Moyal Transformation, Quantum Potential & Algebraic Approach to Noncommutativity

0:00 In the Heisenberg, the Schrodinger, the Heisenberg, the Heisenberg. All I have ever seen in the contractions reduced to one is the Schrodinger. The question is whether the hell is it if I only use the operator? Now, I found that it was very difficult to deal with this. I've struggled with rising waves. I've had all these phases in motion. I've got absolutely no idea. What seemed to be missing was an equation for a phasor. And you can see why the phase is missing, because if you use the pure state, then the phase also turns out. So there's got to be the phase lurking about somewhere. And then, optimal sphere. I've been playing about with typodologians. That's the right thing. And those will work. Perfectly, that's what he was doing with the spheres and the crystals. And of course all that comes out of Clifford Algebras. And what I spotted was that the Dirac wave function could actually be put into the algebra so that we didn't have to use Hilbert's face to discuss the wave effect. We could do it entirely within that. Now in order to do that, we have to be aware that our mathematical objects call left ideals and right ideals. I'm sorry Mark, they're very simple. A left idea is every time you multiply from the left it stays in that subspace of the right. Every time you multiply this from the right it stays in the subspace of the two subspaces. And you can think, this is a left ideal and this is a right ideal, but how can we do that and make sure it's right? Well, I found in Dirac's book, 3rd edition and 4th edition only, not editions 1 and 2. I've had people say, no, it's an African-American, and you look at it, it's edition long, it's edition two as well, edition three as well. Standard care. How many people have registered?

2:30 Standard care is not that. It's got no bar on it. Okay, except when I was in Switzerland last week, the week before that, David Spicer did a lot of work on Paternium, Kevin Alvarez, etc. with Yad and David Finkelstein in the early 70s. And he said, Dirac's a hero of life. I said, look, do you know what a standard cat is? And he said, I don't know what it is, but I know it's in Dirac's book, and I'll never puzzle out what it is. So I said, well, I'll tell you. It's actually what you need to complete the Heisenberg picture. It's a primitive hidden moment. It looks as if it's just symbolism. What Dirac originally put it in there for was because he didn't want anything operating on this side, so he put this thing to stop it operating. There was also a standard Brahma, and these two things come together actually play the exact role of the idempotent that I've been looking for. I eventually found that if I took the discrete vial algebra and went to the limit as n goes to infinity, I had an idempotent in there and it was this idempotent. And if anybody likes to know what that idempotent is, it's simply the delta function in the algebra. Okay, so once I've done that, we're then in business now, because we can now look at the wave function still in its operator form. We don't have to project it into a representation. This wave function is now, all these hats are now operators. So I've essentially got my left ideal, this one, and my right ideal. This one, the epsilon is just a thing, it's just people in the middle. and these things are actually symplectic spinners and it seems as if I've found another route into symplectic spinners symplectic spinners, these are for the guys who don't know anything about geometrical quantisation

5:00 does it? is that the same one? I'm sorry, I don't have it on my screen, so I'm going to explain why. Come on, boy. That's it. Yeah, so we've got, if you know geometrical organization, then there are these terrible mathematical objects in there which I love very much because this is what we've got. but they turn out to be just these left and right ideals. And then, in the operator Schrodinger equation, there is then you've got to take both of them because the left ideal and the right ideal might be different options in this way. Normally we say side and side-star and side-star and side-star and side-star and side-star and side-star equations. You can't do that. You need both of those equations. Okay, if you do both of those equations, sum them, and you simply get the Libby equation. Because that's just a derivation, and therefore putting it back using rows equal to that to the Libby equation. And that's the conservation probability. But now subtract the two. If you subtract them, you've got the anti-comitant there, and then you've got some rather odd looking thing here, which most people give up. Now if you polar decompose these wave operators, then you find that you actually get an equation which is the phase equation. And remember this is the phase in the sense of a polar decomposition of a matrix. So this is not phase just as in the ordinary phase in one account, it's actually a matrix. It becomes a phase that you represent, becomes the normal phase that you guys are used to when you're projecting into the wave of space. This equation is essentially a conservation of algebra. And you'll notice, there's no quite potential there. I've got to produce the quite potential from this equation. I do that not by, see what I mean, I'm in the algebra, I'm in a representation free structure. Now I want to find out a representation.

7:30 So I'm going to project from this algebra into the representation space. And when I do that, I've done it here for a general representation. I get these two equations. Now this is the phase. If you construct a Y function out of this, this is going to be the phase of the equation. Now, let's take an example. If you use the oscillate, everything I do is a sub-adjective space. You get the consummation probability, And ta-daaa! Outbox. This is the classical potential, this is the kinetic energy. In other words, it comes from this extra equation which has to be used. Now you can do the same thing with mental representation, because another thing that Heisenberg criticised in the Bowen approach was, it's one sided, it's only in the X representation. If it's not, you've got a D-representation, then you get a permanent attention. and it's horrendous trying to work with P-representations but you're often to be trying it and bring up to the cubic term and put it in touch with you so you've got to do the momentum representation now this kills off Shelley Goldstein and his mechanics of everybody's stuff the P-relation calls the guidance condition that seems to have derived it from first principles which I don't believe at all, here we have people with Bohmian mechanics fighting with each other, I'm sorry, but it doesn't work well. If you go to the momentum, he was very angry with the songs, when you go to the momentum representation, you have x, so now you're constructing a phase basis on the next one, in which p is the eigenvalue, but you have to construct an x, and the x is constructed by that relationship there in the x representation that is a guidance condition what the hell is that a guidance condition nothing is guiding anything as you know it's the if you're in a p representation one is operating with x

10:00 I, it's one second. Where do you get the trajectory's from, because you can't get the trajectory from here, because the trajectories are assumed to come from the guidance condition. They don't come from this, they come from this tree. And if you look at what the current is space, you then find it's just ds by the x, so that's it, you can use that to find the projections. They're really the streamlines, or they make the assumption that the particles are most likely to follow the streamlines. Okay, so, now then, this is where it becomes... As I said earlier, this really, it's not going to have developed of the work in non-conjugate geometry by Alan Conn and Mike Peller and a few others you find that you want to put the main emphasis on the algebra. Why do you want to do that? Because that is the thing which is the equations of motion essentially, if you go to the Heisenberg picture, the equations of motion are really in the algebra. It's the algebraic elements, the operators that are divided at the time, not the eigenvalues. The eigenvalues are sort of a fold around the floor. So you really should take the non-commutative structure of the algebra as basic. Now what Galfand did in the 40s was to say that, suppose I forget about the non-commutative, just look at commutative geometry. What Gavafang showed in the 40s was that if you start with a commutative algebra, from the relations in the algebra you can deduce or you can encode the properties of the underlying manifold. You can fold into logical properties, metric properties, differential properties in the algebra functions. Normally we do it the other way round. We start with a manifold with a topology and then build up the algebra functions like that. But what we are showing is we do it the other way round. Now one idea, the idea of non-committative geometry is to start with a non-committative structure and deduce what the face space is.

12:30 And the only trouble with that, there is no being in the face space, and that's essentially The only thing you can do is make, they call it shadow magnitudes. The only thing you can do is make shadow magnitudes. And what you're doing when you go to an X representation or a P representation is you're constructing shadow magnitudes. So the face spaces that have been used in the Bohm theory in particular, I'm not sure I should except the Big Mac Theory has this possibility in it, is that you're actually forming shallow metaphors and therefore you have to take that into account when you're trying to understand what the theory is. Now this of course is really opening up, let me see, I don't know, or you can do the Dirac equation in this part of the world, which I don't want to tell you before. And then really where I was skating off to now I'm going to the clouds. Is that there is a basic problem of quantising gravity. Not every problem of greed. At least I'm still greed. Can we quantise gravity? So I hope you realise the reason why I went into the Bowman interpretation in the first place was The reason why I did it was because I have geodesics in general relativity and I have trajectories in the Balmer theory, at least as if they might be brought together. I failed of course. But this other idea is to really be much more general. How am I bringing in Einstein to help me in this? I've always loved to have a big name. Perhaps the sense of Heisenberg's method points to a purely algebraic description of nature. That is, the elimination of continuous functions from physics. Then, however, we must give up, in principle, space-type continuum. And then you've got Wheeler's beautiful. He was a poet, John Wheeler. Not day one geometry, day two quantum physics, but day one the quantum principle, day two geometry.

15:00 In other words, take the algebra as you place it, the non-computed algebra as you place the expression, and then project out. Ok, alright, our conclusion. class. Looking at face-to-face accounts of quantum processes, you've got the two approaches. This gives you cell structure, gives you mean values of the cells, but this probability distribution, it's not, it's a quantum probability distribution, but notice everything's exact, exactly the same with Bohm, you have a point structure, x and p are either mean values or actual values and you've got the frontal potential and we see what the frontal potential is, it's actually the result of projecting something which doesn't really go into the right space. So it's a bit like general relativity where if you project the geodesic into a flat space time it looks as if there's a force prison. And therefore this potential is going to be a force prison, forcing the end of our presentation. I've got at least three questions, I'll stick with one. Why is it better to work with Because the essence of the bones thing is in the quantum potential. That's what I found in my experience. Just working with the bones condition gives you all the... it's equivalent. But I feel I get much more intuitive feeling of what's going on if I complete the picture by using the quantum potential. Right, so there's no sort of fundamental difference. Fundamental, there's no mathematical difference. There's a fundamental conceptual difference. It's just that I prefer to do it in both ways. I'm a great advocate of doing it in many ways as you possibly can to get a better understanding of what's going on. I can never understand people saying forget this, forget that, forget this. Particularly when you get the same results whichever way you do it. Okay, so you want to pick the way which gives you what you're looking for, and not be frightened of it, because when you see what you've got, you can then go back and then say, alright, now let me go back to the standard way of doing it, oh, okay.

17:30 In other words, I'm not, for God's sake, I am not saying that we must all learn Bones theory and forget about everything else. That's not what I'm saying. I'm saying don't just say I'll forget it. If it helps, use it. If it doesn't, forget it. Sorry, I'm so fed up with this thing. I don't let that inhibit you from asking the question. Jeff, one of the features of this modern potential, as I see it, is that it, in a sense, is self-consistent. Now, is it any relation for being some representation of self-interruption? I don't look at it that way. I mean, I look at it, to me now I'm looking at it as if because it only appears when you project into the wrong structure. In other words, I feel it's much more important to work in the Heisenbergs, because that's where the essence is. I would almost daringly say that quantum physics doesn't work in space and time. It's beyond space and time. I'm really going out on a limb here, but I just want to spin it when you know what's happening. I'm not the only way it feels that. I mean, people are crescent. But then you have to have all this conservation problems. Because that follows the beautiful space. No, but it doesn't follow from the sky, it follows what I've got in my algebra, my algebraic relation, that's anti-commutation equation, that's the conservation of energy. Conservation of energy is the uniformity of space, it's one and the same thing. Not in quantum mechanics, so you've got to be careful. It is in classical physics, I can't, come on, esoteric. Yes, classical physics, yes. in quantum mechanics if you look at it in one picture where the energy is not defined otherwise you know precisely the timing of it what happens to the conservation of energy? this is what I say you have to abolish if you abolish space you have to abolish

20:00 you have to abolish you've got it encapsulated cognitive structure. That was the whole point about Gelfand. But if the space-time properties are embedded in your, but in such a way that when you project them into a X representation, you get this apparent force, the apparent competence. It's because essentially, you're not, it's so obvious, you're not essentially using the momentum that you could calculate using way functions. You're only using the real part. and then you square it up to get the kinetic energy if you're only using the real part of it you're missing something and that missing something is the solar energy one last question at the moment your view of reality is an algorithm you try to go beyond that not really dealing with that? My view of reality is not naive classical physics would ever walk around and contain the core space and stuff. But do you feel that the ultimate reality can only ever be an algebra or ever found? I would like to say that it would be a, it would be ultimately, now you're really asking to bear myself, ultimately it would be in terms of a process. Now you're going to ask and what do we want to process? That's another hour's lecture to even get anyone you're convincing you that this might be the right way. Let me try and do it this way. Yeah, let me discuss the processes. You need objects. No, you make objects out of processes. They're the invariant features. Look, if you take a simple, naive example of the Heracliton idea, There they thought vortices would be the objects that you, that would be the objects of your space. But I want to use another, what is quantum mechanics talking about? Is it really talking about properties? Remember this experiment that I cooked up from, I think I got it from VAR. Suppose you want to collect together, you're living in a quantum world. And the only way you can see things in the quantum world is to have spectacles which look at colours, or another pair of spectacles which look at shape.

22:30 You can't see one pair of spectacles that looks at colours and shapes at the same time. So now I want to play cricket, so I want to get, I've got two shapes, spheres and cubes, and I've got two colours, red and blue. I want to get a red ball, a red sphere. So first of all, I put on my colored glasses and I collect all my red objects. I put them here, all my blue objects, whatever, the other colored objects. Then I put on my shape glasses. And I say, here's my red collection, so let me pick out of here the shape spheres. And then you'll come along to me and tap me on the shoulder and say, and I'll go and put one of my other glasses so I can find half of them have turned different colour and I don't know how you look at that as being a property of an object And that's not one of the councils. I don't like that. I'd like if somebody could take that. Is that possible? I'm sure there are many more questions. Thanks very much, Basil. Thank you.