A view from nowhere - quantum reference frames & uncertainty (contd.)

0:00 We can derive a lot. We can derive the uncertainty of relations. We can derive the momentum as a generator of translations and position. These are often assumed rather than derived, but you can actually just derive them. I mean, very often you see the bio-relations derived by beginning with the assumption that position is the generator of translations and position. I'm sorry, the momentum is the generator of translations and position. But you don't actually have to make that assumption in order to get the result. So that's interesting in and of its own right. For my purposes, though, what's more salient at the present time is that if we take the correct translations, and I'll come back to what I mean by that in a minute, probably the correct transformations, or the correct transformation of properties to define what we mean by position and momentum. I'll explain what I mean by that in a moment. Then we actually just get the uncertainty relations between us, in the context of quantity. And that might be the problem. So, let me then relate this to some of the, relate where we are now to some of the things I said in the first half of the talk. Let me say that a frame, a reference frame, is legitimate for defining a given observable. Just in case the following is true, it's a member of a class of frames that are connected by appropriate, and I'll say what appropriate means in a minute, appropriate transformations. And so what I mean by that is, by appropriateness, transformations that preserve facts about the observable in question up to a transformation. So, for example, if I say, I'm walking along the room right now, and I say that John has a velocity of minus one meter per second in that direction. And John says, absolutely not, right? well, we all know how to agree up to a transformation. So if we pick, and we all know what the appropriate transformation is to apply in order to preserve truths about velocity.

2:30 And so what I want to say is that a reference frame is legitimate for defining what you mean by, let's say, velocity, just in case it has the property that when the right transformations are applied, it preserves truth about that particular observable velocity. And the other crucial ingredient of being a legitimate reference frame is that among the reference frames in this class is what I've been calling the absolute reference frame, the one that is in and of itself legitimate for defining the observable. So if you think you know which reference frame is inertial, let's say, then frame is legitimate to define velocities, and then I say, the other frames with respect to which it is legitimate to define velocities are the ones that are related to that one by the appropriate transformations. In this case, Luke. Okay, well, I'm going to simply dictate that the appropriate transformations are the Galilean transformation. And, of course, I said from the outset that we don't know which frame is this absolute but we're going to imagine for the sake of discussion that we do and then see what follows and then erase that knowledge if you like and see if anything remains. That procedure will be more clear when I actually do it. Okay. So the point then is that the observables are actually defined by the need for them to transform properly from one frame to the other. So actually what I take position to be is translates properly under translations and boosts. So I'm taking the transformation to be fundamental and observable, if you like, arise as the things that transform properly after these transformations, which I simply dictate . And I don't really know where to come from . So that's what I want to say there. And then the final point, then, is that when we think of observables in this way, when we think of position as being that thing which transforms properly with respect to the Galilean group, and similarly for momentum and so forth, then the uncertainty relations are a necessary consequence.

5:00 So that's the connection with reference strains, sort of generically. But there's a question still remaining, which is, why, how do these uncertainty relations arise? So we can prove it mathematically. we can prove that if I require a position to be this thing that transforms properly, blah, blah, blah, then the uncertainty relations follow, but we don't really have any intuitive grasp of why that should be so. What's the connection between defining observables with respect to reference frames and so forth, what's the connection between that and obeying the uncertainty principle? All right, well, I lied, I am going to need that 10 minutes, but I won't need more than that. So the idea then is going to be this. This guy here is in our imagined absolute reference frame. The one that we know to be legitimate for defining, let's say, position or lenta or whatever. He's going to describe what she's doing in here in his language. That is to say, she's going to write down some things like X's and P's or whatever. And he's going to transform from those to his coordinates because we know that his coordinates are kosher. And then we're going to ask, are there interesting qualitative features after we've done that, after we've made that transformation that arrived? And he asked, of course, yes. Well, it turns out that the correct transformation, this is due to an article in the 80s. I don't know the exact date. Harvey, do you remember the exact date? I don't remember. It's in the 80s. A lot of the literature on my reference frames, and there's been more of it recently in quantum mechanics. Not a lot of it, but there's been a bit more of it. I classified it into the confusing and the confused. And a lot of it is confused. This one is confusing. It's not confused, but it's confusing. It's very hard to get through and figure out what's going on. I think they are wrong about a couple points, but this is my take on what they're doing. say no. Well, this operator actually does the job. It actually translates from the coordinates that the person inside the frame would write down and the coordinates that the person outside the frame would write down. So the person inside the frame takes position to be given to them by the four walls of the room. So the person outside the frame says, aha, you're

7:30 not actually measuring position absolutely. You're measuring position relative to Q system reference. And this transformation actually does get to the right answer. It also happens interestingly to add a term onto the Hamiltonian that basically represents the drift of the lab with respect to the person, the external observer. And here's the upshot. I'm not going to show you details because they're not illuminating anyway. I'll just show you the result of the case of a position measurement. Again, very simple model. And actually, some of the things I'm going to say, if you were more careful, it wouldn't be true. But I mean, I actually have blocked it out. Okay. So this very simple model, we can see some interesting qualitative features. Right down to interaction Hamiltonian, the position measurement, something like this, so that's going to couple the position of some particle called system one. And I and T means interior, so that's that person inside the frame. That's what they write down when they say, I'm measuring the position of this particle one. And it's going to couple it to, even though it says momentum there, it actually gets coupled to the position of some apparatus, so pointer. And it turns out that the following relations hold. H prime is a transformed Hamiltonian, So that's the one that the external observer writes down. The external observer transforms this Hamiltonian plus the three-point into H prime. It turns out H prime commutes with the total momentum of the system as seen by the external observer. That's good. It should be that way because all the interactions happen inside the lab. So that's just really a check to make sure we're not crazy. it does not commute with the external observer's view of what the momentum of the lab is and it does not commute with the external observer's view of what the momentum of the measured system is now what's kind of interesting about this is that the lab doesn't actually show up in the Hamiltonian so this actually arises because of the transformation the system zero is involved in the transformation so then you ask yourself following, and this is the last two sentences, you imagine you're in the following position. I'm inside the lab, and I have to at least imagine to myself that my lab is in this class of legitimate reference

10:00 frames to start. I have no way of checking that, but I have to, that's an assumption I simply have to make. I didn't mean to say I have no way of checking it, I mean I do have a way of checking it, as I described earlier, I can correct it later on if I get anomalous explorations or whatever. But I have no way of being absolutely certain that I am in this class of absolutely legitimate frames. So in the end, I just have to give myself that assumption. And then I say, suppose that's true. What would someone from the absolute point of view say about what's happening to my lab frame when I make a position measurement? And I've assumed my lab frame to be legitimate for the sake of making a position measurement. Well, I know that that person would say that after my position measurement is done, my lab has been disturbed. The lab's momentum has been disturbed. In other words, it ain't an inertial system anymore, even if it was when I started. And in particular, momentum for the lab, the thing that's defined in my reference range, is not conserved. But if momentum is not conserved in this interaction, then I cannot define momentum with respect to it. It's not legitimate, I claim without further argument, to define momentum relative to something momentum is not conserved and that is i say the explanation of why when you make a decision measurement momentum is ill-defined there's a lot of a lot of contentious stuff there so i'll stop H prime is the Hamiltonian, is this Hamiltonian transformed into the coordinates of the external observer. So this is the stuff that's sort of absolutely true, if you like. And what I know to myself is that these relations are going to hold no matter what that external observer happens to be, what is actually the brain happens to be. So, I mean, this, referring to the very last page, there's still going to be a constellation of momentum for the system as a whole, right, for the lab plus the product. So isn't this just the same thing going to happen in classically? I mean, if I tried to write, if I actually tried to model the classical experiment that I do to, you know, look at, make a position of a particle,

12:30 or it shines the light off it and see where it went. I mean, they'd have to be... I mean, the frame by itself would change its momentum so it goes from the particle that moves a little bit as it goes from the ground to the ground. So isn't something like that just happening? Yeah. I mean, yes and no. The difference is this. I mean, well, one thing to point out is that in the interaction Hamiltonian itself there's no interaction between the part of the lab not directly at least, right I mean, well, I don't know what I mean I mean, it just doesn't interact with the there's no interaction term that involves the system zero the lab, right so right away, you know, it's not quite like the classical case because in the classical case, in order for me to bump the lab I feel like I've got to actually interact with it So that's one disanalogy from the classical case. And the other one is familiar, right? That I can, in the classical case, right, even if my lab gets bumped, I can always go to a bigger frame and sort of detect the bump, if you like, and then everything becomes fine. I can communicate that information back to the person in the frame, and there isn't a problem. Now, in a sense, I can do this quantum mechanically, because I can use that information to retrodict the momentum of the particle. You know, right, the uncertainty relations are going to be valid only at this moment in time. Backwards in time, they're not valid. I can always retrodict stuff that I couldn't know at the time. But the point, I guess the other just analogy, maybe it's a little bit less clear, is that in quantum mechanics, it's less clear because it's a bit difficult to state because of the fact that relations aren't going to be, aren't actually valid backwards in time, if you like. But if you imagine this procedure of going to a bigger frame and settling all questions, quantum mechanically, when you go to a bigger frame, you still can only settle one question or the other. And that would be borne out by a converse analysis, which I didn't do, about what happened in the momentum measurement. Yeah, we're completely happy about that. Yeah, we're back to you. Well, could you say,

15:00 could you try to find a sign like that? The, of course, P0 int and P1 int don't commute with H int interaction, or H int total. I'm sorry, say it again. From the interior perspective, P0, i.e. P0 int and P1 int are also not constants of the motion because these things interact. That's right. So can you say more about why? Well, P0, P0 int commutes with the Hamiltonian. There's no P0 in Hamiltonian, right? So it commutes with that because it's the zero subsystem, not the one subsystem. It commutes with that. So the really important of that trio I put into you is the second one. Absolutely. Because the third one is not surprising. Right. Because P1 doesn't commute with the interior H. Right. And this is the one from which I take the lesson that the frame is no longer, it's a frame itself. That is, say, system zero. is no longer valid for defining the length of it. These other ones are just sort of, I don't know, they're just checks on standards. So can you use this formalism to relate the uncertainty principle for the int variables and the x variables? Because you haven't written it down quite, but there is, from the file relations, a standard form. And you could say them for any Q&P, know the relationship between the derivations, because it ought to be round the corner from this translation, you, that you had on the previous slide. Yeah, it should be relatively simple to do. I never applied the translation, but it should be applied on a job. Yeah, I just wanted to do that. Newton and Principia and three centuries of astronomers the sort of hit-or-miss dialectic that you described in the beginning, looking for anomalous accelerations, looking for violations of the third law, and they somehow get what seem

17:30 to be workable and guesses at inertial point. Now, can you, using this very impressive formally can you translate what they did into your terms and in the course of translating maybe improve on what they did and show explicitly what they yeah assume passively or what they didn't assume and needs to be improved can you do that yes I mean the way it would go is is basically as follows what I did when I wrote this stuff down was I said let's just grant ourselves for the sake that this frame we have right now is coaching, right? It's an okay frame for defining whatever observable we have. I'm sorry. So I said at the beginning of this slide, let's just grant ourselves that this particular frame is valid at the beginning of the discussion for defining whatever observable we're interested in defining. And then we show that by the end, it's not valid for defining momentum. Well, what about that first statement? I mean, where did that come from? Well, it comes from exactly the sort of exercise that you and I described Newton and Paul did, what you would do is you would say, well, I don't actually know that this frame is valid for defining momentum, let's say, and so let me see whether it is. And I start making some measurements, right? And then what I would actually do in this case is I would say, well, now what I want to do is I want to say, what would it be like if I were in the absolute frame? How would I define, how would I write down these results? that what I would have to say is that there are anomalous accelerations or what have you, then I know I haven't got quite the right frame. And that's a way of rationally reconstructing what someone who's looking to improve the ICRF actually does. What they say is, well, let's make some measurements in the best frame I have, the ICRF, and then I say, well, what would it be like if I described these results from some sort of absolutely legitimate reference frame, And then they find out that, well, in that reference frame, I would have to admit that there are accelerations that are not caused by a force, and therefore the frame with respect to which I made these measurements must not be legitimate for defining it. And that's not the way it's couched normally, but I think that it's equivalent

20:00 to what's actually done. Any more questions? and I described the following transition. In the early days, I had become a lecture. Bohr uses disturbance in a robust and physical sense. And after the EGR paper, he uses it in what my terminology is, but the disturbances are really now semantics. And it seems to me that we're not, lots of much disagreeing that what we've actually done is to illustrate exactly how the, in what respect the disturbance is semantics. But also illustrating, yeah. It's not a robust interaction from the external measure to the internal person. It's how you have to transform semantically in order to get the definition to come out right. That's the first one. The second is what Gore was trying to do is to use disturbance to explain uncertainty. the criticism that Howard and I make is that in fact in the end it works exactly back to front. That you have to take the uncertainty and use that in order to say, therefore, there must have been disturbance. And again, it seems to me that's exactly what you're doing here. Okay, so, yeah, those are versions of questions I often get, so I'll see if I can rehearse in the clear as manner possible the answers, in my opinion. So, with respect to the first question, I mean, I think I actually agree mostly with what he said there. I would only add that the disturbance is physical in some sense. That is to say, from this imagined absolute frame, something actually physically happens to the lab. It moves. And so there is a physical disturbance, but it's not, as you rightly say, a disturbance of the measuring system. That was my point at the very beginning, by saying this ain't like what you might think the Heisenberg microscope is like, where the system itself is somehow being disturbed. It's a disturbance of the thing that defines what you mean by momentum. if you're the person inside the frame.

22:30 Okay, but it is a physical disturbance of that thing. Okay, so I think there's not a lot of disagreement between us. There may be some fine points of disagreement. On the second one, basically what you're asking, is there an element of circularity in this whole procedure? And so let me first of all state that I certainly do not claim to have derived the uncertainty relations from the mere facts to assume that position is co-variant perspective translations and invariant perspective groups. That would be an absurd claim to make. What I am claiming is, and I shouldn't say I derive them, because these results, except for the last Y, were known for a long time. I mean, they're all due to Matthew, essentially, in the end. But what I am claiming has been derived is the uncertainty relations from those minimal requirements, transformational properties in the context of quantum theory now you might say ah but in the context of quantum theory and certainly relations are already true right so so we have to ask you know is this a big circle or a little circle right and i think it's a pretty big circle i mean nowhere in the derivation do you say anything like the uncertainty relations have to hold and therefore i mean the only requirement that you make is that you work in the context of a hilbert space. Now, it's true, of course it's true, that in some sense you're thereby forced into the uncertainty relations. I mean, that's the nature of the theorem, right? The theorem tells you that you're going to be forced into it. But the question is, how close to the surface of your assumptions are the uncertainty relations? Are they deep enough down when you start that when you find them, you actually feel like you've gained some understanding of them? Or are they so close to the surface when you started that you're not at all surprised at the end of the journey that you ended up with the uncertainty relations. I claim they're deep. I mean, I claim that you are surprised to learn this. Because when I state these results, when I say, let's make sure that a physician has the right transformational properties, and let's define observables in terms of these positive operator value measures, and so on and so forth, the word uncertainty, non-commutativity, none of that stuff was put in at the start. It was there, of course. But I didn't have to explicitly invoke it to get the uncertainty way. I mean, that may be a matter of taste, but for my tastes, what you've seen is a deep connection, in my view, between the need for observables to have the right transformational properties and uncertainty.

25:00 You may think that it's shallow. No, no, that's not... I think it's deep, and I think your response is fair enough. I just don't think it explicates the relation to the explanatory order that we had in mind. That was kind of simple. Oh, right, okay. Okay. Did anyone over there? Yeah. I had a quick technical question. My understanding is how you derive the minimal uncertainty condition that's from the stone's lemma, and then you look at its Lie algebra, and look at its Lie product. My question is, I'm thinking of Sudarshan and some of his work where he maps that back on the classical mechanics. Have you thought of deriving the Poisson brackets using the same formalism here? No, and the answer is no, I haven't thought of it, except for right now. I guess I don't know whether that would work or not. I can't say off the top of my head whether that would do that. It goes a long way to work. It goes a long way to work. Yeah, I haven't ever thought of that. Unitary transformation could be replaced by the United States. Yeah, yeah, well, I mean, I can see how it would potentially work. I just can't say whether it does. But you tell us it does mostly work. I think that's the point about the explanation, I think it's not, it's actually a deep explanation, I mean not the explanation why the uncertainty relation is a deep explanation if you remember that there is a previous theorem to do the thickness theorem about the fact that if you just want to preserve probability or even just feel sorenality relation, then The only operations to do it are unitary and unitary, and therefore it does not be a unitary representation. Once you have the unitary representation of the Gann group or the Lorentz group and so on, you have the fact that that momentum is a Fourier transform of position and you have the uncertainty relation. So I think this is a very deep explanation because it doesn't start with that very much. I mean, it's not that you're talking with the Hilbert space all right, but not even with the whole structure of Hilbert space. I think it's not a circle. I think it's not even a big circle. I think it's a big circle, but I think it's pretty big.

27:30 Well, all my macro results are circular. I mean, I have another question as well. It's not a unitary one. I mean, the answer is it doesn't affect it at all. All these theorems don't rely on the representation being projected rather than unitary or vice versa. I just wrote down things in terms of a unitary just because it's easier to write down. I have to keep track of phases. Can I raise a sort of broader issue here? Suppose that you start with the idea of squaring the whole wave function. I'm sorry, representing. Squaring the whole wave function. Do you have your states? Then you ask the question, what are the generators of special translations? That would give you an operator, which either way are associated with that. You can ask the same thing about translations in that space. It's basically how you get your position on it. Now automatically those things won't be straight away. Your point was, well I didn't need to interpret the position around each other, translations in each other's space. But you're assuming the Galilean rule. I'm going to say, where does that come from? I'm going to say, first of all, well. What are the equations of motion of these wave functions? What are their dynamics? Well, it turns out that if they're free properties, so I take the free-shorting equation, then it's not the Galilean. It's a much, much bigger. So now I'm going to get the Galilean group because I turn interactions up and choose background potential. So the provenance of the Galilean group depends crucially on the existence of people, on these potentials. It's not sort of God-given. Now, but in order to make sense of the Galilean group in that sense, I've already got to have a Schrodinger equation with potentials. I mean, that means I know essentially how to interpret the theme and the X

30:00 operates. And I see that they're generators of translations in each other spaces. So in other words I can see a way of understanding the origin of non-communativity that doesn't presuppose the Galilean group is sort of given a perora which was already taking place even when the group, the spacetime group wasn't a Galilean group. It's a bunch of logic with the such a pre-cognitive. But you have to know or grant yourself that of translations in each other. But that's by definition. The question is how to then interpret these operatives. But it turns out, for example, if you have a free shorting reference, then the momentum, the average value of a momentum operator, for example, would be conserved. So I mean, I'm getting physics out of this. I'm getting understandings of what these operators mean. Operations. How they time with our classical notions of position momentum. Because I have some dynamics. I have a free shorting reference. I'm just, what I'm questioning is, Do we really want to tie everything down to a space-time group that's really a very non-trivial, probably, full-shorting of vision interactions? Okay, well, a couple things to say there. I mean, one is that the fact that I was working with the Galilean group is only essential in that the Galilean group contains the translations of the boosts. If you want to introduce some bigger... I could have easily just worked with their semi-direct product. I could have easily worked with some group that contains the Galilean group as a subgroup. So the particular choice of the Galilean group was not terribly important. What was important was that it contained as subgroups the translations and the boosts. And that's important because, and here I think, well, that's important because I want to define what I mean by position and momentum with respect to those things. right um and you're you're saying you want to define position momentum as the things that generate the translations in each other um and i guess that's a matter of taste which one you which one you want to start with if you like the reason i want to start with the one with one i said is because the conception um that that i have the role of the reference frame reference frame reference frame sorry my mind is about one o'clock in the morning um reference frames play in the definitions, I want to emphasize the role

32:30 that they play in the definitions of spatiotemporal quantities like position and momentum and I can do that more easily in the context where I say reference frames are the things that themselves transform properly and then the position and momentum are the things that sort of follow along with those translations in the right way. I don't think that there's any intrinsic advantage to doing things that I have a particular concern, namely with the role that reference frames play in the context of the definition of spatiotemporal quantity. So if I had some other concern, I might be more interested in the fact, I might be more interested in starting with the idea that position generates translations and momentum and vice versa. I don't see any intrinsic advantage over my way except my initial interest. That was a very urgent brief question. Sure. Not urgent. Not urgent. Well, then I end this. We're just over time. There's going to be a... I think so. I think so. Oh, well, then you can see a prospect in the question. Okay, so first I was just going to make a comment, but the derivation of certain relations that you find are actually in... They're the derivation that you find in Leslie Palenco's textbook. You find them in lots of places. I was just going to say that the defense he makes, sort of a defense I didn't hear you make, but I think one you can make is that it's inappropriate to demand that the quantum, to derive quantum mechanics from a classical theory if you think that quantum mechanics is a more fundamental theory. I wouldn't endorse that. I think quantization is a pretty fundamental thing, so I wouldn't want to go that way. Oh, okay. His argument was that starting with the symmetries is more appropriate than starting with the... Right. No, I understand the argument, but I wouldn't... I don't endorse the idea that because quantum mechanics is more fundamental, we somehow have to get all the classical stuff and the quantum stuff rather than... I'm not partial to that. Let me ask my question. The last slide you showed I think could probably be generalized to any additive in certain quantity. Oh, I'm sorry, what's getting generalized to additive-conservative quantities? The theorem on the last slide that you necessarily... Oh, about the non-community? ...disturbed the monitome of... Well, yes, yes. So any conserved quantity that's of the form A is equal to A1 plus A2,

35:00 where A1 and A2 are the symbols for the distinct systems, and A is conserved, I think. Well, in a measurement of an observable that's appropriately... Right, that doesn't commute with the additive conservative quantity, right? So you're talking about measurements of observables that don't commute with additive conservative quantities. Exactly. And so my question is, what is the connection with the Wigner-Iraqian-Nazi theorem? Because the Wigner-Iraqian-Nazi theorem tells you that if you want to make a measurement of an observable that doesn't commute with an added conserved quantity, you can't do it in a way that, sort of in an ideal von Neumann way, such that this state vector at the end is an eigenstate of the observed will be measured. And it seems to me your theorem is sort of the flip side, which is that, yeah, the system's state vector will be disturbed, but the apparatus's state vector will be disturbed as well. I don't know exactly what the connection is, but there is one thing to say that's slightly different about my assumptions than their assumptions. I mean, you said it yourself. For them, it's an ideal von Lehmann measurement, right? So you get perfect correlation at the end. My Hamiltonian doesn't actually establish... My Hamiltonian, it's a pretty idealized Hamiltonian, but it doesn't establish a perfect correlation at the end. It establishes a pretty good correlation depending on what that function g of t looks like. If it's nicely peaked and very tall, then it establishes a pretty good correlation fairly quickly. If it's sort of messy and spread out, And the correlation will never be perfect. So I don't know if that's a significant point or not, but it is a difference. What was critical in that term was the finiteness of the It didn't work for unbounded operators. It's not the volunteer . Yeah. Yeah, maybe there's . President Shimona? There's good, deep reasons for starting with quantum mechanics and not . I once heard George Mackey say that if you tried to get the classical form of the Hamiltonian as a sum of kinetic energy and potential energy. And you get it from the role of the Hamiltonian as a generator of time translations. And you do it trying to use his machinery of the impermissive theory and the Galilei group. It won't work. It doesn't go through. But if you do it with quantum mechanics, you do get it.

37:30 You get it quantum mechanically, because the Hamiltonian operator is sub-potential and the kinetic energy operator. And then you go to a classical limit, and you get it in that way. So there's no apologies for starting with quantum mechanics. Oh, let me, okay, so I'll just finish very briefly by saying, I said I'll finish very briefly just by saying, and what I said in response to Rob was, I disagree with the view that one must always begin with quantum mechanics. But I don't necessarily disagree with the view that sometimes one should begin with quantum mechanics. Well, people, keep those questions coming until it's a good one. I mean, a very early lesson in any quantum physics course is that there's one version of the interrelation that's a little bit special because it doesn't involve two operators, and that's the energy of time. I'm just wondering, I mean, in this approach, if there's something out of his way that that falls out quickly from a way that's, you know, maybe not the normal way. Well, the answer is yes and no. There's a very nice paper by Jan Hilgevord in the American Journal of Physics in which he questions the orthodoxy view that there's not a time operator. And he doesn't claim that there is some single operator that's the time operator. He argues that time operators should be derived from the physics of a particular situation. And there will be things that will be appropriately served that role in given physical situations. And he brought some very nice results from that point of view. Who is this? John Hildebord? Ask Fred. John Hildebord from Utrecht. and so the answer is if you give me a time operator and it has the right kind of then yes, I can do it and actually I endorse his view I think it's the right view to have about time but even if you don't even if you're not given a time operator you can still do it in the standard way by deriving all the commutation relations in exactly the way that Abner was referring to but my preferred way of doing it The end, are you?