Spin Networks & Harmonic Oscillators

0:00 He's coming tomorrow. He's coming tomorrow. I'm going to push him to the end of the case. I will come up to the end of the case. We've got to the end of the case. Also, I was going to call my Ukraine. I was going to call my Ukraine. I'll tell you about it. Yeah, I heard that. So, I'm going to make the one. Okay, so our next speaker is very interesting, spinetworks and harmonic oscillators. Okay, so thank you for the invitation. So I'm going to talk about spinetworks and harmonic oscillators. So I'm going to try to put some harmonic oscillators in the context of the spinetworks theory. And this is work done in collaboration with ETAVALID, and we are from Celenetar. Okay. So first, let me give you quickly what I'm going to talk about. First, I will remind you quickly what are the species' work. Then I'll show you how you can use harmonico theater to describe our presentation of the group and basically then what you can do with those harmonico theaters to describe but mainly that's why not. And then I talked about applications. So applications, important gravity, but hopefully also some applications important for my own story. And basically here, I'm going to just give a general prediction, here it's going to be specific, and here again, it's going to be very general. That's basically some project that one has to do. So I thought that the main nuance was about So I wanted to start that traditional spinet works with very base.

2:30 So, you know that, basically, if you have a state and a manifest field, and then you're turning on the manifest field, and you make a loop in terms of diameter, you can generate space. Okay, so how you people are doing that, basically, you have a bar meter space, and on this bar meter space you have a gate bundle, which is a booking one. And what you do is you turn around, you make a loop, you make a loop in this space, and we propose because of the gate structure, so you do a loop, when you come back, you're not coming exactly at the same point in the fiber, but you have a group element here which basically dictates the very phase. So, once again, let this phase go to G, psi, this E, I, phi, psi, and phi. You know that phi is total phase, and you can be decomposed in the dynamical phase. That's something which is very topological. Okay, so that's That's the very case. And so what's important here is that you have a loop, you're making a loop, and then this loop is basically, you can see, decorated by this group element. Okay, so, and in mathematics, we call that some hologram. And you have also this picture, for example, if you consider the Marmaran's effect. So you take a state, you split it and then you put the magnetic field in the middle, then you will combine it. And then you take the magnetic field of phase as well. And you can write that this phase given basically by... Again, you have a gauge render, and so you have A in the connection, in U1 again, because you are considering the magnetic field.

5:00 And gamma is a loop here, that's this loop. And basically you know that with a stop theorem you can, you can, there's a flux of the minus-3 in the space, and I can make this loop over there. Okay, so what's, again, what's important is that you can, you can consider this loop here as divided by this group element here. Okay, so again, that's an example of the loop. And then you could say, let's make this Bomber-Arnold effect a bit more complicated. So let's do a beam splitter again here, and then combine it, and you put a score minus 3. And then you combine it here, and then again. I mean, you put many beam splitters, and then you put minus 3. So basically, now you're going to have, for each edge, you're going to have a group element, here still in the one, and this group element, you're going to see that you're going to see that you're going to see here. And so it's a complicated graph. So I turn around from the root, we can go to a complicated graph. So that's basically generating a spinetwork. A spinetwork is a complicated graph decorated by group elements, or you can say also that this is a graph decorated by a representation of some groups. So it's a graph that requires it by anticipation, by ERAP, so it's a group G. Here I'm taking a G, the group G to be compact. And to be more specific, you can see that the line is, so it's going to carry off an annotation, And you can see that this line, if you put an index here, another one here, basically here, this line really corresponds to the J organization, the group element G, and the end of the matrix element of this line. Okay? And now, so, now that you have the edges, you can talk about the vertices. And basically the vertices, so here I have the pro-experient vertex. And so basically the vertex encodes how the representations are combined together.

7:30 Okay, so now if I take t to the essential proof, it's rather the vertical coefficient. And so the spirit works are being produced by Reynolds some time ago. And I think what we had in mind was to describe particles which are colliding. So basically you see the difference between binding and so on. All right. So in the context of quantum information, all the means if you make it non-apillion, instead of U1, you could consider U1 or something like that. And basically the idea is that by taking a more complicated loop, or by taking a spinetograph, you can also encode some kind of algorithm. And I think the book of math is going to talk about that some days. You can really see a spinetograph as a generalization of a quantum spectrum. Okay? So, in a graph in the quantum information theory, it's basically a quantum interpretation. And what is important is that here, if you have basically a graph, a complicated graph, it's really a process in time. It could be a process in time. So, it starts from here, and you're doing your operation, which is calculated by this spinet rock. And then at the end you get a return. Okay? Now spinet rocks, you can see spinet rocks as, in the context of quantum gravity, as quantum 3D space. Alright? So you, that's very nice that you see the same object, the same mathematical object, to appear in both areas. So a spinetor can be a source of a 3D quantum 3D space. And why is that a quantum 3D space? It's because you can see that, I'm not going to put you through quantum gravity and so on, But you can see that if you consider the area operator, the area operator acts basically on the edges of the spin network, and so the edge is divided by our presentation, you can see that the area operator acting on an edge is giving you basically the quantized area.

10:00 So here, I choose S squared is G. So the area, the value of the area depends on the decoration of the edge. Okay, so you can see basically the area, you are, you are, you are ready. And then, so the edges are denizing area, and then the volume, the volume operator is given by the enterpriners, okay, so the vertices, And they are also discrete, so the spectrum of the co-battle is very complicated to calculate. I want to be very interesting. All right. And also, yeah, also an important remark is that here, the spinetwork is not a process in time, it's really something which is partial. So there's a clear difference, a physical difference between this case and this case. Okay, so let's talk now about the harmonic oscillators. So now, let's consider two harmonic oscillators. And so they are quantum, so I consider visualization and annihilation of the oscillators. And the two oscillators are commuting together, so we have all the commutations of this type, and of course we have the regular relation of the scan in this type. Now I would like to quickly show you that by using those two kinds you can generate And the key point is to say that the energy of these two harmonic serial dots is going to be in my lab. If you fix it, then it gives you the representation j. So basically here, t is going to be equal to j, where j is my representation of S u2. And I'm going to define now the other harmonic serial dots of the S u2 symmetry. it will be one half, or here, I forgot about one half, okay, then you have a minus c, then

12:30 you have a minus c, then you have j plus, which means a minus b, and then you can see a j minus, which means you have the ambition to get, and then you can check out easily, But then that you have the usual computation relation of f of f2, which are those ones. j minus j turns, j minus, j equal to j. And then e commutes, the pattern e commutes with the f of f. Or is it j? of J. And now if you consider the Katimir of SU2, it is given by T times equal to 1. Okay? So you can see that really by using those two homincosinators, I'm generating our presentation of SU2. Now, so for the speed network, when you add a line with a geo representation, you can, instead of having one line, one edge, you can alternate it by two edges, but with We have one for each harmonic oscillator, the A and B, and you can see that it's like the left and right, there is something high here. So the key point is that instead of having just one edge, we have some kind of internal structure. So now I'll talk about the edges, what about the antithriners? So, for the end of Reiner, we would like to describe them using those harmonic oscillators. So, you know that by definition, the harmonic oscillators, the end of Reiner, And then also, if you consider general and binary, you can consider the spheres. Okay? So this sphere is really light of patches, of patches which are, each one of those are

15:00 reactiven by the light of presentation, the edge of the edge, okay? And then there's a twiner that encodes how you control the volume, basically by processing the angle between the transpires, okay? So if you consider this sphere, you can consider it as a punctual sphere, okay? So we'd like to construct the symmetry, which is clear, such that it's globally as a three-liner, but also that you have, you keep the area fixed. So here there are area, the total areas, just, you can interpret it as a total energy of all the harmonicos cathars. You have two harmonicos cathars for each edge, and basically it's just in some of the spaces. So you would like to construct the symmetry for the rule of two constraints. And when you do that, you can see that the symmetry is implemented by the following operatons. So basically the operatons which is just acting on the edge and doing just acting on the edge And then we have the other parcels, which are exchanging points of area. So here you can put some points of area here, and put it there, and so on. So this, this operator here is a bike. And that goes to the, the symmetry. This is B. Okay, thank you, thank you. And, and the other one, Okay, so here this guy is just exchanging quanta of area, and this guy is just exchanging from quanta of the normal of the of the area here and so now you just uh you can check now

17:30 you can calculate the commutators for these elements and by calculating the commutators you see that it's uh it's nice to get to redefine aij which is really a and then you can take a legevah with this side and then and then And so you have to put the air machine, and with the air machine, you can do some of this two-state . So by equating the number of, by equating the number of battles you've got, you see that Those guys are mission, this guy as well. And here you have n, so here I start with n function, or n base. And here, because they are mission, you can do the solve, you can get the base. And you see that you have n plus 2 n, 2 because you have e and s times n minus 2 divided by 2. And that's 2 n squared. And you shouldn't forget that there is an extra condition, which is that the total area is fixed. So basically there is one less e-high, for that degree of freedom. So what you get is really that you've got n-1 generators. Okay, so here what you organize is basically that you're having the real graph of n-s. Okay, so that's the symmetry of minus a point. And this shouldn't be surprising, because the punctual sphere is really, you know basically from other studies that here what you're doing is some approximation of the differons. So there are many theories which are telling you that you can approximate the differon

20:00 this one's sphere by such a group of 1.0. So you know that when n is going to infinity, you're going to recover n to n, the piece of the sphere. And then you can do one extra test, which is what's the algebra of oxalabas generated by those So the degrees of freedom are really living in the U.N. or S.U.N. algebra, and basically which this means that the algebra of our servers are stated to the symmetry is just M of C. Okay? But then, you know that there is this non-committistic geometry approach, you know that your space can be a particular space when described as the algebra function. And you know that the algebra of matrices like that can be considered as I have to see mentioned on the T-sphere. Okay? So here, you will separate this algebra, this is algebra, this is So here, what we are doing is making a link between five minutes left. So five minutes left. So here we are describing the antipoiner as a philisphere. And what's important is that also you can, okay, that's okay. Well, we have until 4.30, so if you want, it depends whether you want questions or not. Okay, I just want to move back. So here, I'm making the link between the antipoiner and the philisphere. So that's the precise thing between the speed networks and the frequency of the obituary, which I think is an important step to make, especially the good x-prontumal words. So now, quickly, very quickly, the applications. So the applications are... Some people sometimes do SO3, you could have gone back. Sorry? Sometimes they do SO3.

22:30 Okay. So then you can, of course, in cotton gravity, you have many issues, many disabilities. Here, in general, what you consider is S2. So here I'm considering S2. Okay. So the application. So the application, now, many, many applications, and because I don't have much time, it's very quick. So in quantum gravity, it's very nice because now you can describe the terpoinor very easily with this human algebra. And in particular, if you consider the terpoinor, which has vortex, you can see the terpoinor So, by applying the current state procedure, which is given by the familiar, you can construct a current state for the diagram, which is very important. Now the construction should be cut for us. And basically, in this way, one would sort of one of the programs of quantum gravity, which is . So what's interesting is that you have the . So that's one of the applications of quantum gravity. So you can see there are some things about the dynamics. But now you can describe the dynamics. Now it comes from a matrix theory, and then you've got the micromatron theory. You have a quantum circuit, usually you use Qubit to work on those. Now you could introduce a quantum . Let's see what happens. Also, I'll just move . Then there is also a link with . I mean, what I did here is that you can read that the bomb observable, which are stable, again, as you can provide a lot, and that's make the link with the content-free subspace, something that we didn't talk about. And finally, also a lot to construct quantum-reform, and here we have this sphere, which can be interpreted as a prospect in finding a .

25:00 So we'll maybe just give a one minute break and then we'll have Bill's talk. I don't know, even. I don't know. I don't know, even. I don't know. I don't know. I want the window open. Okay, I'll keep the window open. Do you want to sit on the side? John, is this okay? This is fine. Okay, to remove the table, they have to clear the whole room. I'm going to like this in here, so I can get out here.