Ray Brummelhuis talk

0:00 You notice slightly differently, the way to go is to notice that it is a Turkic operator, so a Turkic operator is an operator with a constant on all the elements, so this is 1 over k minus j, so k minus j is constant, which is when you add on the element, you get the same value, so you recognize that it is a Turkic operator, so it's a good idea to look at this from the point of view of Turkic operators, so this is a Turkic matrix, so Turkic matrix. So we have to make a small regression in complex analysis on the disc of the circle. So we won't work either on R or C, we work on the circle. And in the circle we look at what's called the Hardy space. So you know that the L2 functions on the circle are spanned by all the complex exponentials needed for our IA data. For the Hardy space you only take those with M positive. These are the functions on the circle, which are such that they can take the harmonic, the extension of harmonic functions into the interior, and actually will not be closely related to each other. Okay, but you can do everything on the circle. You don't really have to know, do very much complex analysis in this stage. And you have the orthogonal projection of L2 onto this space, which is just the orthogonal projection of, projection that's insisting to itself and to zero. And then, okay, so now you think a little bit about it. You say that an harmonic oscillator is basically an oscillator which has either values 0, 1, 2, etc., plus or half. So I can all, I can as well represent it on this space, on this space H2 or S1, by just multiplying P1 by F plus one half. And what you get, and this is the same as what you get by differentiating. Except that I forgot the one half of it. Let's leave out the one half, in my slightly, not very, not very practical way, but let's forget about the one half because this is just a concept and it doesn't play any role.

2:30 So basically the harmonic oscillator in this picture is just differentiation, and more exactly it's differentiation followed by projection, and differentiation commutes from there. So, you look at operators on the circle, and there's 12 operators, which are called Turpin's operators, which you get in the following way. You pick an arbitrary function on the circle, you look at the multiplication, you take the function which is an h2, some method of Fourier series on a positive frequency, you multiply it by a, and you project it there. This is called the Turpin's operator. And what you get is basically the Fourier coefficient, where the Fourier coefficient is just the ordinary Fourier coefficient. So the matrix representation of this thing in the natural basis is just connected with the Fourier coefficient. So now once you make this connection, you can sort of look. Then you get the idea that you want to look for the function of the Fourier coefficient, then you get the time operators. There is nothing equal to zero. Zero equals zero. And then if you remember, you already have this pre-analysis. So, the time operator, in this case, is presumably given by the Turkish operator, the symbolism. Which is basically the same thing as we had a few slides back when we just looked at the particle in a circle. It was also there. The only difference is that we have to project that into the positive substrate. And the nice thing is that you can sort of easily verify this idea by hand, and so if I take the commutator and I have to be very careful because there's this projection, this data is not dictated by the projection, but then this differentiation commutes to the projection, it doesn't really matter at all, so you just put it to the other side, and you are back to looking at the commutator of the derivative of that, and you say, well, of course that's a delta.

5:00 Okay, but then you say, you do again the same thing, you say, okay, I'm not going to look at the functions in H2 which are going to be 0 and 5. Now, you need to have a small argument there to show that this is a dense subspace, because this is not really immediately obvious, because I saw you going with boundary values for a more complete function, and this is a slightly more idiot, but you can do this. At least I have an argument which convinced me. So, a side remark. This sort of representation of the harmonic oscillator on the circle is not really that far-fetched if you think of the oscillator in the class. So classically you can either write it in this way, or you can write it in a complex way. We should use that as an idea. This is the main thing about this representation. It's exactly the multiplication in C bar that we interpret as derivative. Or you can do action-angle variables, which would just be action variables over r squared divided by 2, and that would be the angle. So you sort of expect, naively, that in this picture, the quantification of the oscillator would just be d by d theta. Except that you have to be a little bit more careful. It's not really d by d theta. d by d theta is strictly true. Now, you can apply the general theory of circuits, so to just remind you what the circuit story is. So you have T of A on the function psi, and you can draw like that, so what you do is you multiply the function by the psi, and then you take this projection.

7:30 So if you like, then also like if you follow me, the sum of all M positives. So you take A-psi in a product with E-n, that's E-n-theta, but E-n-theta is just the obvious thing, simply power i-n-theta. But these things have been studied, books have been written about it, and they're actually the most interesting when the A is not real value. In the most interesting book, the total A's are the ones on most of the joints. So if A is real value, this is the most of the joints. And then, you know, people mostly get excited on when A is not too valuable. For example, when A is such that if you look at its range, that if you let theta go over the circle, then A describes a certain complex plane, and then when the certain complex plane goes around the point zero, you have something like index theory. And when you take its operators hard, I guess it's something that the basis of, I feel, is the most primitive. But, so like I said, is that the, but if they are real value, they're going to be void. Sorry, but could you just expand a little bit on the connection with the Atiyah-Singer index, then? I don't know about the cohomology. Can I, I'm probably better to tell you afterwards. Okay, okay, sure. I'd be very interested to learn more about that. But if they are real values, a spectrum is just basically the interval from the smaller value of A to the bigger value of A. A is L-infinity, which is really the essential between our A continuance. A can be discontinuous, can be over, jump, all over the place, but the Turkish operator has just a spectrum. And in our case, it is that that we expect them to think of our minds by time, which is where you expect them to come from, or talk about it, where you have this harmonic oscillation and you have this time operation that just goes on.

10:00 You can also, and this is something I wanted to do, but at that time, if you apply the general theory, you can probably also say something about the eigenfunctions in respect to projections. I don't expect to get to them quite explicitly. I don't know whether one will learn something with the mathematics. What can you do? I mean, there's one thing with spectrum with minus pi pi, which is nice, but not a lot. So I'm, again, probably just having something with spectrum with minus pi pi is not enough to say it's good. But what can you do with harmonic oscillators? So, well, if you look in Feynman and Hitze's book about particles and quantum mechanics, at some point they do the harmonic oscillator and they have a lot of chapters in the basics here, but almost every model in physics is a harmonic oscillator, so you get a lot of mileage out of this. So if you take, for example, a pure radiation field in a human body cage, then... So, each of these ATKs, yeah, it's a little bit different. The derivation operator is straight through the Heine space of the circle for differentiation of the two circles. And this sum then should be understood in the usual way to let it act on the K component and not on all of the other ones. Now the time operator going with this means that you get the best amount of the K over here. And all these H-games commute. So all of these operators commute.

12:30 So if you look at the calculation you see that If you want to have a time operator which is conjugate, the sort of the radiation and autonomy, you just have to sum all of them. This will give you a chronologically conjugate time operator. Now, this sum over there is not a direct sum. This is the sum of all the operators which the identity accepts in the case component. And if you look at the spectrum of all these things, you get the spectrum of all these things, but just... Taking the sum of all the spectrums of the individual numbers, so not the union of the sum, but if you take the sum with the 1 over k where k is in z3, so then this sort of diverges, even in dimension 1, so what you end up is the total delineation of the spectrum, which is the idea of unitalical decimals along the boundary line. But then if you look a little bit more deeply, it's sort of inescapable, but I think it assumes multiplicity. Now, if you actually want to see whether you can interpret this operation and something to do with physical planning, you presumably have to, you know, put in some interaction. And this is at this moment the point where we have to stop. So this is where the time to look at is to see whether if you put in, you mentioned the classical part. Well, there are two questions. First of all, I mean, purely mathematically, I might say, okay, I have the conjugate of light to this radiation, but now I'm going to perturb it. Can I still prove the existence of the conjugate of light? So that's one question, but it's a purely mathematical question. You might sort of try to do something in combination. The second thing is that, I mean, from the sense of having an interaction and working and looking at things in, you know, Al-Azharak in...

15:00 For example, look at things in the ideal representation of the time operator which is only associated with the radiation operator and see if you can get something which you can connect time as is usually used in quantum dynamics. That's why I realized that perhaps it's not even necessary to first do this. You might also do this as a classic. What you might have said at the time is a classical phenomenon. Okay, but these things, this is where, this is how far I go. There's a small, there's a small remark over here that, you know, if the Hamiltonian, the harmonic oscillator in the Hamiltonian is d by dt, then, you know, the raising and the lowering of the rate are connected with the shift of the rate. Okay, I'm basically going to stop here. Except there's a parting talk which I had after listening to Bess. Since I came to Thursday, Bess was talking about the synthetic Kierkegaard Algebra, and bitterly complained about the lack of anecdotes in it. At least here I have some representations of the synthetic Kierkegaard Algebra, and there's a lot of them. If you think of just taking this thing over the P on L2 of S1 and Q is equal to theta, well the spectrum of P consists just of the integers and the importance of course on the projection operator up to a particular eigenvalue and the projection operator somehow is the limit of all the... So, I know that you worked basically on this example, this example of the discrete Heisenberg group.

17:30 Well, that has only been a part of it. It has only been a part of it. It was very much trying to convince you not to call it the Heisenberg group. Yeah, I'm sorry. I call it the discrete Heisenberg group. No, you call it the discrete Heisenberg group. So there are really two questions here. I mean, you showed that if you take some sort of a limit, you can be ordinary, so can you take these things, can you get limits in some other way in your presentation? So in your story you have this picture in which you have quite a few many points and a few words that I don't believe you're going to go through here. In this picture it wouldn't be a picture like this because at the other side you would just be looking for a few of these points. So I wondered if there was a nice story that you could go along with this. In the end of your talk, you touched on the harmonic oscillator, when you ask yourself if you want to operate the complex. I just wondered what the sort of situation is with regard to when you're aligning yourself with complex, and you can think of splitting them into real and imaginary parts, but that can't be the whole story because... If you're talking about homomorphic quantities, that's a lot different from talking about the other two or something like that. I'm not quite sure what the story is.

20:00 I mean, there's 500 representations to say there's no quantum. Yes, of course. So, it's not what you get with Twisty. But it's not what you get with Twisty. But it's not what you get with Twisty. I mean, Twisty is not that. It is Twisty. I'm too ignorant to know what the twistor theory is. I think that's what it is. It's the thought when you have a sum. Whereas if you have sums and digits, it's in the twistors you get 2-2. I've probably got to deal with that because I'm a little confused myself. I mean, if you have something different to solve, I mean, if you have something different to solve, it seems to be just a one-word puzzle, which you would get instead of having a positive spectrum, you would just have to exceed it, which means that, well, I mean, it's sort of, you can sort of go on and on, I mean, you know, you have to solve the one-word question, right, which is on how to R. So, then you make this representation of S1, but this representation of S2 you operate with basically I equals B by D theta. So you can, of course, extend this to a bigger representation. So, in some sense, you start with a non-representation, then you put it into a bigger representation. And this would be a representation, usually when you talk about Amar Mosfet, you say that he's a 7th operator, he gets into a level, and then if it's level, you tell everything else. But you don't need to do it. You can go... And then go right down here. And then, I mean, if you do it a little bit, even a little bit more space, suppose I add this constant here, and this is something which interested me for all of you, if you add this constant over here, you can basically do the same thing right. And you continue. So then you can have, so your harmonic oscillator extends to negative values, and you can look at the inverse of it, and...

22:30 The cohomology, I mean, if they do the cohomology from H1, the H1 is a circle. Yeah. The functions on the sphere, I don't know if it's the same thing. Yeah. It's the same thing if you're looking at spherical harmonics. Yeah. The normal harmonics are simply H0s. Yeah. But then if you go down to negative, effectively negative values. Yeah. You get some post-cohomology. I think it's all tied up. It's probably tied up with the K-Singham problem as well, but I'm not sure I know either one of the... Now, very quickly to you, I didn't want to tell this to you at the top, but the thing is if you have a surface operator, right, and now this A is a function from S1 to C, right, and now so if you go from S1 into the plane... So, it may be something, you know, it does something like this, you don't really know what it is, it's a complicated term, but then sort of the basic result is if you look at the index of this operator in the sense of, you know, the direction of the curve, the line of the curve, then this is really the same thing as the index. So if I call this curve Gamma, it's really the same as the complex equation of the square of Gamma in the sense of the number of times it winds around the square of Gamma. And this is sort of a... this is sort of a primitive equation. Yeah, yeah, I see that. And this is why people are more interested in age, which are... in this theory they're more excited by age, which are complex. Any other comments? Okay, well, thanks. We'll catch you after the physics class, right?

25:00 Thank you. Yes, there must be a topology. So what did you say about quantum mechanics? I must say, cool.