirement that we're going to make in order to find out what these E's are, I haven't said what the E's are, right? I've said it's some math from X's to projection. about what they are. And I'm not going to be able to accept the optimum jurisprudence, but how are we going to find out what they are? How are we going to actually write down a real representation of this thing? Well, we're going to require that position be invariant under booths and covariance under translations. That makes sense. And momentum is going to have a converse property. Now, let me be careful and don't get confused at this point and think, wait a minute, position invariant under booths? That doesn't make any sense. This is all at a moment in time. It's all at a moment in time. So how are we going to do that? How are we going to impose that requirement? Well, I'm going to create a bit more notation here. We're going to have some group that's going to represent the transformations with respect to which we want invariance and covariance of a given operator. It's going to be the Galilean group for us, because that's the one that has spatial translations and booths in it. But in general, you can do this theory for any old group. Well, okay, almost any old group.

37:30 I think the rest of the notation I skipped, except just to point out that, so I'll just say it in words, and those of you who want to look at the symbols, help yourself. But the word version is just this. I have this abstract group, like the Galilean group, And I want essentially two, I'm speaking a little bit loosely here, I want two representations of this group. One of those representations tells me how the group acts on the possible values of the observable, that set S I was talking about. So R3 in the case of position. And one of those representations tells me how that group acts on the total space that I'm going to be mapping those sets of possible values on C. And that's essentially what these alphas and u's are. And I make some other technical requirements here significant, but I want to just press ahead. If you want to push me on why I'm requiring various things like this phase, these are the ones, so feel free to ask me a question. And then we can say, well, okay, so a POV measure E, which remember is going to be a map from possible values to positive operator projections, is invariant under this group just in case it has this property where this u of g is mapping the result of mapping delta to a projection operator, right? So this is the map of the group. This is the representation of the group on the Hilbert space. And it just tells me that it doesn't change. When I apply that map, it doesn't change. It's going to mean by invariance, right? Invariance is less interesting than covariance, which means take my group and apply it to the value and then map that to a projection operator, that should be the same as mapping the value to a projection operator and then applying the group transformation to the result. Because the reason for that is that we believe antecedently that this representation of the group on the Hilbert space and this representation of the group on the topological space are representing the same group. So being covariant sort of essentially means it doesn't matter when I actually apply the transformation this map E is applied. It should be the same thing for the cases. That's what I mean by covariance. Alright. This next slide is a bunch of notation about the Galilean group, which I skipped.

40:00 Again, I'm going to say in words, well, actually I said it in words, the cartoon version, which isn't false, it just doesn't have all the details, is at the bottom of the box. And since I'm a little bit short on time, I won't go through the details of the theorems. The point is just this. Give me some group like the Galilean group, which is a compound group. So it's got pieces in it, like the boosts, the translations, the rotations, time translations, and so forth. Well, and so forth, that means the four pieces of the Galilean group. And so it's got these subgroups. Give me a representation on the Hilbert space of one of those subgroups. Let's say the boosts. That induces a representation of the entire Galilean group on the Hilbert space. And if you don't know that ahead of time, it sounds like magic. It sounds like I've only represented part of this group and somehow I've been forced into representing the entire group in a certain way. But nonetheless, it's true. So give me a representation of, say, the boosts. That induces the representation of the entire Galilean group. And the crucial point, for me at least, of the primitivity theorem is to say that that induced representation is essentially unique in the following sense. Now give me a representation of, let's say, the spatial translations, and then find out what the induced representation of the entire Galilean group is. The representation of the entire Galilean group induced by the representation of the translations instead of the boosts, which I did the first time. I get some purportedly different representation of the entire Galilean group. It turns out that it's not different. It's essentially equivalent to the one I had before. That's a very powerful theorem, by the way, which, I mean, maybe the rapidity with which I want to hear doesn't make it clear how important a theorem is, but it is, in fact, quite a powerful and important theorem. I mean, maybe I should make a quick remark about that maybe it wasn't clear from the slide because we would have had to read the details to see it. Systems of imprimitivity there were defined in terms of projection value measures. And then the question is, what about positive operator value measures? Do I have a similar concept? Yes. I do have a similar concept of, I have the analogous concept of a system of imprimitivity.

42:30 Do I have an imprimitivity theorem? Sort of. What you have is, you have Neymar's theorem, which tells me that I can take this positive operator value measure in the space that it's, the Hilbert space that it's acting on. I can dilate the whole thing to a bigger Hilbert space where the positive operators turn into projection operators, apply the incrementivity theorem there, and then sort of project back down. I'm speaking loosely and quickly because the details actually raise some subtleties that I don't have time to talk about. So when you're talking about positive operator values, whether or not you can actually apply it is a bit of a subtle issue. But for the most part, you can. And this is actually an area of recent where I do a number of theorems in the 90s is not the context of that. Well, the fact, again, read this if you'd rather that than listen to me, but I'm going to give it to you in words. The fact that these representations Pope of Galileanism are essentially unique, is really what drives the result in the box here that tells me that the requirements of covariance is sufficient to fix the uncertainty relations. And more specifically, the vial relations. I think I can... You can sort of think of it intuitively in the following fashion. Suppose I've got... Suppose I'm talking about translations, let's say. And I generate a representation of the entire Galilean group by starting with my representation of a small group of translations. And I also then impose the requirement that position the covariant on the translations. And that's going to then induce, if you like, some facts about the position operator, how it behaves with respect to the representation of translations on the Hilbert space itself. Now, why does that imply anything whatsoever about the connections of the relationship between the position operator and the momentum operator? Well, I do the same deal for the momentum operator, except I say it's got to be covariant under boosts. And the point is that when I do that, when I tell the same story about the momentum operator, I get to a full Galilean group that is essentially the same one I got with the position operator. And so what this means basically is,

45:00 I mean, you can think of it sort of backwards and say, well, give me any old representation of the full Galilean group you like, and then tell me what are the relationships between position and momentum in that representation, and it's going to be the same no matter what, because all of them are essentially unique. So, for example, if position and momentum play a particular role in those representations, which they do, of course, if momentum is a generator for installations in position, relationship is, it's going to hold no matter how you work That was a rather quick version, but I'm certainly not going to try to do this in detail. I'm just flashing it in front of you to point out that you can do the same thing with, for example, angles. You can say, well look, suppose I want an angle operator, and I want that angle operator to be covariant under rotations. So this angle should, you know, rotate to that angle. Well, you can do that, and I mean, this is a very quick sketch. In different versions of this talk, I've actually gone through this slide, but I don't have plans to go through it right now. So you just have to sit in that word for it that you can do this. You can define an angle operator. Essentially, you do it by piggybacking on the position operator. You can also piggyback on spins and define an angle operator in terms of spins. It turns out, in that case, you do get, But you do necessarily need POV measures and not PD measures. Okay, last section of my talk. So by my count, I think I need less than 10 minutes. So that's good. So before, actually, before we look at that, so what do we learn from this discussion of confirmativity what's the upshot well I mean first of all one thing to say from fairly minimal assumptions namely simply the assumption the assumption