Clifford Algebras, Quantum Blobs, 2-Time Theories — Part 2

0:00 And you know what, this is the Fourier transform of the Binger transform. Yes, I've got this as a Fourier, yeah. But it's an ambiguity function. Exactly, exactly. So it should be about blocks. And this comes straight out of the Heisenberg group, as you correctly pointed out this morning. Which makes it rather interesting, because what I thought was essentially quantum mechanics might be the Heisenberg algebra, has got nothing to do with quantum mechanics, because I can use it with radar. Well, light is quantum. Sorry? Light is, no, it's not actually light. Well, you see, no, I haven't got anything quantum here. I'm still using the Clifford algebra idea with, you know, the light rays that I talked about last time, and that Roger's always talking about it. Yeah. so I'm very worried where is quantum mechanics what is the essential feature of quantum mechanics which is what you were worrying about with the H-bar messing around with that and Roger raised the question you've got many states and quantum mechanics how do you decide which is classical and there's classical immersion here I've got all classical and yet somehow I've got results which seem to be relevant to quantum mechanics you're looking at individuals from Japanese Yes, yeah. But that's the spirit in which, you know, I've always been working on it. So what happens if you take a problem like this that you are integrating over all classrooms with Jack Beesley? Yeah, yeah, but then we've come to the problem that Morris was talking about. You've got a mixed state, and how do we deal with a mixed state? I think that's a very interesting question. But like Morris, I'm going to say, I haven't been there because of the problems that we're having. But it's interesting to see in a single case that it is just classical physics, and you're building on that. Well, if it is quantum, I want to know why it's quantum. Basically, it's a positive... I mean, if you think of a wave function, then a photon is just Maxwell's equations. Yes, yeah. But you've got a positive... you've got a complex one in a positive frequency condition. So that's what makes it quantum. And that's all quantum mechanics is? From one part of it. Hmm. Okay. now what I want to do is to go on to try to think about this two times, these moments and there is a two point approach, this was done by Singe, I mean this guy got

2:30 this from Singe's work where he was trying to quantise gravity using Hamilton's ideas you remember the two classic books, Special Relativity in general It's the world function, yes. It's related very much to the world function, so that you can tune in to where I'm running here. Okay, well the idea of the world function is that you have a trajectory connected by the world function, but the two points that you choose on the world function are conjugate points. This is the importance of this, they're conjugate points, which means they have special relations between them. you can get it from a variational principle by varying endpoints and if you vary the original points you can get these two equations and I'm sorry but this you probably recognise is grad S equals but now it's minus sign the plus sign is what if you do the variation at the other end and this is the Hamilton-Jacoby equation and this is the conjugate Hamilton-Jacoby equation So you've got this, you know why I'm doing this, because of the... This is a plan-dependent generating function. Yeah, yeah. And it depends on how two of them. So now what I can do is I can bring in a mean time and a time difference. Okay, okay, okay. Okay? And so now I can have a, I do a Lorento, there's something missing here, a Legendre transformation. Right. and then I get these two relations look, the difference between the energies with the mean time and the sum of the two energies with the time difference, and this is my sum remember I was asking you, have I done the right thing here, why am I getting sums and differences the sums and differences come from the classical world as well oh, sorry so you get something like this coming out okay and then you put it together in this way and the thing I wanted to show you was that in classical physics you get a Libby equation coming out with this approach and you also get this energy equation and this is going to be in the case of the Bohm theory with the quantum and Jacobi equation this is going to generalize into the quantum

5:00 which is going with a lot of potential in it. And I'm now using, this is, I think what I'm using here is something more generally than you use. Because not only do I have, I have T and delta T in there. So I've got extra degrees of freedom I play with. Normally you don't play with those freedoms because everybody says time is time. Okay? So my blob is not only in space, not only in momentum space, but in energy and time as well. And this is, yeah, exactly. Okay. Exactly. So whether that will help you? You have more problems to worry about in your... No, no, it's okay. Because you have any way to have a structure on this space. That's where the minus sign was going to take a look at it. Yeah, yeah. I'm just wondering whether I need this. This was just looking at the, treating the density matrix as a probability distribution and getting into all the trouble with the negatives. oh no it's not I know what this is doing but in fact it's what it's what you did I'm just summarising what you did because I've got a mapping from my operators to my function space and there I'm putting that oh yeah do you thing do you are thanks for coming thanks a million for coming There's nothing more of interest to you here, by the way, Lou, but I'll see you in Egypt. I want a copy of this slideshow. Oh, okay. You can make a PDF. I'll send it to you if I... Yeah, okay. Okay, Lou. Yes, now, what I'm claiming I'm doing is that this is still a density matrix, and the fact that I've got a capital X and a capital P here is not violating standard quantum mechanics because you might think, oh, surely capital X, capital P commutators should equal life. It isn't. Capital X, capital P, where they're operating, is equal to zero because I've chosen conjugate points. Like here. Okay, you've got conjugate points here. And therefore, this is really a quantum mechanical object.

7:30 So I'm really using the eigenfunctions. When I'm converting it into a function space, I'm using the eigenvalues of cap X and cap B. So I'm really still doing quantum mechanics. I'm not doing classical physics at all. I don't know if that helps you, but bear that in mind. Oh, and then, remember, we introduced the Moy-El product. and then I think somebody will be very watching to see what a moyal product looks like. That's another way of writing. Yeah, and the moyal product is you've got your A here and you've got your B here as a function of X and P and now you operate D by the X and the arrow's that way, I'm sorry, you probably can't see the arrow's that way on that, that way on B, that way on B and that way on B and then you subtract. Okay, now the interesting thing about these is that your commutator becomes the Moyal bracket. So this is a non-commutative algebra we've constructed here. And then you've also got another bracket. and the other bracket is related to the anti-commutator in the quantum mechanical case and it's known as the Baker bracket and I remember reading about the Baker bracket and wondering why on earth Baker actually introduced it because it's got a paper in which it appears and it seems not to be used and when I put my thing on the web about the two equations so I've never seen the second equation before I talked to it the one I called anonymous and David Farley immediately said oh wait a minute you should have a look at the Baker bracket because that corresponds to your Hamilton Jacobi your quantum Hamilton Jacobi it is just in fact a Jordan product and you can see why people might think it's not very interesting because when these are functions they just commute with each other the ordinary product. But now you've got exactly the same in this Vigno-Moyal as I was getting

10:00 at. You've got a left-handed energy operator and a right-handed energy operator. But it starts in the Moyal product. It's a Moyal product, yes. But you've got a left Moyal product and a right Moyal product, yes. In exactly the same way as I had a left connection and the right connection in the Clipper-Brandon. I was very pleased when I saw this because it meant that what I was doing was not totally crazy. It was related to the classical. And this thing, the way I define it is by having this product. That's differentiating it from time. And this product is this thing that's differentiating it from time. I assume F is psi. I'm sorry, it's F. No, f is the density operator, and then I just split the differentiation up in this, and there's two different ways. So what's the, ah, I'm confused, because where's the anatomia? Is it the D-A-T-T or? No, this would be the Schrodinger equivalent to the Schrodinger time-dependent equation. so I have to put H in there I'm talking about the time what do I do to the time derivative if I have a left operator if I have a left-hungal toning and a right-hungal toning what do I do with the time derivative so my question I guess the same as you what's the relation between f and psi f is made out of the Vigner distribution my memory is limited let me see yes there yeah yeah well is it is it there where is it it's where is it there no no no it's back here somewhere isn't it sorry it's been too long since i did it's here see the f is related here's the roll which i didn't know the roll it's it's yeah it's this here that i'm starting with here it's related to this thing here but it's two point and two time and two position objects okay so then what I do is I subtract the two equations

12:30 and I get what looks like a equation I got a moyal bracket there instead rather than the And then I add these two equations, but of course if you look here, this looks like an absolute swine. What is it? Because you've got a minus sign instead of a plus. OK, well, I sort of almost gave up at that stage, but then I thought, well, you know, let's do the trick. Let's have a look and see what happens if we resize it with Re to the Is, and just flog it through, then becomes even worse. but the interesting thing about the Moyal algebra is the deformation algebra therefore if you use the Planck's constant as a deformation parameter then you find that if you go to first order in H, or second order in fact, you find that Moyal bracket is just minus ds by dt times the f this in the classical limit is just the product, so you've got the way, it's just the Hamilton-Jacobie equation. The Moyer bracket, the Baker bracket, I think, upon, produces the Hamilton-Jacobie equation. The anticommutator that I get from the third and algebra gives me the quantum Hamilton-Jacobie equation. So there's a complete symmetry between the two, and then you can get that. Now, remember, we know this, don't we? the uh conservation i just want to go on now projections i'm not interested in that what i want to do now is should i finish i just want to mention where time comes in yeah am i doing too much yeah okay so what i've got here then is my face space has got these coordinates in but they've also got time coordinates coming in so if I define Poisson brackets by means of this kind of differentiation I've got a series of Poisson brackets in the double algebra but I'm also making the suggestion that there is Poisson bracket for the difference in energy and time and the sum of energies and delta t so I'm introducing I'm saying I want to take these and this is all classical

15:00 these two things, so the problems you were dealing with do not appear here, as far as I know. The kids have no domains, they're not operators. They're just standard functions. And now I play the game, okay, suppose we turn them into operators. Then what I've got here is I've got these things, and they're... So you had p plus delta p and x, this should be x. You had p plus delta x and x plus delta p was your bot. So you're taking linear combinations in this bi-algebra structure. And in fact what I've done here is I've actually doubled everything. and so I've now got a beautiful bi-algebra and what I'm doing is the two equations I've got essentially can be written in terms of bi-algebras but if you follow this through, I don't want to do this quickly it's collapsed, sorry the whole machine's collapsed, it's obviously wanting me to shut up I just lose my power point, I really must go to B-man it just decides it's had enough no comment you've got a chairman there who can shut me up at any stage where? what I wanted to do was that in this structure what I have is this equation here with the Lagrangian That is the equation that Prigogine introduced in Being and Becoming. And that shows you that this time here is proportional to ordinary time. And this doesn't have, even when you're going into the operator form, this doesn't have the problems that you're talking about because this is not bounded from below or bounded from above. It's the difference in the energy. Yeah, but still, I mean... Have I got a problem there? I'm not sure of some analysis, so that I wouldn't be sure that we wouldn't have formulas. Well, Fugershin's claim was that there wasn't a problem that you're worrying about. No, I mean, it's a purely formal equation. I mean, yeah. What do you mean by formal equation? You mean it's because it's been created out of this biological structure?

17:30 So, no, I mean, perhaps there's no problem. Well, I'm looking for you for confirmation that there is or isn't a problem. Not now, I mean, you know. No, but I'm not quite sure. I mean, I came across the same equation when I sort of tested it against this OK. No, I mean, I think it needs looking at the mathematics rigorously at the mathematics, but I'm just sort of throwing these ideas ahead to it. Now, what I can do is I can actually construct co-products in all this, and what I get left with is essentially the Boglioubov transformations. Ah, OK. But they come out of the co-product, the bi-algebra structure that I've generated, and then I define some new co-products in there, and this then gives me the body of both transformation and I use that to actually get a picture of time I don't have any data when I created it but I certainly didn't do it just recently so what have we got here so what's that and that about? what's your statement about? They're the Bob-U-Bot parameters, so you've got your, the Bob-U-Bot transformation takes you through an equivalent vacuum state. Right, so. Okay? So what I'm saying is let me pick a particular vacuum state, one of these in equivalent vacuum states. Yeah. Then I'm saying that you can have the dynamics within that vacuum state. And that's Schrodinger time. Yeah. but there's another kind of time and that is the time between inequivalent vacuum states so you get two types of transition I got angry with Yakima so nobody's ever done anything with two times whether this is correct or not is another matter but I certainly have done it with two times and this is related to entropy because as you go from one equivalent vacuum state to another vacuum state you can actually show the entropy increases how does this relate to well to chris's construction the chris chris items two times because that said also that i mean that that as i recall also um because i mean he certainly recovered the bubble you bob

20:00 yeah it's just using the transformation as the generator of your irreversibilities i think that's pretty much what he was doing but does anybody know it's the obvious thing to do Yeah, okay. I just like that. And also, is there a connection here with the con-revelli formal time? I'm so tired. Oh, okay. But there is, isn't there? There is a relation. What I would like to do is establish a proper relation between these two ideas. But he uses, what does he use? Do you know about this? It's a complicated theory. It's a very complicated theory. And it works only for a type 3 phenomenon. Yeah, I know, I quite realise that. I realise it's out of my game I've been playing. But then when I go to the biology process, I generalise so much that I don't quite know where the original structure was. That's one of the problems. that really needs me to sit there and quietly and go through this whole stuff again. You have been inspired about stuff as much. You lost me a little bit. No, I'm lost as well, but I mean Mike raised it and I've just got it on the slide saying, is there a connection? Anyway, look, I'm really going to stop talking. Okay. Can I ask you to put on the previous slide to go, you know? You know, we really need to go through it slowly if you want to. Because I've introduced a conjugate momentum to the parameter theta. Yes, I'm not against it. No, I am because I'm not too happy with this, the correctness of this. I mean, I shouldn't really talk about it. I don't know, but I just want to stimulate people like Morris to think about these things. Okay, that's it. Any questions? actually if you can cut that because what I'd like to do can we get the beginning well I was going to say what we should do is to thank George this is my real job at this session keeping you here against your will George I thought it was Carly where is Carly

22:30 We ought to thank these young gentlemen as well. That's what I'm going to do. I want to thank everybody. And I thought we might owe it formally rather than you formally. Yes. Good idea. Well, I'd like to thank you, Roger. I hope you've enjoyed yourself. It's a pleasure to have you here. Yeah, we had fun. That's what I told you we'd have fun. The question is whether we did anything serious about this. That's the same, really. Okay. He's very interested to get Roger's judgment on that, actually. He's gone for a swim. He's gone for a swim. Oh, he's gone for a swim. Well, you can blame him. Let's say thank you for Bjorn and for Daniel. Kari, we'd like to say thank you very much for your hospitality, for looking after us. Thank you for coming here. Lord, it's a mild pleasure. It's very, very good of you. What about the two girls? We'd like to thank them as well. Yes, Linnia. Is Linnia and Anna there? Linnia? Say thank you to all the young people. we'd just like to thank you formally for all the work you've done we haven't noticed you but we've always had food and tea there thank you very much and as for not noticing you left house and street clear yourself cheers Well, that was supposed to be a compliment, you realise. Yeah, I know. I didn't mean to be so forward, but... The more you don't notice about the background people, the more efficient they've been. Ah, true, true, true, true. So it is, in fact, a compliment. Yes, of course it was. I'm sorry. You know, the best referee in a soccer match is when you don't notice him. Absolutely true. Absolutely true. And they often say the same about the president of Switzerland, don't they? Buster's a model of democracy because Albanians doesn't even know who he is. That's a successful and well-known country for you.

25:00 No, no. Do you that with the President of Switzerland? Nor do no Swiss people, which shows a very well-known country, dear. You're always busy doing things. Do you have any particularly strong reactions or feelings, one way or the other, about the differences between your route into the twist and this approach of this approach? I'm going on probably Friday. But what did you see as the distinctive advantage of the route that you took through them? I mean, the thing is that... I'm trying to get Rutgers' reaction to your propaganda favour of Skipper Aldridge sorry I didn't mean to interrupt I don't think you can hear what I'm so sorry I just wanted to get his reaction to the different approaches sorry no it's not going to make any difference to what I think about what he well that's for sure it's different but it's different things Suppose you want to use vector calculus, then you see the analog equipment out here would be, they're all sort of matrix operators, so somehow, to me you're starting the wrong way, because you're looking at transformations, you're looking at Lorentz transformations, before you look at vectors, but sort of geometrically it's the vectors which you want to describe, mainly because the thing with one index you can easily go to two indices out of them yes if you start with two you have to go yes one I'm going to treat the one index and objects of some certain eyes or something much yes which is I think where this focus on the ideals of that yes yes what you motivate this all in terms of the timeline process with one way or another I mean it's it's I'm sure the

27:30 motivation to it yes this is you're trying to replace the chunk of the I know he's got this underlying philosophy that the space-time, of the Clifford Altruz is connected with his underlying process ontology, but as I say, that, I don't, well, that, in terms of the mathematics, I don't see what work that assumption is doing. Yes, and I can see that the... Also, so things like the direction you do it, and the notation, and this and that, but there's also a very different underlying drive. Yes, and that's where everyone is trying to... There's also the concentration of confidence. Yes, that's obviously true. You like reals, but you can see that. I take a different view than the... Yeah, because you've been keeping an eye on them. I remember linear was this one to bring them to the floor. So that's different. And also the idea, as I said before, that you don't regard space-time as as blocking as primal. Is that what it is? It's a derived notion. Yes, the twist of space itself is more fundamental. that leads to different things when you curve space. I mean, that's where the five-dollar approach is, which isn't going to happen to that. But the drive is to think of the space time as primal. Whereas most other approaches, you think of not most of the problems of the over space time. You're looking at structures at points. And the whole point of the twister approach is to turn the point into a derived. That's right. And so one of the motivations behind that, although it's like, well, I'm to the four, and you've got to do that, and you still want to do that, and then you don't think of the fuzzy metrics, you know, having views with the anthropocore, to some kind of point, the rural approach. Yeah, so therefore they try to quantize the metrics.

30:00 It landed quite on Japanese lines, and they hatched up in the other way, as well, they flew it back, and the CO, the Air Force, saw them, and put them on charge, because they shouldn't have been flooding. Yes, no, I've seen all of that, as I say, I don't see them urging very clearly. But no, it's interesting, I did want to get your reaction as to what the difference in the conceptual orientation is, and it's pretty much what I had thought it was. But also, you see, okay, a lot of orientation was using it, but it's not one of the measures, it's not very common. I mean, if this had been done in the 1930s or something, maybe a little later, let's see, Dirac's situation is... I think 1930, but they probably weren't known to physicists until quite a lot later. There was a little burst of activity earlier in this. Some people wrote various things in just a bit of notation. It's not in any kind of phase it out. He's got very good deaf age. He's quite curious, you know, his sort of historical reasons. It's partly because, and I always find it very strange, because Dirac, okay, he, I mean, he sort of read this kind of technique, isn't he? Yes, yes. Exactly. On his own. And then when the notation is, like, his character's, then the violin was just developing So he then introduced this alternative or, I would say, deeper way of... just points was two points, you know structures at points. But then Dirac saw this and actually realized its superiority and he wrote a paper on arbitrary spin and wave equations or using two spin and notation which as far as I can make out since everybody else then followed Dirac's original, well not everybody but most physicists have followed the original Dirac four spinners and so when Dirac went to the

32:30 two spinners, they never noticed this part, they may have noticed it. Yes, but he had moved across from four to two spinners, or what the motivation was. They all worked with the four spinners to develop sort of ad hoc lots of different labels, you know, Duff and Keller and all these different ways of approaching the other spins, where it's basically Dirac edited in this paper, all done and passed. OK, there was still misused, but you have to worry about how electro-magnetic interactions. I'm sure Dirac is quite right, but it's, I mean, there are issues. But then, of course, they used two spinners, too. But the sort of body of theoretical physicists didn't. No, no. And they just followed the original, rather than what he'd been around. And I learned my two spinners, and I made this important dirac. It's time to choose. You're trying to learn it in force and you're playing it. Yes, you say, last night. I do find this way of illuminating in some respects. I mean this way, Basel, because I guess the, yes, I mean, I think this idea of getting the flag slayed out of the spherical, hymobolic instructions, that's ingenious, and perhaps because when I tried to study it, it took me a long, long time to figure out where the flag picture was sitting in, But then, of course, that was because I don't have anything I could feel for complex. Yeah. Well, a chap called Payne, who actually pointed out, he called it an axe. But he was actually, I think, only three-dimensional. I'm not sure whether it was Lorentzian spinners he was looking at.

35:00 But anyway, he did have this axe. Yeah. That's a spy. But I think there is a difference in motivation. Oh, yeah. Okay, there are things that people... It's not just that you have a... A sense that the complex numbers are just more fundamental. You ought to be treated as a fundamental fundamental. Greg Basil knows much how to think of the reels as fundamental. It's something more fundamental. Or rather, that is itself a reflection of a more fundamental difference in orientation. But there is also a right-wing connection to what people are familiar with. Sometimes you find something easier just to relate to what they are. Yeah, because that becomes more of a kind of sociological historical accident, I suppose. Of course, there's always this school in Cambridge problem, and so the same thing is about foundations, people are happy to set their own, because it's what they learn. Sorry. No, it's okay, I don't think anything's broken. But also, there's this lady, and all these people who did Tipper D'Algebra in Cambridge, and they've got Tipper D'Algebra in this fundamental way of physics. Yes, but their motivation seems to be, from what I've understood, very different from Basel. Basel obviously regards to this very kind of way of doing physics too, but his motivation seems to be rather different. Yeah, I'd like to know what the difference is coming from... Well, I think it's something to do with this fact that he sees the grasp of an algebra in terms being even more environmental, because it connects up rather nicely with ideas of... Oh, I know what I wanted to try. This is the... Oh, I see. It's usually easier if I do it because I may have to tie in with what was that one or something last week. Like when we had, um... Ah, the rope for pork. The wild boar, sorry, not the pork. I don't know what it tasted like, but he wasn't, he didn't want to eat it. No, okay. It's easier if I can correlate. I think I'll start with a touch here, please. That sounds lovely. And then... Where do we go? Yes, where is it tonight?

37:30 What time do you mean? It's great. We will meet about seven, quarter past seven. Quarter past seven, oh, that's good, yes. It's a nice, nice to have a little bit of a dinner at the lunch place. Oh, the same place, oh, that's nice. Oh, that's nice. Of course, it'll be nice for Vanessa and Max to see that. Yes, because it's such a lovely place. In fact, at that time of the evening, it's really light, so you can walk on the beach maybe. Now, I'm going to try playing this now, but it's a bit quieter. It's sort of better than I talk to each other like this and have done for probably almost as long as, well, not quite as long as you've done, it's been a good 20 years or so. No, I'm trying to adjust the speed. I think that's just because you're writing on the board. It's been on Blackboard Talks. Oh, yes, it's a Blackboard Talk, of course. Like old Talks, all you know. I think we're going to have to wait until they've shut up. We've just got quite a run. that's a little bit more I know what I'm talking about obviously if you can see what you wrote on the board which I did get a good set of notes I could show you those Yes, I know all about that.

40:00 Yes, get on with it, man. Yes, Akihiro Kanamori, K-A-N-A-M-O-R-I, the higher infinite, large cardinals in set theory from their beginnings, Second edition, Springer ISBN 3-540-00384-3, which I think was in the Springer yellow sale. If not, then I would certainly like to get it anyway, especially for the concluding postscript on the continuum and on the history and conceptual background to the theory. Thank you.