Quantum algebraic approach to phase space

0:00 We've got quite the time now, partly in collaboration with David Bone, on problems connected with the foundations of quantum theory. And today he's talking about some of this work, and specifically I think his title is quantum algebraic theory of generalized phase space. Right, that's the title, thanks. Can we get this thing set? Yes, I'm sure you can. I dare say there are people here who do people better than I do. Is it plugged in? Yes, it's plugged in. ...is to talk about some small portion of the work that I've been involved in with David Bohm, and it forms part of a much more general work involving the investigations of the foundations of... Very recently we have been looking into this problem of trying to reformulate quantum mechanics in a way that doesn't require a spacetime manifold. Rather what we do is we try to introduce the general idea of process and then to use the essential algebraic structure of quantum mechanics to see if we can build up the idea of particle, locality, spacetime. And eventually to obtain classical physics as some sort of limiting case of this structure. Now, the reasons why we want to do this are twofold essentially. First of all, I've always been rather concerned that quantum mechanics seems to presuppose classical mechanics. In other words, you can't start your course with quantum mechanics without ever mentioning classical physics.

2:30 Somehow classical physics seems to be an essential part in the description of quantum mechanics. And one feels that if quantum mechanics is a fundamental theory, then we really ought to start from quantum mechanics and abstract classical physics. Secondly, and this pertains to the idea of abstraction, is that if we're trying to quantise gravity, as I think many people do, then the indications are that we have to radically alter our ideas of space. We've expressed very similar ideas. We have to go beyond space and time. And Wheeler has even given a name to this particular structure. He calls it pre-geometry. We were in contact with Wheeler about what he means by pre-geometry at some stage to get some guidance. And he wrote back saying, you probably know more about it than I do, which is not the sort of thing that encourages confidence in going on. Anyway, I'm not going to talk about those particular deep problems. We have some ideas. I was hoping that I might try an expose here on this, but I felt that we just haven't got it together enough yet to be able to be saying an even plausible letter. Now, of course, the problem is, as you all know, that when we do theoretical physics, the thing we always start with is a spacetime manifold. And very rarely do we start, but there is one exception. Thermodynamics doesn't require it. I've been very fascinated in reading just in the last year is the work of Prigogine and his co-workers. And one of the new ideas, or at least to me it's a new idea, but Prigogine has been working from the point of view of classical mechanics, and what he has been trying to do is to build in irreversibility in a fundamental way, in other words by trying to generalize classical mechanics, so the irreversibility comes in as a fundamental idea rather than coming in as some sort of coarse-graining idea or some averaging over a reversible underlying.

5:00 What transpires in his work is that he suggests that the type of mathematical description you need to handle these questions involves super-operators in super-Hilbert spaces. We extend the idea of the quantum with a phase space, and in that phase space one finds that one has to deal with super-operators. Now, we have come to a very similar idea coming in from the foundations of quantum. And what I want to do in this talk is to try to get over the idea and see how it relates to the type of thing that we're doing. I want us to start by concentrating on the relationship between quantum mechanics and classical mechanics. It was while we were asking such questions, can we reformulate quantum mechanics and classical mechanics in a way? The question we want to know, can we reformulate both quantum mechanics and classical mechanics in new ways? So that we can more closely exhibit the similarities and the differences. You see, what strikes one when one comes into quantum theory is that the language that you're using is very, very different from the language you use in classical physics. In classical physics you talk about trajectories, positions of particles, etc. Whereas when you come to deal with quantum mechanics, you in fact deny the possibility of these trajectories, etc.

7:30 We are not only dealing with an arbitrary pronunciation, but what the analysis is in principle. In other words, we must give up the ideas of trying to analyze what is going on in the case of the individual and to rely instead upon the wave function, but in doing this, of course, we still use the idea of particle momentum and position and so on, but of course we use it in a different way. We use it by actually extracting that information from the wave function. No way that the individual, no unambiguous way in which the individual is at a function, so we have very difficult ways, a very difficult trial of trying to see precisely in what way quantum mechanics works. Now, I feel the trouble with this is that we can always reformulate in a way which will bring out different aspects of that, because I know that it is possible to reformulate, which brings it much closer to classical physics. This is a very old way of doing it, but someone told me that John Bell had actually recommended that I did talk. He said that he feels that one gets a much clearer perspective of what's going on if you do use this method and the method I want to very quickly to show what I have in mind in this comparison of mechanics is the idea of the de Broglie-Bohm method, as I say quite old now what it does is to say all right if we want to compare classical physics and quantum mechanics then what we want to try and do is To retain the idea of a well-defined particle, a well-defined particle, even though it might be unknown, and then find some way of introducing some new field, a real field in the sense of an electromagnetic field or some field, to see if we can reproduce the effects of quantum with the idea of a well-defined particle.

10:00 Real and imaginary. You will then find two equations. One is a simple concept of mobility idea. The second, the Hamilton-Jacobi theory. But the only difference is, if you don't like the Hamilton-Jacobi theory, then of course you can always write it in the field. Now then, what we can do with this is actually look at situations and see what this particular theory looks like when applied to certain physical situations. You can't have interference and talk about particle trajectory. Well, this is absolute nonsense because we can actually calculate the trajectories. We can take a two-slit experiment, say. For example, let's take a Gaussian two-slit because it makes the calculation that much easier. And we can actually calculate the trajectories for the particles going through two slits. And you'll see that because of the presence of the quantum potential, the trajectories do not behave like classical trajectories. And they change in such a way that they bunch up and give the correct interference pattern far away from the slits. In other words, you can reproduce the interference pattern and yet have...

12:30 I'm asking for clarification. By trajectory, do you mean solutions of that differential? I mean solutions of... yes, yes. And of course... And the point is that that doesn't correspond to other variables. The quantum potential. Yes, the chief... Well, that is if you still... if you do not detach the quantum potential from... In which the quantum potential arises from some other, the way Vizier tries to do it. At the moment I haven't done that, but that is a possibility lying behind. People like Vizier want more seriously than I, because they feel that there is something. But then it would be a category. In fact, you can see the sort of thing that we have to have. Get those kinds of trajectories, it's a quantum potential which looks like that. Actually calculate the point. Now you might say, well why do this? Is it of any interest? Well it's rather interesting that it was John Bell actually... Looking at this kind of, not that he had this information present at the time, but looking at the analysis of the quantum potential at 52, that he actually began to think about the question of non-locality, because what you find in the quantum potential is that it's extremely non-local, and remember there's this argument about potential inequalities, then in fact we can begin, which show that there is something non-local, so that by putting

15:00 My point just simply is this, by putting the formalism in this way, by quantum mechanics in this form, we can begin to bring out quite clearly the differences between classical physics. That's the only reason for it. Some people feel that this is a replacement of quantum theory. They feel somehow that what we want to do here is to replace quantum mechanics by this thing. If you're a practical physicist, you wouldn't be so daft as to try and replace it by this, because although this gives exactly the same results that you get from it, So this is only the point here is that you're trying to compare classical and quantum physics and to see what the essential differences are and to see if you can exploit those differences for some purposes or others such as the long run. Now there's another reason why I don't like this particular approach and that is that the relativistic theory is not very good and actually exists. Now here again I think Bell would disagree with me because he doesn't use the spinner problem. Okay, so the reason why I mention this method is because it shows us that you can in fact reformulate a theory in a way which will bring it into closer comparison with classical physics and then you can see the similarities and differences. Now what I want to do is I want to extend that theme now and to see if we can in a phase space. Now the reason why I want to do this is because one of the problems that has always been left behind by quantum mechanics is what is known as the measurement problem. There is a wide opinion that will not in any way solve the problem and that some people see it as a problem of other people.

17:30 Now I want to suggest that some of the methods that I'm going to use later on which ties up with the Priggish actually has a way of dealing with this measurement, the usual way of adding knowledge. So basically it's this. We've got a system and an apparatus. And before they interact, our wave function is just a product. What we do is, if we assume an impulsive measurement for simplicity's sake, as occurred, so far so good. The problem comes not when you look at an eigenstate of that particular observable apparatus, rather when you look at a linear superposition of states. The result of the interaction with the apparatus produces now a linear superposition, and we never see a micro-ratus in a linear superposition of states. What we actually do is we do the following. We say, alright, what we really mean by that is that we see the system either in a state of that nature, a system with time sigma 2, etc. Nobel calls this a very ethical way of dealing with the process. The process is simply this. What you do is you say, all right, I get a particular reading and I cross out all the results in that pure state that don't correspond to the actual reading I get and so on.

20:00 Now what is required in this pure state to mixed state is in fact a non-unitary transformation whereas in quantum mechanics as it's formulated so far we do not have a non-unitary transformation. It's essential to treat the macroscopic being one of an enormously large number. Are you sure you're not slipping?

22:30 I'm sorry. It's John Bell's turn. I think the argument in that sense, perhaps they are. But the point is that the infinite system is in mutually orthogonal spaces. And so what happens when you make the supposition Is that some of the state vectors responding to the measurement apparatus in the sum you've written, in that same sum, lying completely orthogonal spaces? Are you saying that you can get a situation where only one vector lies in one Hilbert space? It can happen, yes. But not in general? With suitable, you need suitable, just an apparatus called... What are the properties? Well, it needs to have a certain instability which is well defined with respect to the interaction. In other words, what you need is something which can trigger off your measuring apparatus into two completely Hilbert spaces. When you have that, then you get a super selection which leads you to mixtures that get down just to the state of the... It's a very special argument. I think measuring and creating is something very special. They've got to have a very, very strong, they've got to have macroscopic to microscopically state. In other words, macroscopic changes have got to give way.

25:00 Perhaps you ought to continue with the story. I just wanted to say that I think there is a perfect... Can I just, my answer is... I haven't seen it, but I know some people don't think it's... Let's say Bell has objected to it. All right, Bell has objected to it. Let's say Bell has not studied it. I don't know whether he has. I think because that quotation is certainly after Hepp's work. I'm sorry, I just took it. So I'm assuming that for the moment. And my trials show that there are other ways of doing it. Okay, so given that Hepp's argument is claimed to be refuted, Let's assume that that Wigner quotation is in fact still stands because right now what I want to do then part of the theme that I'm trying to develop in this talk is that it is useful to actually try and reformulate theory so that you can what I now want to do is to go on to look at quantum mechanics in a phase space to see if we can see if there's anything interesting that we can learn by looking at quantum mechanics in a phase space. And then seeing the differences between all and trying to exploit those differences.

27:30 To do this then we are going to use this theory to apply to pieces of apparatus and so on. It would be useful if we now forget about the wave function description and actually go on to use the density matrix. There are several other reasons for doing that. Now in order to use the density matrix we are actually discussing the theory in a configuration space. Now the transformation that I use is very simple. And that is that suppose we take a density matrix like this, and I'll do everything with one variable unless I have to, then what you can do is you can just simply Fourier transform the thing, in other words you can get your density matrix in that form, now if you use a change of variables and you go to the mean position, then you can rewrite your density matrix in terms of a distribution x and p, you can then get a relationship between the representation of that density matrix in... The phase space was designed to see if you could reproduce quantum expectation values in a classical phase space, and in fact it fails at doing that because the f of x p actually turns out to have negative values. But as far as I'm concerned, we are not going to use that f of x and p as a problem.

30:00 When we have such a, I'm sure this is all, what I've done here is let's take the one-dimensional, use the Wigner-Boyle, and then the point is that if you use the approximation, this difference here is equal to, then you find that your, if you use the Wigner-Boyle transformation on that, then you get the, and this is exact, you get the same result both for quantum mechanics and for classical physics in the case of the three. And what we can do is we can do the following, a small step from there. And we can say, all right, look, let's suppose we take that equation as it stands, there, equation two, and let's now do a Wigner-Moyal transformation on that so that we can write density matrix equations. And now if we look at this, we'll notice that straight away that it's essentially a non-local equation. In other words, it's not like a classical Livy equation, it's not a point-to-point transformation, but rather it's a non-local transformation, it's phase-based due to the format. Second point is that if we make a Fourier transform of this, we can then see that if you Fourier transform that, you'll see that what we've got here is essentially, the distribution essentially involves a set of discrete jumps in phase space which are averaged over which are weighted by this. So we've got in here also the idea of a jump in phase space. Now, of course, we could do the opposite to this. This is taking quantum physics and putting in phase-based formulas, but equally well, we could take classical mechanics, i.e., we could start off with our equation, and then we could try to put this little bit of equation into configuration space.

32:30 All that involves doing is taking the inverse transformation, working the thing through. And here we make use of an approximation namely we write the density matrix where r is some slowly varying function of the x's and if we do that we then find that this term here actually turns out to be that and if we use the Hamilton Jacobi results we will find that that just becomes p prime so that we can get an equation. Now I want to draw your attention and that is that this equation is not factorised unless it's the free particle. Now what do I mean by not factorizable? I mean that the equation cannot be written in that particular form. Now the difference is that when we go to quantum mechanics, of course, all the equations can be written in this factorizer. Now the importance of this factorizer is the following. What we've actually done is that we're now working with the density matrix treated as a vector. Suppose that we look again at our function here. That is now a vector in this. And therefore, the question is, can we choose some f, choose some, biggest place in the probability distribution will give us something like. Before I can do that, to ask is, what meaning are we going to give to these f functions? Because we cannot treat them as, or as densities, rather.

35:00 Now let's recall that really what we're dealing with when we're dealing with something is we're dealing with something which is a constant in order to go further what we want to do is some of the ideas that in his plasma theory when he was dealing with plasmas one of the methods that he used was the following he wanted to try and find some form of general collective And the way he did that was to find these coordinates as approximate solutions of Liouville's equation. Then, if these F's that I'm talking about here satisfy this Liouville equation, then we can get some collective coordinates of our system in terms of this form,

37:30 which satisfy this particular equation here, which is quite odd. Then, the F naught are the constants of motion, and notice that when we've written it in this form, The constants of the motion, the solutions, are actually obtained from an eigenvalue solution, somewhat similar to the stationary state eigenvalue solution that we use in quantum mechanics. I mean, now, instead of having the Hamiltonian as our operator, we have the Luville operator. Okay. Now then, the claim is then that these eigenfunctions that we get here actually satisfy, actually represent some form of stationary state of this classical motion in phase space. Recurrent process going and this is described by means of these collective coordinates and in general terms of course we can write that position of these a general recurrent process in R. The next question then is now can we convert this over this idea over into quantum mechanics. Now remember that constants of the motion are in fact complex and therefore in quantum mechanics we can write a matrix If we use the Wignham YR transformation, then that means that we can analyze our f of x that we've got here, then I can satisfy that equation, and then what I want to do is, I want to say that these matrices actually are some form of, they've characterized the actual type of motion that we're dealing with, and therefore we're dealing with some kind of generalized quantum. In order to deal with this, we say, let's introduce, and this is the new feature that I'm towards here, let's introduce a new matrix, which is essentially an extension of the density matrix, which will satisfy the following louvial equation which we've had.

40:00 Now the point is, can I actually discuss a probability function using this characteristic function in such a way that I get rid of my negative probability? Now the claim is, and we can do it in the form, just to illustrate, of the density matrix. I'll come to that, I'll show you how it's related to the density matrix in a minute. Consider, suppose we take the density matrix, this characteristic matrix, to be represented by the square root. Then let us define a twiddle, which is the complex conjugate, and form a quantity like this. I'm going to call that the statistical matrix. Why? Because when I put this into, when I put three into there, I find that this thing actually reduces to the density matrix. In other words, what this characteristic matrix is doing, it's behaving like a kind of a square root of the density matrix. The phase space, well, more generally I've taken a very particular form of that to actually get there, but of course we can generalize the matrix in that form. And then if we define the expectation value of an operator in this way, you can show, if you work through it, that you can actually write your expectation value in terms of a trace, Wg, with W, okay? So now we've actually, we can show that W always has non-negative. The next question we've got to ask is, can we find a non-negative probability density in phase space?

42:30 And once again the answer is yes. If we use this psi and then do our Wigner-Moyal to get an f, this is not the same as the f of x I've been using, originally this f of x here is actually the Wigner-Moyal transformation, then you will find that a psi star of psi will have negative probability, will have non-negative probability. Now the question is will this work? In other words, can I make expectation values of this kind in my phase space? And the answer is yes, provided we have g to be the simple form of x and p, but as soon as we generalize that, so operator to deal with p squared, etc., then we find that this does not give the expectation values, rather what we've got to do is we've got to show that if we allow the, it'll only work properly if we go into some high temperature limit. And you can see very I'll explain briefly, very quickly, why that might be plausible. If you just use this very simple argument here, that it's high temperature, you find your density matrix matrices are actually becoming diagonal and becoming local, because if you put in your weighting function there and you let T go to infinity, then the density matrix essentially becomes a delta function. In trying to get that distribution out is this idea that when your system is a theory which looks very local terms, then you can, so in other words, the argument you can do this provided when you go to high temperatures, a system will behave in a way.

45:00 Now this seems to me to offer the possibility which was not present in solving the collapse problem, not many of the theories, in the chimney, to see whether you can... In fact, there is a difference, but they've looked at it experimentally and they find that the advantage of using this particular approach, this only becomes classical when you have, because it's when the system comes into the measuring apparatus, collapse shouldn't be, the linear superposition, the linear Schrodinger equation is okay provided it is isolated and not, because if you want to bring in the collapse of the wave function in that way, okay, well once you've got this distribution working in a phase, Then of course you have the possibility of actually bringing in a more generalized motion mechanics because you can now introduce the density operator in your phase space by through an equation like this and this means that we can now actually use the case we can develop different functions K with different derivatives in here in fact I've got an example here to show you what what is going on if we If you use the inverse Wigner-Moyal transformation when we introduce a term which is proportional to a dx squared, then you find that your equation is no longer factorizable because you've got this extra term on here, and it's this extra term which gives rise to diagonalization of the density matrix. For example, if you go to the Fourier representation, Fourier transform...

47:30 In this case, the equation looks like that, which you can then solve straight away to give you this exponential term, which shows you something decaying going on there, and therefore you've essentially got a high unitary transformation present. And when you look at t going to infinity, then the density is going to go to zero unless p equals p prime, so that if you take that particular case of p... And so we have a possibility not changing pure states in the mixtures, but we've got describing. Quite arbitrary which particular, what's missing from that is that you need some sort of law which will tell you which terms you need in your k function there and unfortunately there is no real investigation of that. Now it's here that I would really like to show how this is related to the work that Prigogine is doing and how it introduces the idea of an operator which is related in some way to time.

50:00 Because what I've been doing in this particular mathematical session here is I've been looking at this generalised phase space and I've been looking at particularly to a non-factorial Prigogine to exactly the same conclusion, not from the phase space geometry point of view, but from classical mechanics. Now in order to motivate this, I'm not quite sure what's the best way to do this, but let me try this to see what's all real. Let's suppose we go right the way back to classical physics. Then we know that if we've got a variational principle, a Jacobian equation, not so well known as in the beginning of the world, now then suppose we go to a new set of coordinates as we've been all the way through, then we can find that we'll get, and we change our s function, in this time we're talking about cap t, the same previously I've been talking about, then you can get a set of equations which, trivially, and this suggests that we've got Poisson brackets present. Which are satisfied in that way where I've now defined the Poisson bracket in terms of and also there is an extension of this which says that if we take the energy difference and we take the Poisson bracket with time then that also equals unity.

52:30 Now then we can do exactly the same in quantum mechanics. In other words we can instead of using Poisson brackets as I've used there let's use P and P prime and I choose the, I'm dealing with the That means that I can then get a set of commutation relations which, when I now change to those variables, become that. In other words, what I have is, I now can find a representation for those variables, which are of this form, and these are in fact the super operators. And Prigogine writes them in this particular form here, namely he shows that they are matrices. In the case of x it's q plus. But we've gained one important thing, that when we're dealing with ordinary quantum, we've always tended to write and it doesn't exist because it will only exist if we have a spectrum of p and x which goes from minus infinity to zero. The problem with energy is, if we go to the phase space, the time, if we go to the phase space, then, and so the suggestion is, so if we go to the energy representation, now since it's e minus e prime,

55:00 It means that we now have the infinite spectrum. So it means that what Prigogine does is to show that there are classical systems. There are test events in the middle. And Corbage has shown that these operators actually exist in the quantum mechanical case as well. So why does Prigogine use this and what is the relation of this and time? What he does is the following. He says, but I can construct very simply in the following way. Let me take a particular example. Let me take an example of a Baker's transformation, which is a particular type of transformation, which is not a point-to-point transformation. Wait, sorry, it is a point-to-point transformation, but it doesn't have any continuity in it. What it means is that if you've got a space Q and P, which is a square like that, then if you take every point in this half, half there, you take every point in this half, then you do it again, you take every point in this half... So as you get exactly what I've made, you get now a movement in this space which is not a continuous movement. And then what he shows is that you can define a complete set of functions on this space, and that this transformation B actually induces a unitary group on there, with the relationship that u is a measure of time, n is going to be a discrete term now,

57:30 so that we can then define an operator u , the generator of that transformation. And if we have a EN as the eigenprojection of T, we can write T in that form. And what Prigogine claims is that this actually gives an idea, gives a measure of the age of the particular distribution. And his whole idea is to say that we can now introduce an age operator which you can use on the particular distribution. Because the distribution has a one-weightness in it, you can then talk about that age, not time. So, one thing that I would like to add to this is that this would have actually, it seems much more appropriate if you talk about it as the age of a distribution, if you've got a linear superposition of these terms, mean age of a process and so on, so that you can actually talk about the age of the development of systems, and in fact just recently I saw a paper in FISREV with Prigogine and Misra and someone else who actually talked about using this operator to discuss the age of evolution. What I would like to do is just to show how this also comes out of his particular work. This is only just a very rough sketch and an outline of the approach to equilibrium. And then we'll see that this age operator, the entropy, his assumption that the entropy is some monotonic... Now we know, and therefore it's related to the question of the approach to equilibrium.

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