Symplectic Quantum Cells in Phase Space & Gromov's Non-Squeezing Theorem

0:00 fantastic working conditions we have here it's a great pleasure and honor to be here for the second time okay let me first oh yes before I introduce myself I didn't need that chairman to interrupt me after one hour so we can use you have our Oh no, I prefer the other one there. Do you have a pillow? Okay, I'll find it. Okay, let me first introduce myself. Thank you. Thank you very much. I got my PhD at the University of Nice in 1978. Nice was at that time a stronghold of the infamous Bourbaki School. And my supervisor was Jacques Chazarin, a specialist in partial differential equations and pseudo-differential operators i was his last math student because a few months after having supervised my my thesis he started working with artificial intelligence my phd was on partial hypoelectricity for of sub-differential art language, Hermander's theory, it's called. I got my habitation at the University of Paris 6 in 1992, with the mentorship of Jean Leret, who was a very famous French mathematician.

2:30 I was the last person he supervised. He died a couple of years later, and actually my habitation was on a collection of papers I had written with Leray and papers I had written before, and actually I was very impressed by LeRae's work on the Maslow Index and semi-classical mechanics. He's worked on so-called Lagrange analysis and LeRae helped me very much to understand things in his book. It took me actually five years to read the first chapter in his book and I haven't finished it yet because it's very condensed and very complicated. Thanks to Bazit Hailey but I'm not his last one. That's a highly, okay, my interest in mathematical physics and especially in quantum mechanics has been growing ever since, and I thank Basie very much for all his advice and help. Yes, yes, yes. Dangerous. Dangerous, okay. Today I'm going to talk about, well, it's going to be much ado about nothing, actually. about emptiness and face space and uncertainty principle oh I apologize my transparencies were lost or stolen in the plane from Brazil to Europe so I have to make a reconstitution today so but anyway I'm a mathematician and of the old school so I will use the whiteboard and body language My permanent affiliation, so to say I say, is at the University of Potsdam in Germany, and, well, starting next month I will be in Vienna. I will work with the Feistinger group on signal analysis, or time frequency analysis, which is a version of quantum mechanics, actually, quantum mechanics in phase space. And, okay, I'm going to talk about Heisenberg's uncertainty principle from a symplectic point of view, from a topological point of view, so to say, and this will allow me to invoke

5:00 the principle of the symplectic camel, see why this is called the symplectic camel, and to produce some kind of quantum phase space, and there I would be very grateful if I got some response and criticism and perhaps encouragements and all this because I'm not a physicist I'm trying to be one but and there's a fundamental link with all this with Bohmian mechanics and when we start this by reading Basel's recent paper what is it yes it's here the relationship between the Wigner-Moyard and Bohm approaches to quantum mechanics, and this paper opened my eyes on the fact that Bohm mechanics is already contained in the Wiener-Moyal formalism. So it would be also interesting to see in which way Heisenberg's concept of the principle to be used to derive or to study quantum mechanics, I don't know. Well, let me first begin with some simple things from geometry. I'm going to talk about phase space ellipsis. So what's a phase space I'm going to use these standard notations here. The x's are the physicist's q's, a position variable. And the p's are the momentum variables. I work in n degrees of freedom, that is the phase space will be r to n, OK? Working two n variables. And what's an ellipsoid in phase space? set Bm, defined by a relation like this here. This is the equation for an ellipsoid centered at the origin in phase space, and more generally, if you center your ellipsoid at an arbitrary point z0, the equation will be this here. M is a positive definite matrix, symmetric.

7:30 That is, in addition, we have m equal to mt. And to say that this matrix is positive definite means that all its eigenvalues are real and superior to zero, or equivalently, this expression is positive, or zero when z is equal to zero. Okay, a related concept which is extremely fruitful and interesting is that of symplectic capacity of a phase-based ellipsis that we can explain. I say that the symplectic capacity of any of these two ellipsids is 2 pi over lambda the largest eigenvalue or the modulus or the largest modulus of the eigenvalues of the matrix JN. What does this mean? Okay, you know, there is a fundamental matrix in symplectic geometry which is sometimes called the Vi matrix or also the matrix. And J here plays the role of the complex number I. You can see that by noticing that J squared is equal to minus the identity. And to this matrix you associate an antisymmetric form, which is called the standard symplectic form. It's defined the following way here. If I take two vectors, z equal to x, p, z prime, x prime, p prime, well then, sigma of z, z prime is just z prime transposed, j, z, okay? Or in coordinates, if you prefer, This is just p dot x prime minus p prime dot x, where the dot is the usual scalar problem. This is a b-linear form. It's linear in both z and z prime.

10:00 Take it separately. It's anti-symmetric because if you switch z and z prime, then this thing will change side. Notice that in the case n equal to 1, this is just minus the determinant, the usual determinant. And in addition, it's non-degenerate in the sense that if this is equal to 0 for all z prime, while then z must also be equal to 0. But it's a kind of scalar product. It's not a scalar product, actually. It defines a structure, which is called a symplectic structure on phase-based. And the symplectic structure is essential in understanding Hamiltonian mechanics, for instance. So it's a natural phase-based symplectic structure. Yes, okay, so let's come back here to the notion of symplectic capacity. n is a positive definite matrix. Follows that jn has eigenvalues which are of the type plus or minus i lambda j for j equal to 1, 2 and so on, n. Why is it so? This is quite easy to see. Because since n is positive definite, I can write it as the square of its square root, which is very well defined, okay? You define the square root of n just by defining its diagonal form, by taking the square roots of the eigenvalues, and then that's it. And it follows that jn is equivalent to n square root jn square root, of course, because you know that if you have two matrices, A and B, one of them being non-singular, then AB and BA are equivalent in the sense that they have the same eigenvalues. Now, look at this matrix. It's an antisymmetric matrix, of course, because if you transpose it, you can transpose only the J, and J is and symmetric, transpose J is equal to minus J. So it follows that all the eigenvalues are of this type here. Now, you take the, in addition, of course,

12:30 assume that each lambda is positive, and it follows that you have here a sequence of numbers, real numbers, positive real numbers, lambda 1, lambda 2, lambda n, and lambda, the lambda beginning here, is the largest of these numbers. This is called the symplectic capacity of this phase-space ellipse. I'm going to explain to you why, and why this is interesting here. Because one proves, it's not obvious at all, that you cannot send two large balls inside these ellipsoids using only symplectic transformations. This is a variant of the principle of the symplectic camel that we explain because I So I think it's going to be essential for the understanding of the uncertainty principle and the symplectic overview. First, what is a symplectic transformation, or a linear symplectic transformation? Good. Let's take a matrix S equal to ABCD, a 2N, 2N matrix, in block form. The blocks have dimension N here. We say that S is symplectic if, and only if, SJ, S transpose is equal to J, which is equivalent actually saying that this is S transpose J S. So this is the definition of a symplectic matrix. actually to write explicit relations for the blocks A, B, C, D. For instance, you find that you must have A, D, D minus C, D, D equal to the identity. You find that A, B, D must be symmetric, and so on. You have quite many relations. in optics. These relations are called the Luleburg or Luleburg relations, because

15:00 sublective matrices play a fundamental role in optics, linear optics. Well, an alternative way of saying this is to say that sigma of s z, s z prime is equal to sigma for all vectors z, z prime. So in other words, a matrix or an onomorphism of R2M is symplectic if and only if it preserves the symplectic form sigma. From this follows, by the way, that symplectic matrices form a group, and the group is often known by SPL. This is the symplectic group. There's a crucial role in halitonium mechanics. It's one of the classical Lie groups because it's closed in the group GL to an R of all invertible automorphisms. Being closed, it's a Lie. We call it a classical Lie group. It's easy to describe its new algebra, but I won't do it here because we won't need it. And why is it so important in Hamiltonian mechanics? Well, take a Hamiltonian function. The term is a flow. Okay? That means flow. Now, if you calculate the Jacobian of this flow at every point, you get the matrix. And this matrix is always symplectic. That's another way of saying that Hamiltonian flows consist of canonical transformations, that's all. And a symplectic matrix is just a linear canonical transformation, that's all. It falls from this here condition that you must have determinant of S equal to plus or minus one. In fact, this is a little bit delicate to prove, the determinant of a symplectic matrix is always equal to 1. Hence, symplectic matrices are volume-preserving. Volume-preserving, and it follows that Hamiltonian flows also are volume-preserving. That's Lewis' theorem. Okay? Yeah. Come back to this.

17:30 Good. any questions here please feel free to enjoy it fine so let's take the camera now okay well okay let's have a look at this ellipse right here you can ask the question okay i'm not going to no no i don't be afraid to do any anything here on the on the screen Once I did that in a conference, I did not realize that I was doing that. Okay, so here, let me take an ellipsoid again here, and with symplectic capacity, we go to 2 pi over lambda. that we write this like pi r squared. These are not the words that you find. The r by r squared is going to be 2 over lambda, so r is going to be square root of, yeah, square root of 2 over lambda. This is right. Yes, okay. Suppose now I have a ball. Did I do something wrong? No, no, no. Okay, take a ball with radius R. And the natural question now is, can I find a symplectic matrix, S, and a translation, allowing me to push, to squeeze this ball inside this ellipsoid? Well, symplectic matrices, or the automorphisms that they represent, can distort, of course. Generally, linear automorphism will transform this in an ellipsoid or something like that. So, you could ask, yeah, under which conditions of R can I distort this ball? Using translations, of course, so far nothing is going to happen here. and symplectic matrices to push this ball here inside, inside the ellipse. Okay, the answer is that it is impossible if R is larger than this here. And it's always possible if R is smaller or equal to R.

20:00 I want to take that, okay, but using all the linear transformations, it's very rigid, perhaps we could do better using general symplectic transformations, canonical transformations, non-linear ones, okay? But a canonical transformation, F is a canonical transformation or, as they are called in mathematics, symplictomorphism. if, as I said, is in spectromorphism, if its Jacobian matrix is in SDN for every Z for which it is defined. Now, okay, the thing is the same, and this is a very deep theorem which is called Gromov's non-squeezing theorem. It is impossible to squeeze a ball like this inside this ellipsoid using very general canonical transformations, unless the radius of the ball is smaller than this number r, which is here. This seems to be in conflict with the usual interpretation of Louisville's theorem, because Louisville's theorem says that Hamiltonian flows or cyclotomorphisms do not change volume, but not changing volume does not mean not changing shape. So, say, ah, perhaps using cryptomorphism, especially constructed, one would be able to just form this in a kind of sausage so I'm stressed it inside the ellipsoid, but no, it does not work. By the way, if you use general volume preserve different morphisms, it's much easier, you can do it. But simple morphisms, that is canonical transformations, are of a very special nature actually. A variant of Gromov's non-squeezing theorem is the following. Take, and this will be very close to the uncertainty

22:30 Take a plane of conjugate variables in phase base. That is a plane of type XJPJ, the same index, okay? They're called conjugate variables. And consider a ball, again, with radius R in phase base. Now, if I project this ball on the plane here, I will get a circle. of course, a circle with radius by R squared, this is obvious, okay? Suppose now that I start distorting this ball by using canonical transformations. For instance, a Hamiltonian flow. This ball can be stored while keeping a constant volume, of course. But what is is going to happen, suppose it is the historic ball here. The projection, orthogonal projection, of the distorted ball on this plane would always have an area of at least by R squared. So in other words, the area of the projection will not decrease. As you can see, this is a variant of the uncertainty principle, but in the classical regime. Because if I take another plane here, X, J, and P, K, with J different from K, then the projection can become arbitrarily small. But again, the complete planes play a totally fundamental role here. Can I just ask if you're excluding all dissipative processes here, aren't you? Yes, yes. I'm talking about Hamiltonian dynamics. Otherwise, it's not true. I must have Hamiltonian flows. Yes, yes, of course. But still possibly no meaning. Of course, of course. But nevertheless, this principle puts, from my point of view, severe limitations on the possible chaotic behavior in Hamiltonian processes. You know, Penrose, Penrose's book, The Emperor's New Mind there, and several pages about Deuwin's theorem, and Penrose says more or less, yes, okay, Hamiltonian mechanics cannot be true mechanics, it does not apply to the real world, and then he gives examples just of a terrible distortions, of course, of flows which are volume-preserving, but of course he could not know when the book was written that Hamiltonian diffeomorphisms have a very special structure.

25:00 I mean, this principle, Brombo's non-squeezing theorem, was discovered in 1985, so this is very recent, actually, nobody knew about that before, and I mean, the first citations of this in the mathematical literature appear only in the 90s, and in physics, I think there are very few people who have realized that this can be of some importance, and of course, well, okay, I'm not physicist enough to to be able to judge whether this can have forage in consequences or not, but what I'm going to show today is that it applies very well to the uncertainty principle, which you can already see here, okay? Okay, good. So let's now turn to the answer to the principle. Okay. I think that everybody knows this is what this is about, of course. This is the naive version here. the textbook example you find in every books of elementary quantum mechanics here, it turns out that the true, the sharp principle is this one here. I will explain that this is because when you write this you don't take into account the covariances, that is the correlation between the variables here. This form was due, I think, to Robertson and Schrodinger, actually, I think. It was very famous sources, but Schrodinger had proposed this sharper version, which was then proven by Robertson in 1932 or 1935, or something like that. Okay, now, here I'm taking a big shortcut. it. It turns out that this here is equivalent to something very algebraic, where the standards in the matrix J intervenes. What is sigma? Sigma is what is called the covariance matrix. We'll define that in a few seconds. And this inequality is equivalent to this inequality.

27:30 This means that sigma covariance matrix plus i h-bar over 2j is a positive semi-definite matrix. And this is the symplectic formulation, or one symplectic formulation, of the uncertainty principle. It turns out that this is equivalent to saying that the phase-space ellipsoid defined by this relation capacity superior or equal to one-half of h. That is, it says that we cannot squeeze a ball with radius smaller than square root of h-bar inside this ellipsoid. This is the so-called Wigner ellipsoid, which is immediately related to the Wigner transform, as we would see, and being related to the Wigner transform is related to Bohmian mechanics, but it's a short clip. I know, but that's it. But, but, nevertheless. Okay. Where does the black people come from here? We have quantum mechanics now, and... Well, it comes from the fact that we have a phase space. Some physicists of the older school talk about mock phase space. They don't believe in phase space in but all young people like you, for instance, know that quantum mechanics makes sense. I mean, physics makes sense. I'm talking about the phase-based formulation of quantum mechanics, you know, the big, real formulation. I understand that physicists don't say, ooh, there can be no such thing as a phase-based quantum mechanics. grammatically it can be, and actually you can calculate it face-to-face quite actually as much as you like. There's no problem with that. What do you say that the precise form of the uncertainty relation is equivalent in the grammatical sense to physics? Sure. The first is something about expectation by this of operators. I'm going to precise what I mean by this.

30:00 Let me see. I did not actually invent this expression. It's commonly used in quantum optics. It's commonly used in quantum optics, and quantum optics is a part of quantum mechanics, I think. You know, in quantum optics, there are no problems, no, no, but there are no problems using this in the study, especially in the study of entangled states. This is very, very, very common, because the covariance matrix plays, of course, a fundamental role in this thing. So let me, let me talk a little bit about operator theory, so you'll see where this comes from. Good. So how do you prove this? You see, perhaps we have to remember how one proves the usual uncertainty principle. You know that the usual uncertainty principle is a consequence of non-commutativity. I know that non-commutativity is very fashionable nowadays. It's not that... I mean, it's obvious that non-commitativity plays a crucial role. You know, it was the tone of voice. Okay. The multiplication of covariance matrices is maybe non-communicated. Ah, I don't know. No, it is complicated. I'm joking, of course. okay so let's take a quantum observable i mean a self an essentially self-adjoint operator okay i'm not going into functional analytical details here this is not the place of worship so i suppose that a is defined in a dense domain here in l2 okay i'm not going to be very rigorous here is just to have an idea. Suppose, ah, yes, I'm working in some Hilbert space, so suppose I take an element of this Hilbert space, which is equal to L2 here, different from zero, and then I define the expectation value of A in the state psi, as we say, I think, in quantum mechanics. I don't like the bracket notation, but that's a personal taste of

32:30 music again. Okay, the brackets here is the ordinary L2 product. I think this is the standard definition, okay, of the expectation value here, and then you define the delta A by the following formula, which we are aware of, but it's always relative to a certain state, okay, delta A squared equal to the average value of A squared in this state, minus a squared, this is the standard definition of the standard deviation, let's call it and so on. Good, let me also introduce the covariance, the covariance of two operators a, b in the same size, so I have a second operator, having similar characteristics. This is one-half of the average value of AB plus BA. This is a definition, okay? It's a definition. And the definition is taken by analogy with the situation in classical statistic on the capital. Good. And then, yes? No? suppose that A, B, and B, A are defined in the common domain and so on. No, no, no. No, no, no. Then, we have the following result. Proposition or theorem or result. Suppose that the variances and the covariances are well defined and so on, then delta A squared psi delta B squared psi is larger than or equal to the covariances of squared minus 1 fourth of the average value

35:00 commentator AB squared in the thing. Sorry. This is the most general version of the uncertainty principle here. Okay? And you can notice that this number here is negative. Why is it negative? Well, just because AB, the commentator of AB, is anti-symmetric, of course. It's anti-symmetric. So the average value, oh, sorry, the square should be here. So this number is pure imaginary, so when you square it, you get a minus sign, you then cancel this one here. And how do you prove it? I'm not going to prove it in detail, but the proof is quite classical. It's based on Kosh-Schwarz inequality. First, First, we can very well assume that the average value of a is equal to zero, okay? You can center on this down here. In which case, it's the same thing here to prove that, or what you get is actually that a squared psi b squared psi It is larger than or equal to, I'm just doing that to show you, show you something here, to A, B, Psi, Psi, and this should be square. Yes. Okay. Doing some manipulations, actually. You show that it is sufficient to prove this. Now, if you prove the classical proof of Heisenberg's inequality, the first one, the naive one, one has the following. One writes that AB is equal to one half of AB plus BA plus one half of AB minus BA that is one half of the commutator and one puts this inside here so this leads to of course what we get is then taking the moduli this one gets one half of the

37:30 of a B sine squared minus, where did we go? Yes, minus one-fourth of a B limitator here, squared sine. And one says that, okay, this term here is positive anyway, positive or zero, so this is going to be larger than or equal to this here. And then one takes a equal to the position variables, b equal to the momentum variables, and one gets this. In other words, by forgetting about the covariance, one does not get this relation. Observe that this here is not invariant by linear transformations. Well, okay, you can swap p and x, that's all. You can rescale the p's and the xj's, then it won't change. behaves very badly under general linear transformations, and even under subjective transformations. Whereas, this one does not, because we have this. So, in a face-based perspective, the only valid or good form of the uncertainty principle is this one here. I'm not going to prove this here because of lack of time. difficult. Actually, what you do is that you rewrite this inequality, taking into account the definition of the covariance matrix here. Sigma XX, that's just delta X squared. Sigma XP is the covariance of X and P. This is the transpose of this, and this is the variance And that's all, in the general case, it's the same thing. And by exploiting the fact that this is positive, semi-definite, you show that this condition is equivalent to, let me write it here, to these relations together with delta pj, delta xk for j difference from k It's linear algebra. It's purely linear algebra.

40:00 But what then becomes apparent is that we have something, something which is invariant by symplectic transformations. Why is it so? I claim that sigma plus i h-bar over 2j daughter or equal to 0 is equivalent to st sigma s plus i h-bar over 2j positive semifinal. And the proof is almost obvious if you remember the definition of a symplectic matrix. I can left multiply this by ST and right multiply by S. This will not change this inequality because SST is positive definite itself. That's one way of saying things. But then, you notice that STJS is by definition of a syntactic matrix equal to J, hence where it is. So, this proof is almost a triviality. So, it's not obvious at all, if you look at this, that this should be simply the environment. But it is, if you notice this here. Now, by doing some calculations, not very difficult calculations, you evaluate the symplectic capacity of this ellipsoid, here, here, here, take a look, ellipsoid associated to the inverse of the covariance matrix, here. This is what I call the Wigner ellipsoid. b sigma minus 1. It's defined by simple conditions. One half of sigma minus 1 z if you're equal to 1. But using this condition here, it's easy to calculate the eigenvalues of g sigma

42:30 minus 1, then to order them, and you find that the symplectic capacity of this ellipsoid is at least one half of h, and conversely. So in other words, this is equivalent to this condition, which is in turn equivalent to the general uncertainty principle. So this is a very concise and factable form of the It's a topological formulation, and it allows you to define a coarse graining of phase-based, not as people working in thermodynamics do. You know, they take small cubes and all this. Cubes is sometimes convenient, but cubes do not have any invariance properties. If you transform a cube by a linear transformation, you get a parallel or something. Actually, here we're talking about ellipses, which are much smoother. And what's inside these ellipses? Nothing, let's say. These are the smallest units, in a sense, which make sense to talk about in front of mechanics, in phase space. It's nothing, but it's nothing contains lots of information. If you calculate, for instance, the value of a Hamiltonian, you get the ground states, the ground state energy, for instance. I can't just say you get the quantum potential. Can you get it from that? You can. I mean, I'm not surprised because of your paper, I made the link between Wigner-Moyal and on the potential and what's the link of all this with Wigner-Moyal to show you how easy it actually is, you see, nothing, nothing, ah, okay, now I'm going to need the pins. The Winkertransform, I think, well, this is one of the most fundamental objects in phase-based quantum mechanics. It's called also the radar ambiguity function, I think.

45:00 It's also called the short-time Fourier transform in time-frequency analysis, and perhaps it has even other names. Sorry? It's called a Lung-Comb in Cabot's language. You know the program guy, Dennis Gabor, the Imperial College, a long time ago. How do you call it Gabor? G-A-B-O-R, Gabor. He got the Nobel Prize for Homography. Ah, Gabor! Sorry, what did you say? I know, I'm getting older. okay okay okay just like variance of this okay so I think something which is really nice here it works for Psi in L2 of course but actually actually you can define the vaguely transformed by catalyst for Psi being even a temporary solution so the definition is quite general you can vaguely transform Ah, I'm almost here. Now, let's calculate this. Don't write it. No, but Psi is of this form, just from each other. This is a so-called squeezed coherent state, okay? Squeezed coherent state. It's a little bit more general than the fiducial coherent state. You know this. It's a Gaussian. It's a complete Gaussian. x is a positive definite, this is supposed to be the big x here, matrix x, and I'm going to add, oops, something here, y is an arbitrary matrix, the x has supposed to be positive here, because otherwise the integral would not converge at all. Well, if you calculate this here, you find that the Wigner transform of this Queens-Bergen state is of this type here, where G is a positive definite matrix, which is, this is miraculous almost, which is in addition symplectic. Actually, it's quite easy to see that it's symplectic if you use the following factorization, and I forgot to write it here, wait a minute, S, we have G equal to S T S, where S is equal to square root of X, 0, X minus one half, Y.

47:30 and x minus 1 a half. Well, doing this, it's almost trivial to see that the first one who made the calculation is a guy named Bastians. He works in Holland. Sorry? You know him? I don't know the name sounds like that. Yeah, yeah. He works in, I think, Eindhoven. Thank you. Also a guy working in signal theory. You know this, it's a, and, well, this is trivially, simple acting, and if you do the product, you see that you have this here. Yes, but look, look, look, look, look at this here. What is G, Z, Z? This is very interesting, because G, Z, Z becomes S, T, S, Z, Z, that is S, Z, Z. Yeah, but, well, okay, let's compare that with the minimum uncertainty linear ellipse, which I call quantum blobs. I was in the name of that the other day. Suppose I have one half of n, z, z equal to one half of h. you know it turns out that N, and where N is the inverse of the covariance matrix, it turns out this is a theorem also in subjective geometry, that N is in this case humane, as inflective and that I can rewrite it like this here so in other words the ellipsoid defined by this collision, one half of this equal to one-half of h-bar is just the image of a ball with radius squared root of h-bar by a symplectic transformation. So there's a one-to-one correspondence between these quantum blobs and the Wigner transforms of squeezed coherent states.

50:00 In other words, if a coarse-drained phase space, by minimum uncertainty ellipsoids is the same thing as giving all these standard or standard sorry squeezed coherent states and we know that these squeeze coherent states can be used as a infinite dimensional basis for wave functions in the l2 that's got more analysis as you said and all this. So here's the link. So in other words, the uncertainty principle is that geometric statement saying that coherent states are fundamental. Now, you know, by the way, if I take x equal to here, x equal to the identity, y equal to zero, you'll take what is called, I think, quantum physics the minimum uncertain state. I don't know if that's exactly the reason, the same reason we have this. But this syplectic camel makes actually the link between those totally apparent. So the datum of a quantum phase base is actually equivalent to the datum of all these squeezed states. So now of course I do not know which interpretation one can or you would kind of quantum conophases. Can I work with that or not? There's a lady, Daniela Dragoman, who has published a paper a couple of years ago in Physica Scripta, I think, in Physica Scripta, where she used my quantum blocks to propose some kind of axiomatization of quantum mechanics. I But there is one interesting thing here. Suppose you take the sum of two species coherent states. You obtain something which is in L2. What is then the geometric interpretation? I mean, what I mean is, to a sum of coherent states should correspond some kind of set in phase space. With what? Should it be the convex hull containing the both ellipsoids, or should it be the intersection

52:30 of all the ellipsoids containing those two ellipsoids? I don't know. I don't know how to act in my quantum phase space. I don't know even if it makes sense. This I don't know. and actually these things are when I visited two years ago at Berkeley I had already presented more or less some of these results perhaps you remember it turns out that there are very important and interesting links with something else which is called the Schrodinger equation in phase space about which I talked last year here in so I want to talk about that again Schrodinger equation in phase space. OK, many I think it would be very upset by this, of course. But I mean, this exists, and it's very easy to construct one by mimicking the Bergman-Transform stuff and all this. All you have to do is actually to construct an isomorphism, or an isometric, sorry, from L2RNX, the usual human space for quantum space, an isometric of this onto a closed subspace of L2 of R2NZ. And this is very easy to do. You can do it in infinitely many ways using the so-called wave packet, And it does not contradict the Ston von Neumann theorem at all, because many people say, yeah, but you cannot do this because of the Ston von Neumann theorem, you know, about the uniqueness of irreducible representations of the Heisenberg algebra, because Ston von Neumann theorem says that you have a unique representation of the Heisenberg algebra in L2, R, and X. It says nothing about isomorphic copies, of course, of L to R and X. And it works, and this is published the results in two papers in the Journal of Physics A, which is one of my favorite journals, because they are really quick, in 2005. Oh, okay. And by the way, I'm working now with, If you can join the paper with Isidro, Jose, Isidro, a Spanish physicist, because he opened

55:00 my eyes on something, it amounts to have a gauge theory, to use this, showing a question face with, a gauge theory, where the notion of gerb, or gerb, and it's called in English, plays a crucial role, I don't know if you know what, all Freddie knows some of the journalists. It's a generalization of the most, you know, five-way bundle, where you have core cycles, and then, it's a little bit of Czech homology, that's one way of seeing things and all this, and I didn't realize that, actually, he does the things like a physicist, so I have to be behind, of course, and do it correctly, and then. But I think, so it's totally related to this uncertainty principle, so that's why I called my talk, nothing, it's much ado about nothing, but really from nothing, you can do lots of stuff. So, perhaps it's, I don't know, this is great, it's hard, it's hard, it's hard, it's hard. Okay, so thank you very much. And if you have questions now, thank you very much. I'm very happy to answer your questions. Just so you don't know how to add, you have squeezed states. But the squeezed states, yes. but the corresponding geometric... But as you said, of course you can use them as faces for building space for your states line and then to form some states that would need linear combinations Yeah, exactly. Of course I could transport the structure, of course, on my face space, but I mean, is there any intuitive way of seeing things? I'm not sure. So, if you have ideas, they are very welcome. Perhaps optics would give you a little way of looking at it in the cheek and be regarded as a nautical shift. Yes, yes. Of some optical device. Yes, yes.

57:30 And then that's squeezed, stays again. It is not broadened, and it's of course directly related to your blocks. Okay. That's the communication in a way. Exactly, exactly, exactly. Yeah, yeah, yeah, yeah. That's also the basis, which was... I mean, I would have a basis consisting of ellipsoids. A basis for what? Would it be for arbitrary sets in art? This might be much more delicate than it seems. I don't know. I'm thinking about it. And I think it's a cover of the face-space. It's just a cover, it's not a partition or something. No, no, no, it's not a partition, it's a cover. It's a cover, absolutely. It's a cover. You can, by the way, calculate the dimension because you can identify this phase space with the inhomogeneous symplectic group divided by the stabilizer of, I think it's OM if I remember well. So by dimension culture, I get the new dimension of this, but I don't remember exactly. I made some calculations. I don't know if it's useful now, but I mean, it gives a geometric picture of what is going on. It would be also nice to try to see if one could improve the Weill theorem on a number of eigenstates, which is contained in a phase-based region and so on, you know, by form and all this, by making a P2 notion of symplectic capacity, which the notion of symplectic capacity, by the way, has nothing to do with the notion of volume. It's something which has the dimension of an area, Yr squared. And this is a typical feature of symplectic geometry. Symplectic geometry is the geometry of area, not of volume that's why you have so many misunderstandings about the view in general we have about the new real theorems the interpretation of your reals here which is much more restrictive in amitona again since this uh three state works in optics yeah you use the you use the fox

1:00:00 Yes, it's more or less equivalent. What does the capacity do in Oxford, sir? I don't know. I just wondered if you... I don't know. This is certainly easy to compute, to calculate. I mean, you see these states actually, oh, you don't need something as, you don't need them all, of course, to form a basis. You can start just with the fiduciary or standard squeeze state, which is, if I... Psi zero, it's not, I think, normalization constant. Yes, that's right. Okay, this is the standard, squeezed, standard coherent state. It's an eigentension for the harmonic oscillator with mass and frequency equal to one. you define psi z by t hat psi zero well this is a Heisenberg pi operator okay so you get the whole family an infinite family of coherent states and these can be used as a basis I have a problem with the transformation equation which one? right at the top this one here right-hand side. Z is XP. Oh, sorry, I should have, I should have recalled my notation. Yeah, yeah. This is, this is XP. The phase space variable, you know, as a vector. Yes, yes, yes, yes. The integral is being taken over, Rn. Oh, you know, I think most people here know that this is very appealing because it plays the role of a joint distribution, although it can take negative values, except when size goes, yeah.

1:02:30 That's a theorem due to Nelson, no, yes, Janssen, Janssen and Hudson, sorry, Hudson, who showed that this is non-negative if and only if Psy is a Gauss man. In all other cases it contains negative values. So it's sometimes referred to as a quasi-distribution. There was a volume edited by Basil where, a few years ago, there was a talk by Feynman, I think, he was talking about negative probabilities. I don't buy that because then you are paying more words, of course, because the definition of probability... All he was doing was saying I can't see any problem in using negative probabilities provided it doesn't come out in the final result. He's absolutely right. By the way, this is also the symbol, the vital symbol of the density matrix of a pure quantum state. And the symbol of a mixed state is a convex sum of the transforms. I mean, this is an absolutely essential object. I mean, okay, I'm not going to talk about all the wild calculus here, but I mean, it's a natural notion in wild calculus. And as you discovered in a paper, there's a fundamental connection with Bohmland. And it's a fun amount of connection with the von Neumann in the poems. Yes, also as you pointed out, basically. And the poems. And, yeah, this would be very interesting to discuss. How we can start from nothing, face-based elipses. Can you quickly say how the connection to Bohmian mechanics is? One says this in two words or three? Two words. Three words, good. No, I don't think I can say it. Characteristic functions. Yeah, well, it's characteristic functions, but then what you... Oh, yeah, the idea is that you take the distribution, being the distribution, you then form a marginal distribution

1:05:00 by integrating over x. What was it? No, x. so that you then find that the probability is the transport of that marginal distribution you then work out a mean momentum with that distribution and you find that that mean momentum is exactly the bone momentum and then you look at the transport equation for that momentum and it's the Schrodinger equation yeah so that must have really something to do with optics Of course, right? Correct? Yes, of course. But everything is optics. Yeah, it's true, right? Of course, everything is optics, right? Of course. But that's what I'm interested in. And since optics is simplex geometry, everything is simplex. I was very much surprised by Kromov's non-squeezing theory. I didn't know that. Few people have, even among mathematicians, have heard about that. Well, now it's started a little bit in symplectic topology, but it's very new, just 20 years old. So it's a kind of shape-preserving that has. Yes, yes. From my point of view, it shows, makes think at least, that Hamiltonian dynamics has a tendency to be less chaotic perhaps could the thing it is i don't know it is but of course uh atmospheric emotions are specific more or less i don't think you can call them uh hamiltonian so they're not conservative of course does not apply to the tradition it's horribly complicated in the general case yes it takes the 2030 how it goes Well, to prove it in the case of linear transformation, it's not too difficult. You diagonalize and you compare the spectra. You diagonalize, actually, not in the usual way, but you take, oh, there's something which is called the Williamson's theorem. It goes back to 1937, but some people say that actually it was interesting in Biaschow's work. Say that suppose you have a matrix M which is symmetric and positive definite.

1:07:30 Well, of course you know that you can diagonalize it in the usual way. But the Williamson proved that you can diagonalize it using symplectic matrices. And the D here is of a very special sort. It's a type, lambda, 0, 0, lambda, where lambda just consists of lambda 1 and lambda n, where the lambdas are the eigenvalues of jn. And using this, actually, you can prove a Bromwell's theorem in the linear case. No, the lambdas which are here, no, the lambdas are the eigenvalues, no, the absolute values of the ideal values of J.M., the same as I had written down there in my... Right here. And, okay. Is the implication of what you're saying is that if you find yourself a single particle in some Hamiltonian system and you say you make a measurement and you combine that particle to one of your small volumes and then you allow that to evolve classically, that there isn't any difference to quantum atom? Precisely. Mm-hmm. You don't even need the quantum retention. That's very interesting. But don't forget that instead of . I mean, in the classical case, I mean, the uncertainty is due to measurement errors and things like that. Well, it's your original. If you stay in space space, you don't need the quantum retention. That's what my talk said. I'm sorry about that.

1:10:00 And I showed in one paper, actually, that you have, you can write an uncertainty principle in the test case as well. I mean, there's no decrease of the thing. Is this not a challenge when you get more than one part? Yeah. So that's what you can't do in the moment. Exactly. I mean, it seems like in one mechanics you have a universal lower bound, which is H-bar. It's a fix for all systems. I don't know how to interpret that. At the beginning of the universe, God added everything, but he only did it to that procedure. I was tempted to say that. I did not want to see it ridiculous. It's like, it's if, like if, like. It's very interesting because you told me over when you got into it. Yeah, now, the camel, by the way, you know the origin of the camel. It's because of Mark 25, or was it the Peter 27? I don't know. It's harder for a rich man to go to heaven than for a camel to go through a nice needle. So in this case, the camel is symplectic. And that's all, that's all. I think it was invented in 1990 at a congress, or a congress in Nancy, which will be for us about to get the topologies and so on, that's a different way. It's actually a, Maurice, before you close, is it possible to say a little bit about how all of this picture connects with the properties of the covering space and particularly the covering, the double covering in the face space by the metaplectic group No, because it's totally independent It's a totally independent issue But still still perhaps not totally No, the metaplectic structure that's something else What's the metaplectic group? The symplectic group a Lie group. It's connected. So it has coverings. I mean, it's a homotopic group. It's isomorphic to the integers, to Z. So it has covering groups of all orders, right? And in particular,

1:12:30 it has a double covering. The double covering can be realized as a group of unitary operators, which is called the midflectic group. And actually, the Schrodinger equation can be derived from... It does connect what you want to do, because that's precisely how you get the Schrodinger equation. Exactly. But of course, there must be some links, but I mean... One can't. I haven't explored them yet, and I don't think perhaps it would be the idea, so that's why I'm... From the optical point of view, it's obvious that there is a link, namely. The magnetic representation gives you out of a given light distribution the new light distribution. With the Wiener function you detect the new distribution out of the old distribution. And of course both ways characterize the optical system in between. So, actually, opticians have diagrams. They do not talk about the retablectic terminology, but it shows up. This intervenes in the Wiener function anyway. Of course, because if you take W psi, you take SZ, then this is W of S hat minus one one of the two operators, oh no, it's a bit more complicated than that, but anyway, yeah, corresponding to the metabolic group, so these are co-variants from us, I think I wrote something, something wrong, written like this, ah, anyway. Yeah, on some search, it's true. Yeah, on some search, it's true, yes. thank you thank you i'll throw that but that's a but I think no i was just before that We thank both the speakers again this morning. It's quite exceptional.

1:15:00 I should collect the pens. Oh, yes, yes. It's Ernst. We have one sponsor. What about you? We'll let me tomorrow morning. Oh, you're immortalized.