Classical Mechanics from QM via Commutators

0:00 I'm not going to read this paper, because I think people are reading out the paper to do it. I'm going to talk about it, and I'm going to use this concept. It's rumour that people have no time for a matter to be here tomorrow, and if it's true, and I'll make a mess of this, you'll all be able to ask him about it. Let me start off by saying that over something like the last 40 years, I think, certainly pre-David Amcar, Tony's been doggively worrying away at a problem which I think I'll put it in this book. He says, in effect, make any good book on quantum mechanics, for example Dirac and really believe that what they say at the beginning is what they're going to do ignore the way they actually solve the problem themselves really believe what they say at the beginning and he's been cuddling over all sorts of features of ordinary non-analytistic quantum theory that's all proceedings of Amparo, he's shown various stages in the development of this. Now he's got far enough to believe he can put it all together, and it's more conflicting as he hasn't been able to come and talk about it. If he does come tomorrow, I don't feel that he ought to give a talk anyway, so we'll continue as we think. I'm going to read some bits of his paper and as I say I'm going to use his slides during the past year Lou and I have been a bit concerned with the work I mean we have been cooperating a bit but essentially the work's all told so we start off with his abstract and I think it's worth going through fair in mind as we go through we begin with a classical model structures, structure-less particles, coordinates cube, the functions of time we don't say anything about the space in which they are

2:30 and then there come these identities which are the ones that Lou was talking about before C you quantise the identities and when you do that it turns out the quantisation of them are no longer identities they limit the forms permitted for the any variable theta and the Hamiltonian and that's just what Lou was saying if theta is arbitrary to get two possible forms of H and then choosing the quadratic forms and making theta satisfy both of the conditions you find the fourth order equation which Lou mentioned And then he goes on to try and determine what that's all about by, as he says, reverse quantising H. That means going back to thinking what it means. Now he outlines this scheme in a series of flowcharts, and these rules, he says, are simple. The text in oval boxes represents human input, hypotheses, choices, assignments. In sharp-cornered boxes are well-defined calculations, and the text in round-cornered boxes is due to results, with arrows indicating the flow of inference. Mathematics is standard operator-out-of-earth quantum mechanics, operator, amythian symmetry, real spectra representing real-of-earth. the spectrum of reading, the brief or continuous, which will partly bow the prediction, and we are just doing what people have learned from it. The time is represented by a scalar T that commutes with all the operators. So, time in quantum mechanics then cannot be measured directly. Rather, clocks are devices that measure a coordinate of Q, and they are designed so that the Q keeps roughly in the step with T. unavoidable in principle and this you'll find set it up rather more fully in the 2003 and past.

5:00 One further remark about this is that he says that about this time, and I want to return to this later so I labour the point a bit, In this model, the coordinates are functions of time, and so, he says, we may choose any one of them to equal t. But I say, well, yes, so long as the one you choose is a monotonic function of the t you started with. So that's a slight qualification on my part there. Now the overview of the actual process here is reduced with great care and I think at this stage we don't really want to worry too much about that lower cost but we could look at the um as he says it's a classical model uh you're feeding in ordinary quantum mechanics you have this uh variable which depends on the coordinates in the model and the identities are the the first and second derivatives um which are then going to quantize one of the consequences of the way you do it because of quantum identities you have to have a flat space to do that and then you find that the results of the quantization are no longer identities but are they're not identities if you put an arbitrary theta in and if you choose a particular Hamiltonian then you get an equation called theta that's what we're we're going to do later on as we reveal the rest of the story one finds we have certain amounts of tensor calculus here and then we finish up with these field equations at the bottom which can be put into this tensor form of ice I've got some qualifications to

7:30 make about those which I'll make when we get to that stage so if you've got a little bit more detail about the actual D the actual calculations the point is here that you've got this variable time perfectly ordinary things the infinity variables here this theta, and the coordinates are in fact the dimension of space times the number of cos, which is ordinary analytic, and Tony is worried about these two derivatives. Now, let me pause a moment and say, well why is he worried about those two derivatives? These identities, they, or rather not these, but the consequences, the constraints that who we talked about, came up many years ago when he was trying to understand how things could be measurable in quantum mechanics. He had a lot of arguments at that time which were fairly hard to understand but were much more physical than the present way in which this sort of identities being quantised and find the results are no longer identities doesn't actually tell you physically what's happening in the way that he originally got to the identities I think he likes it more because it's nice and neat so now what we have left I'm sorry to have not been able to go through this as cleverly for some people because this is a piece of modern technology which I'm not accustomed to using I didn't even know when I switched it on whether it was going to come out up or down or not but it's easy to use really now this quantity theta which comes in here

10:00 is meant to be something which is a function of the coordinates and is physically significant tongue each slips in the sentence this function might be for example a scalar potential, an element of a vector potential a field strength, or an element of a metric tensor he slipped that in to soften you up at this point because that's what he's going to be so, yes, I think we can pass fairly quickly over this one here is a statement of the usual sort of things about quantum mechanics and how you actually do it, and all the usual business as I said, we have to quantize the identities which we had on the previous slide and this is actually just a more direct approach to the sequence of constraints which he found by more physical means many years ago one of the advantages of the more physical arguments and I refer to them in this way because although I saw this all from him many years ago I can't remember now what the arguments were, but they were of a more physical kind Most people turn a similar idea onto them Yes, I believe it does, I think it is But I think one of the... How I could know about that Sorry? How you could know about that About that similar idea One of the advantages of the more physical approach then was that it made it plausible that one or perhaps two of these constraints would be important but as you went up not only did the data constraints get much more complicated but they were also shown by the argument to probably be of less importance so you could contemplate on the first two which is what he wanted to do for algebraic reasons anyway

12:30 I'll just show you this next one but actually I think this more or less overlaps what Lou was doing of course here we are putting in this is not the clever stuff that Lou has done but actual quantum mechanics which has much the same algebraic form then you'll get this sort of thing and Lou gets a credit in the oval green there along with temple This sort of thing is also in Temple's little book on quantum mechanics, which is where Tony Scott is from. And then you, indeed, get Hamilton's equations out by something which seems rather like magic. so these constraints which Lou told us about come out and you find that if theta is an arbitrary function then the first constraint is satisfied only if the Hamiltonian is polynomial either linear or quadratic and the second and the higher ones are satisfied only by an H that is linear in the P's now what Tony says next is macro physics is characterized by quadratic Hamiltonians micro-physics insofar as it has its amenables of the Hamiltonian method is characterized by linear Hamiltonians given says Tony our objective of explaining the structure of classical mechanics by using quantum mechanics

15:00 will confine discussion to the quadratic Hamiltonians in these sets is only going to look at the first two constraints I've made the qualification here the linear Hamiltonian in quantum mechanics well that's true, but it is a Hamiltonian in which the coefficients are not any longer commuting quantities, so it's not quite the straightforward linear Hamiltonian that Tony is sort of getting by turning the machine here the fact that you'll need the gamma matrices means mentioning now then so you'll get the restriction on the theta which then works out in fact have I got it up here? I'm going to move the ball Oh, no No, I seem to adjust it Ah, and the next one Let's hold on that one So now I've come along to this expressing these conditions on an arbitrary theta if the constraints are to be satisfied Let's read the top bit out We start with a quadratic H or with a general quadratic H that's not just the Newtonian one but it's a more general form

17:30 if you put in the Newtonian form then the equation of the theta turns out to be that thing which is well known to people who work in elasticity for example and the corresponding thing when you put in the general Hamiltonian in which the coefficients of the P's which Susan was talking about, then comes to the generalization of that, which has that for it. And we move down to the bottom of that slide. Oh, let's move it out. Then, what I'm seeing here, when you look at the particular case where n equals 3, that would have a single particle, so the whole dimension is 3 and then Tony gives an argument which I'll read out about that last remark the remarks that the equation well this Anyway, I can't work these things The equation which he gets Represents the motion Euclidean 3 space of a particle of mass M Under a scalar potential But the scalar potential doesn't appear In this equation Now, because of the constraints vary H, you will be varying theta so theta of Q, which is supposed to be a function characteristic of the system, must appear in H and since the only place where it can appear is the scalar potential only here, you conclude that you must identify theta with the scalar potential at that point not necessarily to be understood particularly physically I mean this Hamiltonian is just of the usual form

20:00 but may not, the terms in it may not have their usual meaning now one of the conclusions about but once you put p to equal to omega so that you have this equation you'll find that this omega which is said to be the scalar potential will have more structure than the one in which you're familiar in which the equation is simply that So you've got a good deal more freedom In fact, what you get out is that instead of the omega being a 1 over r term Plus a possible constant You'll have a 1 over r term plus a constant Plus an r squared term and a linear r term In the spherically symmetric case So this is all possible for the field of a point source at the origin. Then he goes on to the more general quadratic Hamiltonian. And when he does that, he gets on to what seems to me to be the most important part of the paper, but also the more speculative one. He has a general Hamiltonian without linear terms, that is, quadratic in the P's, coefficients depending on Q, and then what he finds, and this is not at all surprising actually, is when you apply Hamilton's equations you get equations which I mechanised it as the geotistic equations with the GJK as a metric and since the space is flat the motion is a uniform straight line because of course there isn't any field in the Hamiltonian

22:30 so that's not quite consistent now what interested Tony at this point and what Lou has mentioned is the relationship of this to general relativity well before we get on to that I want to make some qualifications the fact that disregard the fact that there's no potential there the fact that you get with Hamilton's equations or equally with the Lagrange method in a space with a matrix you get the geodesic equations is quite commonplace in analytical mechanics it goes certainly back to hertz and probably earlier than that so one shouldn't get too carried away by the fact that the sort of equations you're getting here just the sort of equations one looks at in textbooks on general relativity because there's a big difference and the difference is in the dimensionality and this is a qualification which I've been urging on Tony for some time and he responded in a somewhat characteristic way I mean in an earlier version I said look here the dimensions are wrong and he then written version in which, as I said to you, you have to answer the problem, you have to mess the problem. It's no longer possible to see where the dimensions are wrong, but they're still just as wrong as before. The fact is, in what way the dimensions are wrong? Because this equation is in three dimensions. Now, the significant corresponding things which look just like this in general maturity are important. And this is just a... Well, you know, because of time and so on. And it's a bit more than that because... you see, he said, well, n equals and he said, let's make the dimensions what we like

25:00 we start in a space station, or if you like but then if you want to make an interpretation you've got to say, well, three of those are space and one is time and then he also is being to go to a two-particle or more system and his original formulation was specifically designed to do that so that it would have been one and six dimensions but now if he's going to say oh, but he's really getting relatively open his way in then I'd want to ask the question well how many dimensions have we got now if there are two particles have we got eight or have we got seven because there should be some relationship between the two times of the two particles well you know you can think of the dot as different from the time in the general relativity space time you can figure that some kind of process that's in And then, yes, in that case, I think, are you recommending, as it were, that... I'm recommending that you think that one of the coordinates is time, but there's still a dot, which is a process in the background. Yes, so that in the Ah, well, we're trying to dovetail your I see what you're saying, Luke Now, would the corresponding thing for Tony have been then to have all his particles four-dimensional ones and plus a dot That's what I'm possibly suggesting I'm slightly worried one way or the other in any case it's that kind of I think this question about time and time in quantum mechanics and the relation about quantum mechanics there's a book come out quite recently by a man called Oliver Johns It's called Analytical Mechanics for Relativity and Quantum Mechanics.

27:30 I'm still, I've got a copy of this, though. You've got a copy here. Yeah, mine's not here because it's so heavy. Yeah, I've got eyes on too much, but I thought it was. These books are published on heavy paper. I think this may provide some help in this topic. he does make some further remarks about dimensionality because well let's go down to the bottom here and the way that one gets into these equations which I've shown you already is by some fairly straightforward tense of calculus you see you've already said somewhere that the space is flat maybe that was an unwise thing of Tony to say earlier on because he now wants to suggest that it might be third, I thought then the QJKs would still be the metric but some of these things which were perfectly alright before were no longer being real in equations so my only contribution to the whole argument was to say well we can always skate around that one I've done it a dozen times at different points in my lectures on general relativity you just say well suppose they're true in what I call normal coordinates which are the ones which have been fixed up to be as near to the flat as they can in the regions we're talking about and then you just turn those conditions into these couple of pages of calculation but no more into the equations which we were looking at earlier I agree with what Peter said although I feel there's some obstacles we've had to jump over on the path the great virtue of these equations compared with Einstein's field equations is that we actually well, to say we have a derivation is perhaps a little bit strong

30:00 because of my difficulties but we've got the germ of the derivation instead of just adding something on when you get to a certain stage Terry also has an interesting argument counting up numbers of equations because the the condition I'll have to find out what this equation 12 is oh yes well, let's not go into the details but he has certain conditions on the number of dimensions n which put together imply that you wouldn't get any curvature from these equations if n were less than 4 So he's got a kind of internal argument for N being 4, apart from just the sort of general relativity arguments. Now, I think that I won't go on a lot further, but... The conditions which come out of this What Tony does at this point Is things get a bit too awful To try and find general solutions of the equation And so what he does is Look at a sterically symmetric solution With a single particle at the origin You then get, as well as the inverse square force You get an accelerated expansion which he suggests is something to do with dark energy beyond a sufficient distance so long as the space is pervaded by a negative energy

32:30 which he works out like that and he makes this hypothesis that the total energy of the universe is zero so that these two different components cancel out and I was very struck when I came to that Back in the pre-ampire days when we used to come up here and meet in a thing which was called the Eddington Group at the time Dennis Chalmer was very keen on this idea of making the total energy of the universe zero because of course it has the great advantage that if it's zero then the whole thing is just a wobble as it were negative and positive have been produced at the same time the other thing could I ask you what his percentage of dark energy is then does he have 2 thirds, 3 quarters, 7th I don't know, no. I think he'd be able to answer this, but I haven't been able to find it in the... Look at that screen there. K2. Yes. Now, K2 is negative, and... how does he compare this with any of the well I think I think people I'll have to say ask them tomorrow I'm afraid I have no idea what did strike me I suppose it was on this sheet but maybe it's a bit of a scott Oh, yes it was this this potential here which rather surprised me because I remember back in the 70s sometime in just such a potential function

35:00 in an abortive attempt that I made at that time to show that Newtonian mechanics incorporated mass principle and my attempt failed because it finished well with homogeneous differential equations which therefore as well as having solutions which might have done the trick for me also had the solution naught so my attempt failed but this is not to say that anything one of his own is in fact he gets on to this dark energy now I there is another slide which I'll put up if you like though I think I'm not at all qualified to discuss it it's the future and that's what you'll be hoping to do I think and you must just look at that and ask him about it instead of discussing the figure I'll just go on to read his conclusions I'll shove it up a bit shall I these are all the things he's going to do when I get found to it So, meaning you should read that for yourselves, I'll just read out a couple of paragraphs from his conclusion and leave it at that. he says it seems that the recipe summarized in the abstract and in figure one contributes to the unraveling of two puzzles first, it forges a link between quantum mechanics and classical mechanics which ostensibly has nothing to do with the correspondence principle second it provides an explanation of the geometrization of physics it provides an automatic link between the coefficients field terms that inform the equations of motion in the quadratic Hamiltonians of classical mechanics and the new field equations the latter include equations for gauge fields

37:30 as well as for gravity the fact that the new field equations turn out to be fourth order is another source of complexity but it may presage a further enrichment of Certainly the new equations are vastly more complex than the conventional ones Nevertheless the solutions of the conventional ones are also solutions of these We might likewise expect solutions of the conventionally electromagnetic equations To be special cases of the solution of new gauge field equations Not yet studied in detail, I promise a notable feature is that we get possible distributions of mass, energy, momentum and stress because of the higher order however these distributions are very flexible but the tensor equation which we found requires that in the corresponding general relativity case the cosmological constant had to be zero A linearized version gives extra terms, and these might be interpreted as accounting for the observed dark matter acceleration. However, this conclusion is highly speculative. so that's what I have to say about his paper and I hope he'll be able to answer problems when he arrives I'm prepared to do what I can now Do you think that the older approach that he had might be in something, or in the library, over there in the library? It could well be. I would say one should go back at least ten years in the proceedings. Somewhere around there, I think. I don't know if the truth is not clear to know whether those would be there, but there are some other ideas. Sorry about my voice, can you hear me?

40:00 You know this idea of the whole universe being zero, you know, adding up to zero. The most puzzle about this, the universe, is this all in one instant of time? It's all time. Yes, always. One instant. It's all time. Sorry? It's all times, ever. all times a very vector physical isn't it oh yes the picture is this really that you have something in this case they're talking about energy it might be something else which one has thought of in the past as being necessarily positive but now people think oh it can be positive or negative you say, well, then suppose that the total is always zero, and then sometimes there's a positive comes up, and so, somehow, there'll be a negative. There's this total business that I can't quite get out of. Who taught us on this? I'm saying, and he's concerned. It's all that he's concerned. It's all about saying that. Yes, but that's what I'd like to see if you don't. You know, if it's conservative, but... a single instant somehow but it's all total I can't see what justification there is for that other than in metaphysics of some sort say some kind of mathematical I'm puzzled that you should raise this problem because I thought you were saying a very similar thing about angular momentum it's not the same thing as you're saying It's sort of like saying Touche Well, it's an interesting question So this total energy still has fluxes So it can move from place to place So if you have a closed volume of some kind if the total in that cold volume at one time was zero, right?

42:30 I mean, at the beginning, let's say at the beginning, the total was zero in that one time. And certainly in the early stages, it's hot. So that means that the hot bulb, which is positive, has left it. So we would be in a very, very, very negative energy state right now if you're in that closed volume theory well that's what I was wondering I don't know is it a closed volume theory though is it a closed volume theory no no no I'm doing an arbitrary closed volume I'm not saying that the universe is closed I'm saying we have an arbitrary I can make it this closed volume right here certainly this closed volume here, 13 and a half billion years ago, was very high. So the heat is certainly positive. I don't know about that. And so this closed volume, if the net was very near zero at that earlier time, would be very negative right now, unless also the negative is formed. And normally our ideas of heat, that's positive energy flowing out, so it would have to some kind of negative flow of energy and then it would be cosmological constant so that's you see the point now yes i don't know what the answer is he said he said something confusing there he said cosmological constant zero yes yes um well the um when he says the cosmological constant zero, what he did in order to get that was to suppose it's non-zero you have empty space on the Einstein theory equation so you have R mu is lambda g mu mu you feed that into these equations and out comes lambda equals naught so he, I mean it's rather it's a sort of thing where So you're drawing a conclusion with your feet on two horses going in slightly different directions. Do you know a way to get the higher order version of Einstein's equations from a variational principle? No, that's a good question, but I have not tried.

45:00 I think why I mean just unless there's some sort of urgent questions I answer that at a totally personal level I went back I went back in talking about this to my attempt to show Marx's principle to be present in Newtonian mechanics and the government for this business And at the same time as doing that, I used the same technique and got some of what I call alternative theory equations in general relativity And often I published that, that very clever person on GR, Troutman, came up to me because he had also found some alternative equations and he said, I don't think yours can be derived from a variation principle and I thought, what the hell but it seemed to him that this was a very serious matter so I'm just talking my own personal psychology so when I got these few equations I never worried for a moment whether they came from a variation principle or not, but I ought to and I've no idea how to get them because of that rather awful state, you know, quadratic terms in the Ritchie tensor at the end linear but derivatives of it at the beginning But in this context, the reason I'm asking is because we know that the fact that commutators satisfy the Leibniz rule I mean, x, y dot as a whole as x dot commutator y plus x commutator y not, that happens to Hamilton's equations being satisfied, plus on bracket level, so there's this interesting back and forth between variational principles and algebraic properties, and so there might be a kind of deeper variational principle if you're related to the same strain. Yeah, yeah, yeah, that's what I'm trying, yeah. Yeah, well, that needs further looking at something, yeah. And I'm sort of, of your feeling that you can always sort of put together something

47:30 that will give you a variation, that by variation will give you the equation of the motion. Probably, yeah. So it's not really necessarily something that you would worry about. In other words, if you have the interaction, you can usually, if you think long enough, come up with a more grounded in that sense. Yeah. And am I incorrect in that? Well, I think, I see what you mean, James, but I feel that must be false, because, and then when you've got the Lagrangian, then you could use Noether's theorem to get certain conserved quantities. so if I write if I manage to write down field equations in which there aren't these conserved quantities then I can't possibly expect a variation principle for it that's just off the top of my head I might withdraw it tonight so the theorem requires that there be invariance of some sort of So you're able to write these conserved things. So you never know anywhere and then you couldn't like to learn anything. I can come over pathological things is what I'm saying. And probably pathological agronisms, but I'm not sure. So the question is circling around the previous question. what does this constraint thing mean and that I feel we should well there are these earlier papers that maybe we'll have him here tomorrow and that will well we mustn't all descend on him at once as soon as he arrives I think do you have a sense if the cosmological concept if this cosmological concept is zero, then his dark energy is the cosmological constant evidently, right? His dark energy is the negative energy evidently. Yes, yes. And he's saying the cosmological constant is zero, so that means that since the negative energy is not zero, his negative energy, which is the dark energy, is something different

50:00 from the cosmological constant evidently. Yes, yes, oh yes, sure. That's not the usual work, getting started, isn't it? It's usually to say gravitation is negative and the rest is positive. Yes, that is the usual way, you're quite right. It's not the usual way at all, if he's doing his negative with something else. No, that's true. Yeah, that's very unusual. That's non-vax principle, which is basically what I said earlier. Yeah, no, it's different. But it struck a chord with me in that other thing. a lot of people say it's zero energy but that's what makes it well thank you for listening thank you for doing it it was a major effort let's see let's hope we see you tomorrow thinking of time now to get ready for me and quit And we resume tomorrow, I'll emphasize the time, when is it? 10 o'clock. Yes? 10 o'clock. Thank you. Thank you. Thank you.

52:30 Thank you. Thank you. Well, we've got a place to ourselves, I think. Yeah. Well, that makes three of us, then. Thank you. Thank you.