Quantum algebraic approach to phase space (contd.)

0:00 Entropy operators don't seem to be able to define an entropy operator properly. Rather, what he does is replace the entropy operator by a Lyapunov function, which has the properties that dv by dt has the opposite sign of v. Once you find that type of system, then you can discuss the approach to equilibrium. And what he finds is the following set of results. Key terms may include an operator m such that m is greater than zero and dm by dt is less than zero. In other words, m is now going to be a Lyapunov function. If you work, this is Miserer's work, if you work entirely within phase space, then you find that d does not exist, it's always zero. If you make d an operator, that is, you use, then you find that d again doesn't exist and that's because h is bounded. And finally, if you now try and say, is M factorizable, you find once again that D is equal to zero. Therefore, increasing entropy function does not exist in the case of systems approaching equilibrium unless the thing is non-factorizable. And so you see here a general idea of the approach to equilibrium, super operators. Our particular work using the quantum phase space suggests that you get approach to equilibrium, i.e. in the measurement problem. The suggestion is perhaps... Question. Do you suggest operators in your...

2:30 They're in Hilbert space, but they're bigger than the original Hilbert space. And the so-called super is transferred in the description from the Taizé star beam. I think that is all. Is that used in this? No, that's just a mathematical description. It is known about the work in algebraic systems. The other thing I'd like to ask a question of, with regard to this last step about the approach to equilibrium. What is there in this that would tell me that if I have There are certain types of systems which I know will not project equilibrium, such as harmonic oscillators, that the passage to this description in terms of the Hilbert-Schmidt space or super operators will also obviously not be possible. Well, the point is not factorizable. If you look on the oscillator, all the things will be factorizable.

5:00 Oh, yes. It seems to be echoes of that here. Perhaps you've studied that stuff more than I have. You'd elaborate on that. Well, for example, it's possible to, you look at the classical description, phase-based description, you try and quantize that, letting decimal brackets go over to commutate, and you can, in fact, set up an exact homomorphism of the thing. Yes, you're right. The problem is that you, this is the Van Gogh quantization. Yeah, right. I'm not sure how it's related. You've had something written up there which looked to me exactly like it. I see. The flaw, of course, is that you don't get any reducible representation of the CCR, so you're much too big a... For example, you have Hilbert spaces, as it were, of weird functions on the four-phase space, whereas quantum calculations... That's right, that's right. Well, I mean, I could see the problem is that I got into it on the other side. Yeah, we've got problems. Also, the work of Lucienerovich... Which is very vaguely linked, yes. I'm afraid I haven't studied that sufficiently. You see, I only came across this about six months ago. It seemed to me to be... Have you seen this before, Lucienerovich? I've seen Miseres, I've seen Miseres. What do you feel? Do you feel there's anything in this approach? I'm trying to argue with the data. I think so.

7:30 I don't think we've really pushed it far enough to see it emerging, but when you mention it, I can see that there's a possibility of it coming out. Your super-operators sound exactly like them. Yeah. Yes, but we have to use approximations to get rid of some of it. I've probably rushed over into quick comments about the temperature, you know, letting the temperature... We're certainly jumping out of the structure that I was in into something else. We're certainly going to a different structure. Were you thinking that there was a one-to-one correspondence with the conventional quantization of the world? Well, no, I didn't say that. All I said was... Yeah, well, we can't reproduce the whole of classical physics without doing something physical, you know, by making certain assumptions when we run through this procedure. And whether deformation does the same sort of thing, I don't know. I haven't checked. Thank you very much.