Theories of Everything with Curt Jaimungal - Quantum Physics Missing Link Discovered... [Geometric Quantization]

Episode Date: January 26, 2025

The classical and quantum worlds are not as apart as we thought. Eva Miranda, a renowned researcher in symplectic and Poisson geometry, explains how “hidden” geometric structures can unite classical and quantum frameworks. Eva dives into integrable systems, Bohr–Sommerfeld leaves, and the art of geometric quantization, revealing a promising path to bridging longstanding gaps in theoretical physics. As a listener of TOE you can get a special 20% off discount to The Economist and all it has to offer! Visit https://www.economist.com/toe Links Mentioned: •⁠ ⁠Eva Miranda’s website: https://web.mat.upc.edu/eva.miranda/nova/ •⁠ ⁠Roger Penrose on TOE: https://www.youtube.com/watch?v=sGm505TFMbU •⁠ ⁠Curt’s post on LinkedIn: https://www.linkedin.com/feed/update/urn:li:activity:7284265597671034880/ Timestamps: 00:00 – Introduction 06:12 – Classical vs. Quantum Mechanics 15:32 – Poisson Brackets & Symplectic Forms 24:14 – Integrable Systems 32:01 – Dirac’s Dream & No‐Go Results 39:04 – Action‐Angle Coordinates 47:05 – Toric Manifolds & Polytopes 54:55 – Geometric Quantization Basics 1:03:46 – Bohr–Sommerfeld Leaves 1:12:03 – Handling Singularities 1:20:23 – Poisson Manifolds Beyond Symplectic 1:28:50 – Turing Completeness & Fluid Mechanics Tie‐In 1:35:06 – Topological QFT Overview 1:45:53 – Open Questions in Quantization 1:53:20 – Conclusion Join My New Substack (Personal Writings): https://curtjaimungal.substack.com Listen on Spotify: https://tinyurl.com/SpotifyTOE Become a YouTube Member (Early Access Videos): https://www.youtube.com/channel/UCdWIQh9DGG6uhJk8eyIFl1w/join Support TOE on Patreon: https://patreon.com/curtjaimungal Twitter: https://twitter.com/TOEwithCurt Discord Invite: https://discord.com/invite/kBcnfNVwqs #science #physics #theoreticalphysics Learn more about your ad choices. Visit megaphone.fm/adchoices

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Sign up for your $1 per month trial period at Shopify.com slash theories all lowercase go to Shopify.com slash theories to upgrade your selling today that's Shopify.com slash theories so you're unveiling something for the first time the audience is in for a huge treat I had a chance to preview it fortunately and I'm so excited to go through this or for you to go through it. Thank you. Firstly, I think it would be great to talk about what quantization is as most people know about the path integral quantization or canonical quantization. So what is quantization? Why is it important? And how did you even get into the field of Poisson geometry initially? Yeah, let's talk about that.

Let's think about the world as we see it. This would be classical mechanics. The world that is following Newton's law. The force is related to the acceleration. This is the world of classical mechanics. We are used to the movement of trajectories of celestial bodies following this pattern. There is a step forward from, you asked me about Poisson, to go from Newton to Hamiltonian dynamics, which is more or less a change of coordinates. And then we can formulate the equations of movement of particles in something called

a cotangent bundle. This sounds very mysterious, but essentially it's formed by pairs of position and momentum. And then the principle that guides the movement of the particles is the conservation of energy. And we think of the energy of the Hamiltonian of our system. And then our system just follows these two equations here, which are Hamilton's equations. This is just a system of differential equations and as the movement of the particle evolves, it follows these equations, these simple equations here. But there was a surprise long time ago, there was an experiment, the double slit experiment that showed that not only light but also electrons have this wave-particle duality. Right? The experiment showed electrons through a double slit and while we could expect,

if they were particles, that we would see this double slit again projected on the second screen, there was an effect of interference pattern. So we see on the screen, we see a pattern that corresponds to wave. So this was quite a surprise. So maybe everything, every equation we had been using was not totally correct. And while here we have Niels Bohr lecturing about quantum mechanics on Iowa, precisely showing, explaining this experiment. So we are at the beginning of a new area. We are jumping from the classical to the quantum reality. And, but this quantum reality looks like something very modern, almost science fiction,

but we could think it's quite old. It all started in the past century with Planck, who introduced the concept of energy quanta to explain black body radiation, right? And he already proposed that energy could be emitted in discrete units. And this idea of having energy in discrete packages or units was around also, of course, in the theories of Einstein, of Bohr, who developed the atomic model with quantized electron orbits, de Broglie, who proposed the wave-particle duality for matter, and Heisenberg, Schrodinger, and so many people. We can see many of these people in this congress in Solvay in 1927. We can see most of the main characters of this revolution, which is nowadays,

it looks a bit old, right? And here, but still we maybe don't understand completely, right? We see the Schoeninger cat and well, the cat is here or is not here. What's the mystery? So we can make some jokes about it, about your card. I have good and bad news. So indeed, maybe the truth is that we are a bit in both worlds, right? Classical and quantum. Already this was observed by Feynman. Nature is unclassical, and if you want to make a simulation of nature, you'd better make it quantum mechanical. It's a wonderful problem because it doesn't look so easy and here we are still trying to understand it. So there is a lot of, at the beginning, there was a lot of discussions about classical versus quantum world, but maybe we have to think in a different way. It's like not about opposing two different worlds, but trying to understand

nature with both phenomena at the same time, the wave particle phenomena. So here we have this picture where we have on one side the the solar system right celestial mechanics and on the other side the subatomic particles. So it wants to to illustrate this classical and quantum world. So maybe they are not so apart? I mean feel free to ask me. Yeah I have a quick question. Yeah. Most of the time we talk about quantum mechanics as being more fundamental or all of the time that the classical world emerges from the quantum. However, when speaking about the standard model, we start with the classical

Lagrangian and then we quantize it. So we start with something classical and then we apply a procedure. Now there's a criticism that that's going about it backward. What we should be doing is starting from something quantum and you recover the classical. So I want to know how it is you respond to that. How do you think about that? Well this is a very good idea and very good question and we don't need to take a decision about what is better because we can do both. Indeed, what you suggest is a very good idea that has been implemented.

You could first apply the quantization procedure, something I still didn't explain, but I will, and then apply it again somehow. And then somehow you can start from classical to quantum and then recover the classical. So you can do the quantization. I'm not going to talk about this. I mean, I didn't prepare the slides about this. But this has been done. So the problem is that you don't always

recover what you started from. Right. So it's not one-to-one. Yeah, and the reason why is that I could say quantization, the process is very capricious. It's almost an art. There is not a unique way to do it. You have to make choices.

And depending on your system, this may or may not work. It's a kind of art, I could say. So what you are suggesting is a very good idea. I think we should approach this towards the end because I still didn't explain the first quantization, but what you are suggesting is something that people have been trying to do. And in some cases, it works very well. In some cases, it doesn't work because in some cases, works very well. In some cases it doesn't work because in some cases not even the first quantization works. So here is a picture of what we want to think about. Right now I'm just trying to explain if our reality is classical or quantum. I want to

just first answer this question without explaining how to go from one to the other. So here we have like two pictures of a girl walking up on one side, we have a ramp, on the other side we have some stairs, and we could think of this as an illustration of reality being at the same time classical and quantum. These stairs are a little bit the symbol of the packages of energy that we see in the quantum world. So in a way we shouldn't think about, well of course if you have to go up you have to decide if you take the ramp or the stairs, but maybe we could take both the ramp and the stairs. Indeed, let's

think of this as a picture, as an art. Let's think as I was talking about art. Let's look at some pictures, right? Here we have some pictures of impressionists, right? And here we have, if you come to Barcelona, you should come to Barcelona. Please come to Barcelona. You'll see this wonderful, have you been here? Not yet, but at some point I'll be coming to the University of Barcelona. You should. So if you come to Barcelona and you have the opportunity to see Gaudí art, you'll see that it's formed by little mosaics, but little pieces of art. But from far away, you don't see that this is formed by little pieces of mosaic.

So this could be the classical and quantum world at the same time. In a way, we could be running up the stairs at the same time as we are running up the ramp. So we are combining classical and quantum world in a way. But this is like a modern approach. This is more like, well, not so modern because Einstein and Feynman were also pointing to this coexistence of both realities on the same. But if we are like sometimes to make difference between systems. If we think about classical systems, we are thinking about Hamilton's equations.

And on the other side, if we're thinking about quantum systems, then the equations are Schrodinger equations, where here we have the symbol of the wave, right? So the evolution, again, both of them, there are many patterns in common. There is an evolution, but it's an evolution of not trajectory of a particle, but probability of trajectory of a wave. Yes.

And we could, and well, this is extracted from somewhere on the internet, we could indeed summarize. We could take, for instance, as you were suggesting, let's take a classical system, the harmonic oscillator. And we could just replace the piece by the partial derivative of x and the x by themselves. And this gives me the quantum version of this harmonic oscillator. Okay? And this idea is a little bit the idea that you were suggesting of going from classical to quantum, but we need to understand what we are getting. In a way, what do we need to understand?

On one side we have classical systems with observables. Observables are the observables of the classical systems. The energy of your system, etc. This could be potential energy, whatever. And these are functions. This Cnfinity M is just functions on, I would say, a manifold. But if you don't want to think about a manifold, think about the Euclidean space. These are just functions. And there, when I have classical systems, the dynamics is governed by a bracket, which is the Poisson bracket, which I will talk about. And now I'm just presenting the ingredients and then I will go deep into them.

On the other side, we have quantum systems. Instead of functions, these observables, we have operators on a Hilbert space. This is what we want. And instead of the bracket, we have the commutator of these operators. That satisfies this formula here. You can see it. And well, here you have another, you see a mathematician, right, struggling to see how to go from one side to the other. Quantization is the art, what is the definition of quantization? Quantization is the art of

crossing the bridge from classical to quantum in this picture. So assigning from classical system, a quantum system to functions, we need to associate operators of a Hilbert space and to the Poisson bracket, we need to associate a commutator. And we think that if we, if we go back to Dirac at the beginning, people thought at the beginning, people thought this was pretty simple because we know what we want, but it's difficult to get it. So we are now trying to see how to go from one side to the other side. It looks simple, but the truth is that this is the joke from Shapiro. I still don't understand quantum theory. And then this reminds me of this famous...

Okay, so yeah, so this is the typical sentence we all heard about that nobody understands quantum mechanics that Feynman says. And he said that in a class apparently, right? Where he's mentioning that, okay, relativity was very complicated, but at least there were more than 12 people could understand it, but quantum mechanics, maybe nobody can understand. Do you believe that to be the case? What is the definition of understanding something? Yeah, that's indeed, that's the question. That's the question. That's very good. Well, I think right now people understand quantum mechanics,

but quantization is a different issue. So you understand both sides of the river. You have your classical system. This is well understood. Well, I mean, still many things to be solved on the classical side because you want to really solve something, you need to integrate it. And this is sometimes difficult for systems such as the three-body problem. The three-body problem, you cannot, it's not integrable. And this is still something that people are proving nowadays. So you think, oh, this is very, very intuitive, right? On

the other side, like quantum theory, I could say it's pretty well developed, right? However, understanding, like trying to put an arrow that connects classical systems to quantum systems, it's maybe the problem. Like, how do we understand this error? If we are mathematicians, we are obsessed with this precision, right? And we want to have a definition that works in each and every case. And this doesn't happen with quantization. As you know, on Theories of Everything, we delve into some of the most reality-spiraling concepts from theoretical physics and consciousness

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in and now back to our explorations of the mysteries of the universe. And this doesn't happen with quantization. So the bridge is not understood or a tunnel or what have you. The bridge is not understood. The bridge is an art. And got it. Exactly. So you want to jump.

Yeah. Now people don't know, but this is a great primer for our next conversation because you spoke about rivers and classical mechanics and what we don't understand so next time just as a preview we're going to speak about Navier-Stokes equation and computability theory. Exactly many things we don't understand there still many many things and there are prices in the way I mean there's money you can earn if you can solve some of these riddles. For Navier's talks, there is, I would say there is even money on the question. Just as an aside now, is that one of the reasons, like what is it that motivated you to study

fluid mechanics and Turing machines or Turing computability? Yeah, it's not money. That's not the reason. No, indeed, I am a geometer, as you will notice today in the presentation. I come from geometry. I'm a mathematician. Mostly I'm interested in the shape of things and the several reincarnations of what we understand by the shape of things. things, right? And indeed this took me to a moment, like we were working with some geometrical objects, okay, to which we were able to associate a solution of the Euler equation, which is an equation of fluids. And then how I came up to this question on Turing machines, it was because I saw on Twitter

a question of Terry Tao. This was not Terry Tao asking this on Twitter, but he was asking this on his blog and somebody was referring to it on Twitter. And then I saw it on Twitter that he was asking about if whether the solutions of the Euler equations could be Turing complete. That was his answer. And then this is why we started to work on this because we were applying some techniques from geometry to fluid dynamics.

Again, doing a bridge. I like bridges, apparently. So I was trying to apply some kind of games, geometrical games, to say things. So I was applying games which were like movement of a trajectory to say things about movement of a trajectory inside the water. So the difference, let's say in a way today we saw these Hamilton's equations and these equations are differential equations. This means that they are in, so you have the evolution of several variables with time, but then you can have a partial differential equation in which you have evolution, in which you have several variables evolving altogether, in which you have involved also the partial derivatives. So it's quite a mess, right? Partial differential

equations, it's much more complicated than differential equations. And therefore, if you can say something about these partial differential equations. And therefore, if you can say something about these partial differential equations, using differential equations, you win in a way, because you are able to use more advanced or precise techniques. So in a way, we were playing this game and we got to a point in which this question

was asked by Terence Tao in order to address this riddle on the solutions on the equations of Navier-Stokes admitting a regular solution or not. This is the riddle. The riddle is do Navier-Stokes admit a regular solutions for all time or are the solutions going to blow up as they say, blow up or present a similarity. And then a way to attack this question, which is still not answered, was precisely using, I mean, the idea of Terence Tao is maybe we can associate a Turing machine to the earlier equations

and then use them as initial data for these Navier-Stokes and then maybe get blow up. It was all heuristics but there was no proof there. Indeed there is no proof about the blow up yet yet And this is why I started to work on that I started to work on that because of that question and not because of solving Navier-Stokes-Riedel I think Navier-Stokes-Riedel will be solved soon But it's going to be quite intense to know how it's solved. So yeah to know how it's solved. So under your current estimation, is it that the laws of physics are computable or not computable?

That's the question. Indeed, I think we were exhibiting lots of physical systems which are computable. I mean, computable is a dangerous word, but we were using the expression to incomplete, meaning that that system could mimic any computation of a computer. So if you think about this, then your system has to be somehow very chaotic, but not in the dynamical sense, but chaotic in the sense that it has to be very complex, very difficult. Okay? And that's more or less the intuition. And I think the question is that there are systems that you are not going to be able to

to mimic with our computer. That's what I think. Indeed, for instance, that we are trying to see if some systems like the three-body problem that I mentioned before, if it could, because this is absolutely very complicated system, if this system can mimic also a computer. And this is something that we still don't know. We still don't know. And this is one of the problems I'm working on at this moment. So answering your question, I think it's very difficult for me to give a yes-no answer.

I think it's difficult for me, it's difficult to prove that certain systems are computable. I think that not all the systems, the physical systems, are going to be computable. I think that not all the systems, the physical systems are going to be computable. Interesting because as you know, there are various approaches of physics which have the universe as a computer. Yes. Yes. So this would say no, not fundamentally. This What's the universe?

this, what's the universe? Again, so the question is that I think that every, that thinking that, I don't know, I think we have to ask this question to Roger Penrose and he would say no. I think I have proof that some physical systems mimic a computer, but it's very hard to prove this. There are examples, but not everything you can imagine has been proved. Okay, let's get on with the presentation. Let's get on. Yeah, exactly. To be clear, asterisks to the people who are watching.

If you have any questions, you look up Eva Miranda. She's a rock star in the field of geometry, topology. She's a full professor at the University of Barcelona. And you're going to have a variety of questions from this podcast and also from hearing the teaser for the next one. Write them in the comment section. Yeah. Great. So now I continue with my class. I realize I prepared this as a class. Well, you see, on the classical side, I'm going to be talking about these bridges. We have like an important theorem, like a Darbu theorem. On the quantum side, look, I use the word principle, which I like a lot. We have Heisenberg principle and we have Darbu theorem. In this

case, one really follows from the other. Let me have a look. Indeed, here, okay, I'm just here, you know, this is really a class, this is really a proof, how to go from classical to quantum, right? Since answering this question is going to be hard, at least let me show something we can do very easily going from classical to quantum. We can associate, so here we have the operators, so we have x and p, right? Position and momentum. And we can associate to them operators. People know that classically to the operator of x you associate just multiplication by x

and to the operator of p you associate this partial derivative of x. And then you think of an operator that it looks very strange because you are putting a partial derivative and you are putting a multiplication by a function. But this makes sense because you are applying it to another function, which here is the wave function. Okay, so well, here just by applying the definition and knowing that the commutator, you apply it like the commutator on x and p of the operators is first applied x to p and then p to x and you need to apply this to the wave function all the time. Okay, then when you do this computation, ping-pong-pong, at the end everything matches and you get Heisenberg principle.

So the Heisenberg principle tells you that the commutator on x and p is like the product of the Planck constant and i, which is a complex number. And this looks exactly like Darbu theorem. And indeed, we could just say that we replace p by the hat of p, q by the hat of q, and it almost works. So this works pretty well in this particular equation. Why? Because this bracket, this Poisson bracket, is the bracket of two functions that are linear. Now what happens if

these functions are not linear? So for instance, then it's going to be more complicated. But I'm going a bit too quick. So what is... now we are starting to talk business. What is what? Here we have a picture of Paul Dirac. Paul Dirac is trying to point at his dream mapping rule. Indeed, what we did to prove the Heisenberg principle is just to replace the function by the hat of function as operator and g by the hat of g. So indeed in his dream the bracket, the commutator of these two operators should satisfy the same rule as it happened and the operator associated to the bracket.

Okay? This is what Dirac thought was true. This is, let's say, the principle of Dirac. But this principle doesn't work in general. It works depending, it doesn't work for any function f and g. But this is a dream. The dream is, to the classical framework, we associate operators on a Hilbert space and the commutator works like this.

And this reminds me of a film that maybe you cannot project, but... So after watching this movie, everybody wanted to do a PhD in physics or in quantum mechanics or something, right? So in this picture, we have, right, this is Boor in principle, I don't know if this is true, but Boor talking to Oppenheimer, right, about algebra being like the sheet of music where you write the music. And the music is quantum theory in a way. That's a little bit also the idea of Dirac. So that's a beautiful, beautiful metaphor. I like it a lot. So, indeed Dirac, that's the dream of Dirac, A, B going to this bracket, okay? And indeed, the quantization map, this bridge, is exactly a map that associates to the bracket of, in the classical world, which is the Poisson bracket, the bracket of operators. Okay? That's the quantization.

And so can you hear the music is can you do this quantization, right? The truth is that this is a nice dream of Dirac but it doesn't work. So, okay, we can hear the music, but not for a long time. So it doesn't work already for quadratic functions. So here you have an example of two quadratic functions, x squared and p squared, and the classical bracket is for xp, okay? And the quantum bracket,

I will talk more about the classical bracket, This is just an introduction. And the quantum bracket or the operator is exactly the symmetrized operator x hat x hat p plus hat p hat x. So they do not coincide. And indeed people have been trying to understand for which functions this works or not. Here you have a little bit the proof of how you do the commutator operator. Essentially, we could do it as we did the Heisenberg principle, or you could really just apply, expand it using this property of commutators. Okay. Now, can you go back one slide, please?

Yeah. So for this, do you have to have extra rules that say you can't have 4xp plus zero, because if you plus that zero and then you make it xp minus px or which classically would be zero that would equal something non zero in the quantum case. You say xp. Sorry. Can you say if you add anything for something that doesn't commute quantum mechanically, but it does commute in the classical case, you could just add that and get a zero in the classical case,

but it becomes something non-zero. So there has to be some extra rules placed atop about the ordering and then you can't do any of these tricky, what seems like trickiness. Exactly. It's trickiness. That's a very good point. That's a very good point. So in a way, things that commute XP commutes because it's the product of functions, but not as operators, right? Right. And then your question is very good. So what do you have to do? Then people have been looking a little bit more about the cases that work

and the cases that do not work. For instance, if you take a function that is x squared p squared, it doesn't work and the way to prove it, it's even funnier than you were describing. You can present this function in two different ways, or at least in this case, at least two different ways, as a Poisson, as a classical Poisson bracket. And then if, you know, if the quality holds at the quantum level, then you have a contradiction. You have a contradiction because then this term has to be equal to this term, but it's not. And then this took to a person called Grunwald to understand that it's impossible. This is a kind of no-go theorem. So it's impossible to construct a quantization map that satisfies the correspondence between classical and quantum.

So whatever Dirac wanted, Dirac dream cannot come true. This is pitiful, right? So it's impossible to construct a quantization map that satisfies the correspondence between classical Poisson brackets and quantum commutators for all classical observables. And now as you pointed out very, very nicely and wisely, you need to choose the observables, right? Not for all observables, but maybe for some observables you can. And that's a good question. And so now the question is, and now what? And now the hat, right? The hat, the quantum operator. So indeed, what do we need? As you said, we need a quantization scheme that satisfies the commutator rules. We know that this cannot happen for any functions, but maybe we can

choose a subspace of functions, of observables. In order to make that choice, I'm going to propose you something. I don't know if you agree, Kurt. I'm going to propose you to look closely at the Poisson brackets. What do you think about that? Sure. Okay, let's go.

So, I wanted to show a picture of a woman in maths. I have been talking about all these men in the Solvay Conference. Finally, a woman in maths. I have been talking about all these men in the Solvay Conference. Finally, a woman in maths. This is Emmy Nether. And Emmy Nether is known for many things. Today she's going to be known for the Nether principle. Nether principle is the principle that conserves quantities, gives rise to symmetries in physical systems. And these symmetries can be encoded as group actions on the manifolds. Today, there are going to be some group actions.

I like group actions a lot because it's a way to look at our symmetries. Now, if we have a rotating object, you see the object rotating, but indeed this is only group, this is SO3. So I like groups, I have to confess. So in a way, integrable systems that are going to be important for us today are going to be a key friend to solve the problem of choosing the observables, these integrable systems, are very close to group action and to groups, to having groups. But let's think about this simple idea from physical perspective that conserved quantities

give rise to symmetries in physical systems. So well, remember I was talking about these differential equations like three minutes ago. We have Hamilton's equations, this is the evolution of your system, and this system satisfies the preservation of energy and the energy of the system usually is this Hamiltonian, which is a function, but it's the energy of your system. Then something very, very interesting happens. Look at this equation that I have here in red. This looks very strange. I'm contracting omega. Omega is a two-form.

It's a differential two-form. And I contract this differential two-form with a vector field. And this gives me a one-form. Look, indeed, this is also an equation that I need to solve because the data that I have here is that I know what is this one-form. This one-form is minus the differential of the energy of the system of the Hamiltonian. And what is the unknown? The unknown is the Hamilton, this vector field, X to page. Okay? So I can look at these equations in two ways. Either I have the vector field and I contract and then I get a one form or I give you a one form and you give me the vector field. So let's play this game. I give you the one form, which is minus the differential h, and I give you these two forms, okay, which is the two forms. I'm going to give you an easy

two form, the one that you have here, which is what it's called Darbu form. So, well, let's go back to this computation here. I give you a one form, which is minus the differential of h. I give you omega, which is this omega differential of h, I give you omega which is this omega and you give me this vector field. Yep. Okay. And you are a good student. What are the coefficients of the vector field?

It's the Hamilton equations. The coefficients are. These are the Hamilton's equations. So Hamilton's equations are just the equations of the trajectories of the vector field. If I compute this vector field, I'm going to get a vector field whose first component is partial of h with respect to p and the second minus partial of h with respect to q. Okay.

So, indeed what happens is that this equation that we write here gives us a vector field that is just a vector field whose if I compute the trajectories of this vector field I get Hamilton's equation. So can I say that these two forms control classical mechanics, these two forms? Yes, I can. Okay, so I'm going to give a name to this form. This is a geometrical structure and it's called a symplectic form. So here I took a particular example of symplectic form, which is this one in this form. I can always locally think that it's of this form, thanks to Darbu, who is not the guy in this picture.

And so I'm talking about symplectic structures and why do we need now another geometrical structures? Because we have always been talking about Riemannian geometry, right? Riemannian geometry is important, also generalizations of Riemannian geometry are important to relativity, so why do we need now another geometry? Well, because we need to look at evolution of systems that come associated with conservation of energy. And this is very connected to the evolution of an area. If I consider just two dimensions, then having a symplectic form is the same as having an area form. So I'm measuring, not measuring with the lines, just the length,

but I'm measuring the area of my, and this is the measure I use, the area. Okay? And something very interesting is that when you talk about Riemannian geometry, you have invariance, and the most important invariant is curvature. You could ask yourself do you have such invariance in Riemannian geometry? The answer is no. You don't have any local invariance. Locally all symplectic manifolds are the same and this is something that Darbu theorem tells us. So we need to look a little bit more close at the Albu theorem, but again, why do we care about symplectic geometry? Because we are interested in conservative systems. If we are mathematical physicists, if we are geometers, we think of Hamiltonian systems

and we think of Hamiltonian systems as vector fields, solving the equation I show. The contraction of this vector field with a symplectic form is one form. Okay. So if I'm a geometer, this is how I think, but this corresponds to many physical systems. Now, Eva, just a quick question. Yeah. The audience may have this, look, there are a variety of two forms you could have chosen for omega. Are you working backward choosing an omega such that when you put in the hamiltonian or the derivative of it

you get back hamilton's equations or is there something canonical about this two form? There's something canonical about these two forms. This is an excellent question. So I chose this form on purpose because if like if I'm in a... If I locally, any symplectic manifold looks like that example. This is magical. And this works only if I'm very close to a point, right? If I have a magnifier, I am looking very close to a point, then what happens if I'm on a Riemannian manifold,

I may have change of curvature locally, and this will be an invariant, but not on a Simpli-Imani manifold. So there's something very important about that. I could think it in two ways. Either I could think that it's locally all Simpli-Imani manifolds look like that, or I could think that this is almost like a cotangent bundle, with having as form the differential of the Liubil one form. Indeed, both things happen and you are right. I chose that in a specific way. So this is thanks to Darbu. Darbu proved that locally, two symplectic manifolds look exactly the same.

So I can always think that I have these formulas here. I can always think that my simplectic form is as simple as that, in a neighborhood of a point. Another question someone may have is, OK, we're dealing with position and then momentum. Why not position and then velocity? And where does mass come in? You're just seeing momentum on its own, but there's no velocity times mass. And why is it that when you all of a sudden have mass times velocity, it becomes

a covector and not a vector when you're multiplying a vector with the scaler? applying a vector with a scalar. And the answer to this question is that the Poisson bracket of position and momenta is a constant, is one. And this is, I could say it's a miracle. It's like this Heisenberg principle, exactly. And if I take another function, it's not going to be a constant. It's going to be, this other function will be a function also of position and momenta. So when you take the Poisson bracket, maybe you don't get a constant.

You get something more complicated. So in a way, the position and momenta are dual coordinates because the Poisson bracket is one and this is magical. Wonderful. So, yeah, let's go to this. I mean, you are asking the right questions. That's perfect. And in a way, here you know in these slides that I prepared that it looks like I'm teaching a class. So in a way I'm saying, well, a symplectic form has something magical that it's a non-degenerate close to form. And this gives me a kind of isomorphism canonical, if you want, between the cotangent and the tangent bundle. I would say natural isomorphism between the cotangent and the tangent. And

the isomorphism is exactly the Hamilton's equation. It's exactly what we did. I give you a one form, so I give you a section of the cotangent bundle, and you give me a vector field. So you give me a section of the tangent bundle. So this gives me this isomorphism. And this is why position and momenta are sort of coupled in a magic way. It's a magic way that they are really coupled. It's like a magic dance between this positional momenta between this tangent and cotangent. And this is thanks to the fact that they are in a way dual to each other. And this is why Darbu was able to prove this theorem here we have. That in a neighborhood of a point, we can always pair position and momenta, here position

and momenta we call them X and Ys, but it's the same. We can always pair coefficients in such an easy way. We have a two-form such that the coefficients are constant once. Right? So this tells us that in contrast to Romanian geometry there are no local invariants. And in the coordinates as we said, the flow of a Hamiltonian vector field is just given by Hamilton's equations. So locally every symplectic manifold is a cotangent bundle. This could be the sentence of the day. Everything looks like a cotangent bundle. This could be the sentence of the day. Everything

looks like a cotangent bundle, so it looks very, very simple. And now I wanted to talk about the Poisson brackets, so let's talk about Poisson. Well, we all saw the Poisson brackets, but what you don't know is that the first time that they appeared was in this paper by Poisson, which is from 1809. And this, here you see the notation of the brackets and the Poisson brackets of B and A looks like a parenthesis, okay, but notation has changed. Here is the first article in which the expression of the Poisson bracket appears. And the Poisson bracket in the modern language indeed gives us something very interesting. Remember I was talking about Netho's principle, right? Conservation of energy. Now I'm going to show you that conservation

of energy comes for free from the following fact. I define, I take a symplectic form. This is going to be a bit abstract, but let me make this effort. I take this symplectic form. This is a two-form. This is a differential two-form. So this differential two-form can be applied to two vector fields. The vector fields to which I'm going to apply this omega, this form, are going to be the Hamiltonian vector field of the function f and the Hamiltonian vector field of the function g. Remember, these Hamiltonian vector fields, we have to find like Hamilton's equation, exactly the same, right? And we define such a Poisson bracket. What happens is that if I take two functions that are the same, f and f,

I'm applying a two-form that is anti-symmetric to two vectors that are the same. Therefore, the Poisson bracket of a function with itself is zero. And you say, and so what? Well, this is exactly Netho's principle. Netho's principle of conservation of energy can be, which was like a big statement, nowadays can be written in this so easy language thanks to the Poisson bracket. The bracket with itself is zero. That's exactly Nethers principle. I'd like to pause here for a moment because people don't realize there's not a single

lecture online, and I've searched in preparation for this, that covers symplectic manifold nor geometric quantization as you're about to cover in such a beautiful manner that not only covers the math, but talks about the importance and the relevance of it and also the beauty of it. Oh, thank you very much. This is fantastic. Thank you. This is a treat. Oh, thank you. Thank you so much. Okay, so that's an example. So if I have a symplectic manifold, I have a Poisson bracket. And now I'm going to surprise you, Kurt, with something. I'm going to surprise you.

I'm going to say, but can we define the Poisson bracket on a manifold that it's not symplectic? And the answer is yes. I'm going to give you another example. Well, here I said the algebras, but just think about matrices. OK? I'm going to consider matrices that are traceless, so without a trace, and such that if I transpose them, they are the same as minus the matrix.

You say, is this possible? This set is exactly what is called the Lie algebra of SO3. what is called the Lie algebra of SO3. But it's a set of matrices, it's an algebra. And it's an algebra because I can do the bracket, it looks like the commutator in quantum, but now I do the bracket of matrices. Like in the same way I would do commutators. I do the commutator of two matrices by applying the product of two matrices

and then minus the product of two matrices and then minus the product in the reverse order. Okay, and look what I get here. I take a basis of this Lie algebra, which is the Lie algebra of SO3. So it's the Lie algebra of the space of all possible rotations in three dimensions. Okay, and now I compute these brackets. Okay, I did it for you. They are not zero. They give me precisely the other vector. Okay, now I can define a Poisson bracket doing something that looks very strange, which is to take the dual basis of these vectors and this gives me some Poisson brackets. Okay, to have a Poisson bracket, I need some conditions. I need to have skew symmetry, anti-symmetry, and I need

to have that this Jacobi identity is satisfied, which is something that is satisfied for the commutator of matrices. And you know what? This Poisson bracket cannot be the Poisson bracket of a symplectic manifold. Do you see why? You need an even number of dimensions by definition. Hi everyone, hope you're enjoying today's episode. If you're hungry for deeper dives into physics, AI, consciousness, philosophy, along with my personal reflections, you'll find it all on my sub stack. Subscribers get first access to new episodes, new posts as well, behind the scenes insights, and the chance to be a part of a thriving community of like-minded pilgrimers. By joining, you'll directly be supporting my work and helping keep these conversations at the cutting edge. So click the link on screen here.

Hit subscribe and let's keep pushing the boundaries of knowledge together. Thank you and enjoy the show. Just so you know, if you're listening, it's C-U-R-T-J-A-I-M-U-N-G-A-L dot org, KurtJaimungal dot org. Just a question. There's something perverse happening here with these matrices, because matrices are representations of linear transformations. So you can multiply them. But then what is this plus that we're doing? We're just adding, like sure we can do this mathematically but physically.

Yeah. What you are doing when you are doing the algebras, you can visualize this kind of product by... This is a very good question, the question you are asking me. There is the notion if I take... Let me try to reply in a convincing way. I'm going to take now instead of taking this Lie algebra, I'm going to take something that is called a Lie group. The properties are different, are the properties of group. Okay, I take the group of rotations in three dimensions. This is SO3. These are matrices A such that A transpose with A is the identity. That's the definition. When you do this then we could even make a picture of this. These are rotations.

So when you are doing these rotations, you can also put the tangent space. You can think of the tangent in a physical way. So these would be the velocities. The Lie algebras can be understood as the velocity fields associated to this space of rotations. This deserves another lecture. But you can visualize. You visualize these Lie algebras as the space of velocities. You think, oh, but velocities are matrices., never mind you can put these matrices as a long vector if you want Okay, these are velocities of something happening on a group which is SO3 Which is the which is the group of rotations in three dimensions?

So I was asking you a question which is a a tricky question, is can this Poisson bracket correspond to a symplectic manifold? The answer is no, because if you look at the dimension of SO3, it cannot have a symplectic structure. A symplectic manifold, and this is something I didn't say, because it looks like the cotangent bundle of something, it has to have dimension even. So you cannot have dimension three. It should have either dimension zero, two, four, six, eight, whatever, or with even dimension. However, there is a close connection between Poisson and contact, which is now in this picture of the dual of SO3,

I can think of it as R3. And R3, we can fill it up with spheres of different radius, and these spheres of different radius have a symplectic form naturally associated to them, which is called the Suryok-Korst and Kirilov symplectic structure. And this can be understood physically because this Poisson bracket has what we call a constant of motion, which is the sum of x1 squared plus x2 squared plus x3 squared. This means that this is a function that is preserved and this is what is called a Casimir of the Poisson structure. So this means that when you take this function equal to constant and this gives you all these spheres that contract to zero, these spheres do have a symplectic form associated to it. So it's a magical world and it's bigger than symplectic geometry. So then a natural question is how do Poisson manifolds look like?

And that's a question that you think that, okay, if Poisson wrote the first Poisson bracket in 1800 or 1809, you think this is a question that people should know the answer immediately after. Do you know when this was proved? 1970s, 80s? 1983, and it was Alan Einstein, the person you have here, is the first person to give the equivalent of this Daru theorem for Poisson manifold.

And the theorem, it's very difficult to understand. The theorem that Alan Einstein proved is a theorem that tells you that a Poisson manifold locally is the product of a symplectic manifold with some transverse manifold that can be as complicated as the one of SO3. In this one of SO3 you have these concentric spheres on this other picture, this one to be always this symplectic foliation.

The idea is that this is the product of a symplectic manifold with a Poisson manifold of which you know very little. You just know that it vanishes at a point. But if the Poisson manifold satisfies some condition here, look that I have jumped from notation to use by vector fields and nobody got very nervous about this. This is our notation, right? This is our notation. Instead of vector fields, in the same way that you can go from one form to two forms, you can go from vector fields to two fields, and this is a good way to work with Poisson vector fields.

That's the standard. It's a notational thing. But if you don't like the notation, it's okay. Just think that a Poisson is a product locally, yes, locally of a symplectic manifold and a person manifold that vanishes at the origin. And in the case, uh, this person manifold is linearizable. Then the theorem of Alan Meyerson tells you that all the examples are a combination of example one and two. Uh-huh. Right? But instead of having, well, example two, you have a specific Lie algebra, the one of rotations on the tangent to rotations, right? Maybe you have other kinds of Lie algebras or Lie groups. It could be all kinds of Lie algebras and Lie groups. And indeed, that's a magic theorem that was proved in 1983. It looks like, okay, it's been a while,

but you see, okay, what happens then to Poisson geometry, to people working on Poisson manifolds? How many things have been proved? Many, many things, but it's much more complicated than symplectic, right? So in a way, it's tricky. Okay, now why I'm so obsessed by Poisson brackets? Because of course we want to go from Poisson brackets to commutators, but I want to, you

know, you asked me a question, I want to reply that question. You told me, okay, in this choice of functions that do not satisfy this good property of Dirac, how do we know? How can we choose the observables for which the bracket works well or not? And here comes the big, big surprise. Let's talk about integrable systems. Integrable systems, the name indicates something, the name indicates that you can integrate them. Indeed, that's their origin. But for today, it's going to be systems on

a symplectic manifold. So now we have a manifold that I told you that this has to be even dimension, dimensional with these two forms. And I'm going to take a set of functions as much as n. Okay. So half of the dimension of the manifold such that these functions plus on commute. So they have this property. Why do I want to do this? Because indeed I want to find a kind of in a canonical way, a kind of position and momenta.

And this will be very useful to answer your question. So I want to look at these integrable systems. In these integrable systems, often when you take all these functions together, you call them moment map, which sounds like, why do we call them moment map? Because they are going to be like this position and momentum. You require some technical conditions that these functions, somehow they

are generically independent. But the main point is that you require that they pass on commute. Why don't you give me an example? And here I am. I'm going to give you an example. I'm going to take the coupling of two simple harmonic oscillators. So I take a phase space, the cotangent mandrel of R2, with this symplectic form.

And here I take this total energy, this is the total energy of the system, this is just kinetic plus potential energy, I didn't do much here. And now I'm going to take a level set of the energy. Okay? The energy. Look, if I take a level set, this is an equation in dimension four. When I have the sum of x1 square, x2 square, y1 square, y2 square, What is this? That's a sphere of dimension 3 inside 4. I have a 3 sphere. Okay? And then you say, okay, this is very cool. But why do you need

this 3 dimensional sphere? Because a sphere, you can rotate it around and stays the same, right? It has symmetries. So I'm going to use the symmetries of the sphere to find another function that was on commutes. Well, I realized that I have rotational symmetry on this sphere. So the angular momentum, physics is telling me which function to pick. Physics is telling me, take the angular momentum.

Well, the angular momentum is this function which we call L here. And you can check that this L commutes with F. How do you do this? Well, I did here the computation for U. XL is this vector field. And then if you apply it to H, you get 0. So this is an inter-operable system on the harmonic oscillator. Another example that is going to be quite striking to some. I'm going to take an holomorphic function on C2, and I'm going to decompose it as its real and imaginary part. and I'm going to decompose it as its real and imaginary part. It is well known that this real and imaginary part

of holomorphic function, they follow the Cauchy-Riemann equations. The Cauchy-Riemann equations can be written as here. These are some relations between the real and imaginary part. And indeed, something that it's very, very surprising is that this is the same as saying that H and G defines an integrable system. This is mind blowing, okay? For which Poisson bracket,

while you take omega zero and omega one, just the real and imaginary part of this, of this, uh, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this,

this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, So now I want to understand because I want to use these integrable systems to explain me how to pair position and momenta. And I'm taking here an example. This is a two sphere, right? Now I take rotations in this two sphere.

Rotations are just, if I take a rotation, then there is a moment associated to this rotation, which is just the hate function. Indeed, the hate function is just the Hamiltonian vector field. If I take here as a symplectic form, TH, Tθ, then the Hamiltonian vector field of H is usually partial of theta. And look, this expression tells me that I have a kind of paving between the rotation theta and H. So I have a kind of situation of positional momenta. This is a two-dimensional example. Indeed, this works always. This is a theorem by Arnold.

Indeed, well, it's called Arnold-Liouville-Miner theorem. What's magic about this example here is that there is this kind of duality of position and momenta and now it's the height function with respect to these rotations on the circle. And this is an integrable system in dimension 2 because in dimension 2, which is the sphere, remember the definition of integrable system is I need as much first integrals as half of the dimension of the manifold. This is one in this case. So in dimension two, one function defines an integrable system.

Now, the surprise is that this is always the case. So if I have an integrable system on a bigger dimensional manifold, I have a kind of product situation of these spheres. Indeed I have these per-coordinates, this position momenta, are called action angle coordinates. Because I have the action, which is this momenta, and the angle, which is these rotations. And indeed, if I take a look here, if I take the integrable system, the level sets are a circle. In higher dimensions, there will be products of circles. This is what we call a torus.

And the magic thing is that the symplectic form can be written in this simple way in a neighborhood of the sir of this of the of what we call the fiber of the of this moment. globally? So this, this theorem tells you, but maybe not globally, the example of the sphere, all of it, I can write the symplectic form as d h d theta, maybe not globally, but in the neighborhood of this orbit or it's an orbit of the, of the torus at the same time, it's a fiber of the integrable system. This fiber of this integrable system, first is going to be a torus. This torus is called Liubil torus. Ok?

Who is here? Ok, Liubil is up there. And next to Liubil we have... These torus are put one next to the other. This is a fibrillation by torus, and this theorem tells you that in a neighborhood of what we could call an Immanian manifold, in a neighborhood of this torus, everything is like a product of this, so this was only selling me the fiber is a tourist and a neighborhood is like a total several copies of the doors put together what we call a vibration of Torah okay this story is just a product of circuit that's all and the funny thing is that the symplectic form can be written in a kind of easy canonical way, which looks like the Darbu theorem, but it's more, because the Darbu theorem is in a neighborhood of a point.

This is in a neighborhood of the whole torus. And it's written as this action-angle coordinate. It's very funny, this. So that's interesting. This is what's called this action-angle coordinate. It's very funny, this. So that's interesting. This is what's called the action-angle coordinate. And this is very, very useful because if you want to do quantization and you have these integrable systems that I'm going to be able to use,

then I have a very good way, I have a lot of information about the symplectic form and I can use it to quantize. See? Now let me talk about history a little bit. Miner was in this story. This was called the Arnold Ubell theorem, classical in the books. When I was a postdoc in France, I learned that indeed Miner had already given a formula related to it. And Miner had an interesting story to tell. He was not only mathematician, he was astronomer, and this was very close together at that time, but he was a member of the resistance in France.

So he was a very active member of the resistance during the Nazi occupation. And so he's very well known for this. He died quite young at 55. And he was the first one to give this formula that you see here. And this formula is the formula that gives the action angle coordinates. And the idea was very simple. He was saying take the symplectic form in a neighborhood of the story, can be read this torus, this invariant manifold is the differential of alpha, and now I take this one form and I integrate over the homology of this torus. And this gives me a function, which is the action function.

So this is a historical remark about this. And well, many things are understood about the integrable system also globally because this is telling you that an integrable system comes together with a torus which is acting on it, indeed. The torus is acting on it by rotations, and indeed we could think that a neighborhood is a cotangent bundle of the torus, and the torus is acting by rotations, and you are lifting it to the cotangent bundle. That's what the action angle theorem is telling you. But you could now think, okay, I want examples of integrable systems.

I gave you a couple of them which have a physical meaning. But now if I take an action of a torus on a symplectic manifold, and this torus has dimension half of the manifold. Then, this is an example of integrable system. And these integrable system, these are called toric manifolds, that they are very nice. They were studied by many, many people in algebraic geometry. Toric varieties are very important in algebraic geometry but also in symplectic geometry because a thoric manifold is a manifold, a symplectic

manifold which has an action of a torus which is Hamiltonian. And then the classification of thoric manifolds is described in a beautiful way. Look, if we look at this example that you have here on the left, which is a sphere, if you look at what I call the moment man, which is the hate function, the image is just an interval. If I go from dimension two to dimension four, I'm not going to be able to make a picture because I'm in dimension 4. I can only project. And then I'm going to take the action, this action here. This is CP2, okay, which is a perfectly simplecetic manifold. And I consider this action. I cannot write CP2, but I can write the image of the moment map. And the image of the moment map is a triangle. And don't you think that it's a coincidence that you take a very strange set,

like CP2, which I didn't define, but it's a classical object in mathematics, and you have a kind of bridge again. Now it's the moment map, whose image is just a triangle. Right. And you think, is this magic? Indeed, there is this theorem of Delsande, which tells you that there is a correspondence between toric manifolds and Delsande polytope. So the image is always a polytope.

I thought it was Atiya. Didn't Atiya prove a theorem about convexity? And Atiya, that's fantastic what you are saying. Atiya proved the convexity theorem. That's very good. Atiya proved that if you have a torus, here I'm considering toric actions. This means I have an action of a torus,

but it has a dimension of the manifold. If the action of the torus may be its lower dimension, then I have a torus action but I don't call it toric. For torus actions, Attila proved that the image is convex. That's the theorem you are mentioning. And Guilliam and Stenberg also proved it. Okay? This is called many times that in a Gilliam and Stenberg convexity theorem. And this would be, if you want, a particular case when the rank is maximal, not only it's a convex, the image, but the image is a polytope. So this links this very complicated geometry to the geometry of polytopes and linear algebra, if you want, linear geometry. So you could

play linear geometry and things you do to these polytopes have an interpretation on the original simplest. Super interesting. And this is a beautiful, beautiful, beautiful. We can maybe use these polytopes to quantize. What do you think? Well, I'd say yes, with an asterisk, and I know that you know what all those asterisk conditions are.

Let's do it. It's not a coincidence I'm talking about this polytope. So then you see that in this quantization, you can see that this Attila theorem plays a role, right? And indeed, Dalsan theorem. It's going to be interesting. And I mentioned, and Pasán, I mentioned the three-body problem as non-integrable, okay? And well, the two-body problem is integral. This would be the Kepler problem, right? And, well, the two-body problem is integrable. This would be the Kepler problem, right? And there you can find two integrals.

But the problem, what happens with the n-body problem? N-body problem is very complex in general. As I told you, connecting to the stirring completeness, it's not known if it's stirring complete or not, the three-body problem. Right? It's not known if it's too incomplete or not. The three-body problem. So all this has been a short lecture of simpletive geometry and Poisson's manifolds very quickly. I think I never did this before. This was challenging but I have enjoyed it. And now we have to be back in quantization. It means that everything we have learned so far will tell us something

about this bridge between the classical and the quantum world. So we feel like in this film in Karate Kid, right? Karate Kid, the kid had been washing the cars and you were washing the cars and washing the cars, he didn't know he was learning the movements to win the competition. So we are doing the same. We have been washing the cars. When I was showing you this Delsan polytope, and I was saying this is a polytope, this is a polytope, this is the car on which I will do the quantization on my manifold. So now the big question is, can we define Hilbert's space and represent

the algebra of smooth functions as operators acting on it? And the inspiration is, well, think of the quantization of cotangent models, right? Instead of sticking to action angle coordinates, which I have seen, we could take this action angle coordinate, some kind of distribution of what we call Lagrangian foliation. This is a way to frame the problem. I didn't say what is a Lagrangian sub-manifold, but in this integrable systems case, I'm going to make a remark, which is when I take these Pi's equal to constant, this form, the pullback of the symplectic form on Pi equal to constant, which is exactly the Liouville-Thoreau I, is zero. So on this Liouville-Thoreau I, the symplectic form is zero, right? Yes.

A Lagrangian's manifold is a manifold, is a set of your symplectic manifold, such that satisfies this condition, that the pullback of the symplectic form is zero. And it could be more complicated than the example of action-angle coordinates. This is true. But today I will explain, I will make choices and I will choose the easy path. And the message that I conveyed at the beginning, I, I would like to say, of course, quantization is a science, right? But this is a strong statement. I would say now today that quantization is an art because we have a definition of quantization,

but it's very difficult to accomplish this definition. So it's more like an art. It works in certain cases, but it offers requires a touch of insight and inspiration to guide the way, like a haiku. So here I have the quantization haiku, right? Chad GPT helped me to do this quantization haiku. Forms intertwining, polarized path, quantized dream, Hilbert spaces bloom. I like it. So in it, it's good. So we have forms, we have polarized paths, quantized dreams. Polarized paths quantized means it means that I need to use something called

polarization to do the dream of quantization. And then this gives me a Hilbert space, which is the Hilbert space is going to be defined associated as sections of a bundle. This looks very complicated. Let me make it simple. In this picture you see these kind of choices of position, momenta, which is important. In this quantization process it's important, the message is, if I have a classical system, I want to look at the psychotangent mode, because then I can try to quantize. And here, well, I was inspired and I think, like if you think of the art of war, right? The art of war gives you some good advice. If you know the enemy and know yourself, you need not fear the result of hundred

battles. Okay. You need to know your enemy. Okay. So who's the enemy here? Well, the enemies here is that we need to, uh, the enemies here is how do we do this choice of polarity, this choice of polarization. And if we do this choice of polarization and maybe we can start

doing the quantization, can we end doing the quantization? So the enemy here, I could say the enemy is the quantum world. This is a big sovereign statement. Don't put this as title. Don't put this as title of the enemy is the quantum world. That's great. That's a great clip. It's a great clip. The enemy is the quantum world. And the yourself here is what? The classical world? I am one of these warriors, right? I mean, these warriors on the quantum side should be

same warriors, though they don't look as belligerent, right? Because once you cross the bridge to the quantum side, it means that you already know where you're going, right? So you can be cool. But in a way, let's say I describe this, the art of war is like, the art of war is dangerous because I don't want to talk about war. But I want to say that there is something common in, you can have beautiful ideas in quantization, but sometimes they don't, I mean most, no, they are not going to work in full generality. I know, right? So, you know, when Feynman says, nobody understands quantum mechanics, this is our overstatement. Right? Everybody knows that this was an overstatement. And he was

what he was trying to see, like, if you want to connect classical and quantum, it's going to, it's going, you cannot do it in general. Right? You have to, you maybe can do it in general. You maybe can do it in some specific particular cases. And today I will present some specific particular cases on which you can bridge, on which the dream comes true and Hilbert spaces bloom, as the haiku was saying. And the second saying that in the midst of chaos there is also opportunity. Indeed there is a lot of symplectic geometry has been done motivated by this path towards quantum world. In understanding this Lagrange and foliations, in understanding if we could have this kind of general position momenta, a whole part of symplectic geometry has flourished. So indeed there is also opportunity because in this chaos towards understanding what is

the quantum, the reach to quantum, symplectic geometry has evolved and we have understood many, many things concerning the rigidity of objects, of Lagrangian foliations and polarizations and many, many interesting questions per se in sepcic geometry. So I'm going to talk about the cotangent bundle. Why? Because we want to go to this cotangent bundle. All our lecture is going to this cotangent bundle. All our lecture is going to this cotangent bundle.

I think the cotangent bundle, which I had already in this example, there were p's and q's, here is p's and x, I'm changing notation, but there is a Liouville-Wam form, which is the one form such that this form, is the differential of theta. And this is what we call the Liouville-Wam form. And it's classically known as Liouville-Wam form already in classical mechanics. Very classical mechanics. And then, remember the dream. The dream of Dirac is, I want the quantization is a is a way to assign to functions

Operators, okay, so here there is an assignment and to functions we associate an operator and the operator has several ingredients I Need the little one for and I need to apply it to the Hamiltonian vector field. So now we are in this Karatikit moment. The Karatikit moment is we explain what the Hamiltonian vector field was and now apply this Liouville 1 form to this Hamiltonian vector field and I knew this operation and I add the function.

And you know, this looks very strange. And of course I do multiplication with the plan factor and I multiply with complex because I want to have Dirac formula. So I'm walking towards Dirac formula and the surprise is I have Dirac formula. So that's a dream a little bit. I have, but there is a dream, but there are problems. There are, you know, there is a dream, but you wake up in the middle of the night and oh, I wasn't sleeping, right? So Dirac formula holds and this is fantastic, right? But there is a small problem. Here we have these operators, okay? And these operators are functions, I mean, these operators are on L2, R2n. So the Hilbert space that I'm considering are integrable functions, which is called square integrable functions. Usually this is called. But it's of all R to N. But what happens,

that the physics intuition tells me that this is wrong. Because the physics intuition tells me that if I have a position some moment I should have just the positions So I should have just L2 in Rn Okay, so we need to cut down the variant. So this is at the same time a Problem and an opportunity so we go back to Sun Tzu. This is again chaos and an opportunity interesting We have been seeing that finally Dirac formula, Poisson bracket goes to commutator as Dirac wanted, right? But the prequantum Hilbert space is too big.

We need to cut it down. And we have a solution for that. And the solution is look at these pictures. Okay? The solution is we need to reduce the variables. We need to choose some positional moment. And is it always too big by a factor of two? It's too big by a factor of two of by factor of n. So if I have r to n, this is rn cross rn. Yes. I need to kill n variables. Right, right. I need to kill half of the variables.

How to do it? Choose your favorite pictures. In these three pictures, I'm trying to depict a grid between position and momenta. But the grid, when we think of a grid, the ones that we used to write, we think of the situation on the left but maybe What we have is the situation on the right this kind of wave of water Right that wants to be indeed a foliation a partition of my space Into a space I have a space of dimension 2n and I have a partition into spaces of dimension n. This is what a grid gives me. The position of a momenta gives me a partition into grids of Rn times Rn. So, I need to think more as the picture on the right, which thinks of a foliage. And now I'm going to use one of your sentences in LinkedIn yesterday,

which is, everything is a Lagrangian manifold, no? Yeah, I'll put that link on screen. It was also a Twitter thread that went viral, where I was explaining Alan Weinstein's, quote, everything is a Lagrangian sub-manifold. And I also placed on LinkedIn. Yeah. I realized it went viral.

Yeah. This is amazing that all your, everybody is responding. You are like, you're making me going viral. You're making me go viral. So we are doing, we are making, we are making geometry go viral. This is fantastic. And physics, this is great. So that sentence indeed is due to Alan Einstein. Everything is a Lagrangian sub-manifold.

And it's amazing because it's so true. And I'm going to go one step. We need a little bit more than a Lagrangian sub-manifold. We need a lot of Lagrangian so manifold. So we need something like a foliation, which is a partition into Lagrangian so manifold. This could be a foliation. The second picture you have here, indeed is a geological foliation that you can find when you go hiking, right? So this is a partition into subspaces of half of the dimension of the manifold, which makes you think of this grid of action coordinates, which makes you think of positional momentum. And here you have different kinds of foliations, the first one wanting to be a ciphered foliation, and these are different ways to do partitions.

Examples of Lagrangian foliation, we can think very naturally like the one in the middle. The one in the middle reminds us of this action angle coordinate theorem where we also have a vibration by Vitorai, right? The fibers of an integrable system indeed define a polarization. And this is great. A polarization, by the way, it's a word that I put here. Polarization is a Lagrangian foliation for us, but this word is used in the terms in quantization terms. Okay. I see. But the polarization for us is just a Lagrangian foliation. Indeed, it could be more complicated because if I want to explain polarization in general,

I will need to complexify the tangent bundle, but then nobody will end up looking at the end of the podcast. It will be too much. Today, I will just take real polarizations, which is a Lagrangian foliation. Yes. Now, what's the difference between a regular foliation and a Lagrangian foliation. Okay. Yes. Yes. Now what's the difference between a regular foliation and a Lagrangian foliation? Yeah. A regular foliation.

Some, for instance, this is, let's say there is one difference, like that the leaves of the foliation, if you say regular foliation, this is foliation with leaves, which have all of them the same dimension. And if it's Lagrangian, then you have two additional information to keep in mind. The dimension is half of the dimension of the manifold. So if you're in a symplectic manifold of dimension 2N, if you're in a symplectic manifold of dimension 4, you go to dimension 2. If you have a simplectical manifold of dimension 2n, your Lagrangian foliation will have leaves of dimension n. And the second condition is that when you take the simplectic form and we say you pull

it back to the tangent of the foliation, To the sumani fold is zero. Yes. This is what was happening in these examples on integrable systems. So integrable systems gives you examples, thousands and thousands of examples of Lagrangian foliations. Integrable systems are great. You can choose your favorite one. Here I put several pictures.

And now, and now that's the moment, the most important moment. Now we're going to define what I call geometric quantization, which is my bridge to cross from classical to quantum. And I need to make some choices to cross. I need to take things with me. The first thing is going to sound a bit strange, which is I need to take a symplectic manifold. My symplectic manifold has a symplectic form, this omega, which is closed.

The fact that it is closed, it means that I can take its class. And its class, because it's a two-form lives in something which is called the second cohomology group. And this cohomology group we ask that this is integral and if you don't like this expression I have another one for you which is the formula up there. You can integrate over surfaces these two forms, any surface that you take inside your simplectic manifold, take any surface, and if you do this integral, you ask it to be integral, an integral class. This sounds very strange. This means that the area is integral. Exactly. You can think of this

comology class, if you are in dimension 2, this is just the integral. You are asking that the area is integral, that this is an integral number. Why do you need this? This is very surprising. There is a relation between the first picture and the second one. The second picture wants to be a line bundle over the manifold, which means I put a line bundle, which is a complex line bundle, so a copy of the complex numbers over every point of my manifold. I know, this looks very strange.

We wanted to quantize, and now I need to put over every point of my manifold a copy of the complex numbers. Yes, I agree, but life is complicated. We need to do it. Okay? So we need to take a complex line mandel with a connection. A connection is an object that I need to make derivatives, but I'm not going to be very worried about it. But I need this connection such that its curvature is exactly a multiple of omega. I says here, I need to make this multiple by a complex number I this is a detail Okay, and when you have such a thing because the class is integral Custom proof that you have a line bundle

Over it sides of the curvature is exactly omega So you need this integral condition to have this line bundle such that it's curvature is omega. This is very easy to prove, but it looks very, very strange. First time you look at it. I mean, from a physical perspective, you know, I had like my first PhD student, Romero Soya was a physicist and he did the thesis on quantization. And he got really, really interested in this condition. This condition is magnetic from a physics perspective. So if you have these two things, you have what I call a pre-quantum line model, what is called a pre-quantum line model. And then you need to take a real polarization.

We have been talking about it. We need a real polarization, so this is Lagrangian foliations. So take your integrable system, your favorite one, the ones I described before. Maybe the double oscillator, maybe you want to take Cauchy-Riemann equations. Take any of these examples. Take a thoric system. All these are perfect examples to give you real polarization. So in particular, you can take them as Lagrangian foliations. And why do I need them?

Because remember that you observed before anybody else, and we observed that the condition that the bracket goes to the commutator, the condition that the bracket goes to the commutator, the direct stream, doesn't work for every function. But if I take functions that just depend on the elements, on the function, so if I take a Lagrangian foliation, the leaves of the Lagrangian foliation don't depend on all the variables, just on half of them. If I take a function that just depends on half of the functions, then the commutator works very well. This is almost magic, but it works.

And then, so, I'm going to look at my Hilbert space, and this looks very strange. As flat sections of this line model. Oops, wait, this gives me a headache. What is a flat section? What is a section of a line model? Well, it looks very sophisticated as words, but this is just a function, a complex function. You can think of this such as a function. But I think of these functions that will give me this quantization model as sections of the bundle. And these sections, I need them to satisfy some equation. And this is how we are going to get rid of these variables that our former

example of the cotangent bundle didn't work well because there we didn't take the polarization. We need to take sections that are flat, it means that I need to derivate in some directions that are zero. And I need to derivate for any vector field x that is tangent to this polarization, to this foliation. And so this is going to give me some equations and this is going to work You say this looks very straight. This is exactly what we called geometric quantization the ingredients Okay, this gives and this gives you what this gives you the Hilbert space, but I still didn't talk about the operator I'll do this later But I want to look at an example. Do you think this is

a good idea? Let's look at an example. Some calculations. Please. Let me do some computation of these flat sections. Because I want to relate this to Bohr and Sommerfund. I talked about the Bohr's, about the role of Bohr in the electron of the hydrogen, right? And this is going to appear here in a very sophisticated way. I'm going to say that, well, when I have this foliation, every element, every piece, this is like cutting your space into pieces, every piece, we call it a leaf of this foliation. This looks like a poem. It's a leaf of this polarization.

And I say that this is Borsh-Homerfield if it admits sections which are globally defined. And I need to give you an example because otherwise you don't understand anything. I'm going to consider the cotangent bundle of S1. So S1 times R, this is the cotangent bundle of S1, with this symplectic form, differential of T, with differential of theta, here the Liouville form theta is TD theta, okay? And here I take as polarization these circles, the circles on the base, the foliation by circles on the base. Then I want to look at this equation I saw here, flat sections equation. The flat section equation, I can compute it with this formula because when

you have this connection, you can relate the connection precisely to this theta, to the connection of the connection one form associated to the symplectic form via this differential. So in a way, this connection tells you that you can do the derivative of the section modifying with respect to this connection 1 form that is associated to the symplectic form. So if you look at this equation here and you consider who is theta, who is t differential of t, then you get that the flat sections are given by this expression. These are functions. So you consider sections.

The sections are functions that take values in c, okay? But the section is a function from your manifold, so it depends on t and theta, and it associates the function a t multiplied by the exponential. This leaves on a circle the image, the exponential of i t theta. Well, then if you want this to be well defined when theta goes around, observe, if you want this to be defined, when you go around this theta goes around 0 to 2 pi, when you go around the circle. It turns out that this only makes sense or closes up when T is a multiple of 2 pi. Otherwise, you can make the exercise, if T is not not 2π then this function is not well defined

because it's multivalued. So in order to close up, this only makes sense when t is a multiple of 2π. And you think, okay, why do you call this the Bohr-Sommerfeld leaves? How is the connection to the hydrogen atom of Bohr and model of Bohr and Sommerfeld? This is exactly the model of the hydrogen atom. You had this model where you have the orbits at a constant distance, which is more or less the same. at a constant distance, which is more or less the same. Well, and the connection to this is the following. And this is, I'm going to call this as a theorem, but it's very easy to prove that

if you take a polarization, okay, with some action coordinates of action and all coordinates, okay, then the Borson-Merfer leaves can be read, and attention because this is too beautiful, can be read just from the integral points of the polytope. Which polytope? The polytope of the sun. Can you believe? Okay. So there was that polytope of the sun. Now I take, which was telling me that the image of a biotorous action was always a polytope. Now I take the points inside this polytope, we have integral coefficient because this polytope I'm going to draw it in some Rm. I take the points which are integral, like the green points in this picture. Well, this theorem is telling me that these points are in the image of the F1, Fn.

So if I take the pre-image of these points, what I get are bore-sorbifer leaves. This is incredible. So I get leaves of the Lagrangian foliation for which the sections are well defined everywhere. So this gives me an idea. Because how many of these points I have? Very few. I have a finite number of these points, no? Inside the polytope. Can I count them? Let's count them. So I'm going to call quantize counting these points. And you say, oh, but can you do this? Yes, I can do this because I'm a mathematician. So I give us the definition. Bore-summer from quantization is the quantization which counts these points. But is this the good quantization? You're going to tell me. Okay, so we are in a magic moment now. We see that different objects from different perspectives are all meeting together. So we call quantize counting this world sum of

four leaves and now I'm going to try to explain whether this makes sense. The question is what is the representation space in this case? Remember, the representation... So this is a little bit of the summary. The symplectic manifold is quantizable if you have the integral over any surface of this integral, which is equivalent to the class of the symplectic form. It's integrable. And the condition that we need, this condition we needed to have this line model and to have the connection. Okay. And then I define, I

associate to this connection some equation that if it has a solution, I say the set of, But if it has a solution, I say the subset of leaves of the polarization that admit this solution, this is what we call the Borson-Maffin leaf. So we declare this as a way to quantize. What does it mean? That I define a Hilbert space which is given by the number of BOR somerphen leaves. But then I need to go on. I need to try to understand if this makes sense. What is the representation space in this case? One, the prequantization operator, the one that works for the... is precisely the one that is in this expression. The one that worked already in the cotangent bundle works in general with all these conditions of pre-quantum conditions that I call.

You associate to the function just the nabla of x sub f, nabla of x sub f, multiplying by the Planck constant and i, and I add it to f. Then what's nice is that the pre-quantization operator satisfies the commutator equality. This is fantastic. Now the thing is that operators, we need to think of them on the space of a small section of L. This looks very, very strange, but these are functions. There is something called half forms that it's useful to get sections that are square integrable, so for which you can do the integral and the product and it works. But today I'm

going to ignore them. Why? Because it's too technical. Okay. It's nice. One day I could talk about half forms, but this is too much. But today I will ignore them. In practice, in this picture that I think it's very good to keep in mind that the worst homophonyl leaves are the integral points inside this lattice, inside this polytope. If I do the half form correction, the effect is that I move by one half these points. Which is very interesting from a physical perspective. So, I could quantize counting the Borsommerfeld leaves, okay? And, well, I could just declare this as Borsommerfeld quantization, but indeed, this quantization makes sense

from differences. This coincides with something that I prepared this last, but I think now it's too much. We could define the quantization using shift cohomology. Shift cohomology is something very algebraic, which is beautiful. But in a way, the idea is that we are counting the sections over this discrete set of Borsammerfeld's leaves. And this has very good properties, topologically, to do cohomology. So it was the Snedtinsky who proved that this quantization It was the Snetiski who proved that this quantization gives you the dimension of the Borsammer's relief. So this idea of counting the Borsammer's relief, you can make it more formal using something

which is called shift cohomology. And indeed, this is very interesting. This is something I did for a long time. And I'm going to skip this because it's not so interesting. I'm going just to focus. I mean, it's interesting. It's interesting, but it's too much. Of course. Yeah. Sure. I'm going to focus on the case of the torus, on this case that I find very fascinating, which is the case of toric manifos, which is, for instance, the sphere and the image of the sphere.

It's just an interval. Okay. So what happens if I consider this torus, this foliation has singularities? Well, nothing happens. The quantization is always given by the integral point. This is something that was proved by my friend Mark Hamilton a long time ago. And this is why with Mark, we started with Mark who is Canadian, we started to look also at other polarizations more complicated. For instance, consider that you take a simple pendulum or a spherical pendulum, the kind of singularities that you get, if you get an harmonic oscillator you get a singularity that looks like the one that you would get

on rotations on S1. But if you are on a simple pendulum you get other kind of singularities. On your spherical pendulum you get some singularities which are called focus-focus singularities. So with Mark we work with these hyperbolic singularities. Okay? And then the interesting thing is that when you consider these hyperbolic singularities, you get infinite contribution. This is what the computation of the shift cohomology gives you. So computation, mathematical computation tells you that the quantization of something that you expect to be finite is infinite. Of course, this means that this definition of shift cohomology is not good.

So we need to correct it. Just a quick moment. Yeah. You're a collaborator here, Hamilton. I assume it's not William Hamilton unless geometric quantization allowed you to travel backward in time. I mean, I'm old, but I'm not that old. Not William.

I mean, I, yeah, but I'm not that old. Not William. I mean, I, yeah. Unfortunately, not William Hamilton. Not the one of Hamilton's equations. Right. Indeed. Yeah. At some point we made some jokes about this saying that the, I only look for collaborators name of somebody who is really doing Hamilton's equations.

Right. So he has the right name to be on business. Yeah, it's true. Your next paper is with Jacobi. That's what I want. If you find somebody called Jacobi, please introduce... I will let you know. So this is a call. Please, old Jacobi, write to me. We will write a paper together. That's nice. Yeah. So with Mark, we found that if you include these cross-singularities, something very

well happens. You have infinite quantization, but this quantization doesn't meet the expectations of a physicist. So, of course, you need to correct out, and it's possible to correct out these contributions. And these contributions are not good because there is something, there is one of the problems of quantization is that, okay, now I made the choice of polarization and you didn't ask me because I have given too much information, what about choices? So polarization, if you change the polarization, do you get a different answer? Yes, this is terrible. So if I take a two sphere and I

take the rigid body like here, you get infinite with and you take the shift cohomology as quantization, you get infinite number, you get a Hilbert space of infinite dimension, this doesn't make sense. And if you take the rotations by spheres, you'd get a finite number, right? But in this particular case, you can correct this problem just by killing this infinite. There is a way to do it, essentially. For any dimension, if you have some reasonable singularities, which you're always going to have, by the way. If you have an integrable system, you need to have a maximum and minimum, therefore you will have these singularities.

But you may have reasonable singularities like hyperbolic, focus-focus, or elliptic singularities, then you can always find a model that meets the expectations of physics. That's the summer. There is some other quantization that I didn't comment, which is called the Keller quantization, in which the polarization is not Lagrangian foliation. But this is too much. Interesting. And now you're going to ask me, and now we're close to the end, what about quantization of Poisson manifolds? In general, because I said not all Poisson manifolds are symplectic.

So here I have a very simple example of Poisson manifold which is not symplectic. I just take the two forms, indeed these two forms, explode on h equal to zero. This would be the situation here on this sphere, the equator. The area explodes when it gets close to the equator. But it explodes in a very controlled way. These forms are called B-symplectic. B stands for boundary because they were introduced by the study of simplectin-manif simplex and manifold boundary. And for these forms you can compute, for this particular example here,

and this is an exercise I leave to the readers, you can compute the Borson-Maffer leaves. And you have infinite number of Borson-Maffer leaves on the north and on the south. So your Borson-Maffer quantization initially it seems that it should be infinite. However, in this particular case there is a change of orientation of the area on the north and the south hemisphere. This makes that this orientation affects also the sign of your Borson Marfan leaf. So in a way you can paint them with two different colors and I chose the red and the blue because I'm in Barcelona and these are the colors of Barsan. Okay. Okay. And this is also the color of the of the screen. So you can paint them in such a way that the infinite number cancelled out and you get a finite number. And this quantization coincides with some other quantization we did with Victor Guillemin

and Jonathan Bitesman. So it meets our expectations and it meets the physical expectations. And for more general portions I'm working on this problem with Richard Nest and Jonathan Bitesman who is in this picture, my beloved collaborators, both of them. Now, of course, I explained many, many things, but one of you is going to ask me about topological quantum field theory. So, I'm just going to say two things before finishing.

First one is Dirac. I have been talking about Dirac as the dream we couldn't fulfill, but I could imagine Dirac telling to Feynman about geometric quantization when Feynman, he gave his first model of quantum field theory somehow because in his lectures of computation somehow he gave the first model of quantum computer. So I can imagine Dirac telling to Feynman, I have an equation, do you have one too? This would be a good way to define topological quantum field theory. The second way to define topological quantum field theory would be the sentence by Nelson. First quantization is a mystery, what I explained, by second quantization is a phantom. So topological quantum field theory, it is a phantom because quantization is not a phantom. But topological

quantum field theory means a phantom from two categories. The categories of co-borders of manifolds, of manifolds with boundary and with the fine co-bodies of these manifolds. And vector spaces. And I'm very interested in topological quantum field theory. This is surprisingly connected to this question you asked me about Navier's talks. Interesting. And we'll go back to it on the next episode. But for today, I'm done. What's next?

Choose your your end of the story and These are the three possible ends. Professor. Yeah. That's absolutely fantastic. What a magic moment to our magic moment map. What a wonderful way to spend a What what is it today, a Monday? It's a Monday, yeah. It's Monday the 13th, so it's just one of these not so magical days.

I'm extremely thrilled that you were able to present this. Thank you. Thanks so much for inviting me, having me here. I have to make a small confession to you, Carl. Last week I was in Copenhagen and it was extremely cold because who goes to Copenhagen in January? I go because Trinxarnes is there. We had to collaborate on a project we have about Poisson manifolds and approximating them with symplectic manifolds and so on.

I really wanted to show him the slides. I had prepared partially these slides already, so I showed him. And we started to discuss, really, discuss is a very nice way to say it. I would say the women were fighting over the definition of quantization, right? And then, indeed, this is how I also thought about Sun Tzu, because it's like the art of war. Like out of this long discussion that we had on the blackboard, we got a new result. Interesting. Yeah. So now I'm writing it down, thanks to you that we started to discuss the notion of quantization, indeed related to these Poisson manifolds, right? Not only simplecty. And I could make sense

of many of these. We could make sense discussing together of a generalization of the geometric quantization of simplecting manifolds for Poisson manifolds. So this is thanks to you. Kurt, I think you are going to be on the acknowledgments of this paper. That's wonderful. I think, no, you are on the acknowledgments of this paper. I have not written it, but I'm going to start writing the acknowledgments and I include you. So thank you for this. The next time I'm in Barcelona, I would love to talk about computability and decidability

in physics in person. So that'll be so great. Thank you. Yeah, yeah, I'm looking forward to it. Thank you very much. New update! Started a sub stack. Writings on there are currently about language and ill-defined concepts as well as some other mathematical details.

Much more being written there. This is content that isn't anywhere else. It's not on theories of everything. It's not on Patreon. Also, full transcripts will be placed there at some point in the future. Several people ask me, hey Kurt, you've spoken to so many people in the fields of theoretical physics, philosophy, and consciousness. What are your thoughts? While I remain impartial in interviews, this substack is a way to peer into my present deliberations on these topics. Also, thank you to our partner, The Economist. Firstly, thank you for watching, thank you for listening. If you haven't subscribed or

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Every dollar helps far more than you think. Either way, your viewership is generosity enough. Thank you so much. Eva, a quick question is, what is the definition of integrable systems? Because this term keeps coming up over and over, and then the way that people define it, I've never seen a specific definition of integrable systems. I see people say this is integrable, this is not integrable, but I've never seen an outline of exactly the definition. Yes, the definition, I mean there are several types of definition that what we call Liouville

integrable is the following. You have a manifold and you assume well you have a system okay and This system is going to have a first integral that you even don't think of it But it's there which is the energy of your system This is a function that you have you say that it's integral if you have other functions that Commute with this one. What do you mean by commute? Poisson commute with respect to this Poisson bracket. Okay? And this in a way, how many do we need? We need as much as half of the dimension of the manifold because you are writing things in position and angle, so you have an even number.

So you need half first integrals that commute. And why do you need this? Because in a way, the idea is that to have these commuting functions is more or less equivalent to having the action of a torus or your-hmm. Or your manifold. And if this torus is as big as N. And do you mean half functions minus one because you already have the energy function? Oh yeah, you already, yeah, you take N minus one additional to the initial one. So in total N. You need N in total?

I see. Okay. And then why is that? Because you can associate a torus to this combination of functions that commute, more or less like this Arnold Liouville theorem, and then the reduction of your system by the torus amounts to a point. This means that you cannot make it smaller, because it's the smallest you can get. So the definition of it, and this is equivalent to being, I mean this is the definition of integrability and is related to having explicit models of integration of the equations. Interesting, okay.