Géométrie et Physique — Part 1

0:00 I'm going to be située dans la lignée des travaux de Cavallès. Je ne vais pas du tout vous présenter une nouvelle théorie, une ligne ancienne, mais plutôt, je dirais, une grille de lecture géométrique de la physique. I would say a little Magritte lecture, but I think that it is quite universal. The idea that I developed is that the physics theory is the geometry. That is to say that all the concepts of the physics theory can be expressed in terms of the geometry. And the physics, for me, my opinion is that it depends on the metaphysic. It is to say that when we build a theory of physics, we talk about some metaphysic principles that could be the principle of inertia, the principle of symmetry or other, and that these principles are always a geometry geometry that will determine the way in which the theory in question can be formulated in a geometry. Je vais donc présenter de la géométrie, bien entendu, mais aussi comment les diverses théories, alors diverses théories, je parlerai de la théorie Newton, évidemment, je parlerai de la relativité, restreinte surtout, générale aussi, de la physique quantique et puis un peu des nouvelles approches qui sont les théories des corps, la supersymétrie, la gravité quantique, la géométrie non-commutative. and also I will try to show how, even if these are different theories, considered as different, as well as the differences of formulation, I will try to show how we can assemble them in a common, with a common language geometry. and the dream would be to have a sort of meta-langage, a sort of theory, not not a theory universe, but a cadre universel in which we could enter all the theories that exist today, with the idea that, in exploring well this cadre, which we might find within the theory unitary that we look at the moment. Well, it's not possible to associate geometry and physics,

2:30 because we can do that at least at Pythagore, maybe in a way a little bit more scientific than today, but also Pythagore, Platon, Galilée, and of course he wrote very explicitly, all this is geometry, but it seems that the first one who has really formulated the theory geometry of the world, is Newton. But it's not geometry, because it's more based on an existence of a sort of material universe, with its tourbillons, and then we know that it's a theory which has not been retened. So, I think that the fact that we date the birth of modern physics, often at the principality of Newton, is not at all something that is an algorithmic, but it is a fact that really, the modern physics is born when it has been geometry. Well, what did Newton do? Well, he introduced really a model for physics. This model, which is explicit in the Principia, which is one space and one time, all two geometrised. The space is geometrised in being assimilated to the space, R3, because at the time, all these new geometries, as for example, the geometries in a manner, So if Newton wanted to geometrise the space, there was no choice. There was only one there. And the time, he was assimilated to the right real. So we can obviously talk about this. There is already something here. For example, there is a structure topological, that in this structure topological, there are dimensions. the dimension 3 for the space, the dimension 1 for the time, that if, for example, we are interested in the line of time and we know that it is the dimension 1, there are two topologies also, the ouverture fermée. So, it puts already, it puts already in place a case in which we can solve problems. So, it is not at all anodin, even at Newton's level. So, this space has already properties which are already symétries. And, of course, this notion of symétries, you can doubt it that it will not always appear

5:00 in all the theories. Evidently, it was already in Platon, since you know that Platon was, I would say, in the sense of the term, by the idea of harmony. For example, you know that he thought that the world was, The world terrestre, the world sublunaire, was composed of four elements, which were composed, which interaginated. And he had associated with these elements the polyèdres fondamentals, the solides platonicians, which are the polyèdres réguliers, which are the cube, the octahèdre, the tetrahèdre, the do-décahèdre, these four arms, which in fact, I think at the time, were the best way of mathematics and geometry to represent the symmetry. which is very powerful to represent the symmetry, it is the Torino. At the time, well, here, we represent the symmetry by these polygons which have the symmetry or the sphere which there are even more since we have the symmetry but we have not the tools more sophisticated. the symmetry is already in Platon and here, in the space and in the time newtonian, we have the symmetry. What is the symmetry? In a very general, a symétrie of something is a invariance of this something, of this space, in some transformations. The problem is to know what are these transformations. For the collieres, it's rotations. For example, a cube, you make a rotation of 90 degrees, the cube becomes something that is identical to the beginning. The sphere, you make a turn of any angle, it remains identical to it. Well, the space of Newton, you can see it, it remains identical to him. You can see it, it remains identical to him. So it is homogenous and isotrope. And this homogeneity, for example, is very important. It means that all the points of the space are strictly equivalent. There is no point privileged. So, just in saying that the space is homogenous, it means that there is no centre of the world. There is no more borders or borders. So, it exclut, by this simple enoncer, which was implicit, this simple enoncer exclut the most of the models that we had before encountered. Isotropie means that all directions are equivalent, there is no direction privileged. Today, we could say that, for example, that there is no rotation.

7:30 Because if there is a rotation of space, it would mean that there is an axis of rotation. a direction privilégiée. Aujourd'hui, on sait que l'espace est en expansion, cette expansion est forcément identique dans toutes les directions. S'il y avait une direction où l'expansion est plus rapide, ça serait une direction privilégiée, donc ça briserait l'isotropie. Vous voyez que simplement énoncer homogénéité et isotropie de l'espace, c'est quelque chose de très important qui est riche de contenu. Alors, rotation et translation spatiale, aujourd'hui, on sait que ce sont des transformations isymétriques, that these transformations can be composed of a transformation in a unit which is simply a rotation of an angle of zero or a transformation of zero, and that each transformation is asymmetrically, and these are the actions of the group. So, we know, and in general, all the transformations that we are going to use by the next, we know that they form the group. And so, the group, it's the tool, and more precisely the group of Lie, it's the tool mathematic which, today, serves to express the transformations And if we are interested in an invariance of certain transformations, we are going to say that there is an invariance under a certain group. And you will see that in a large part of the physics, it comes to discover the groups that are invariant to tell or tell system. So I say group here, we will talk a lot about the next. And what is important, I will talk about it also, but I anticipate, is that when there is a group, there is an association with the Argen, there will be a way to treat the physics of an algebraic, and then also an association with the geometry, it is to say that a group of lines, you will see that it is a variety, a variety that generalizes the space, the space is a variety, the space-time is a variety, but the notion of variety is much richer than the space, so there will always be a sort of correspondence between space or variety, group and algebra. the three points of view, algebraic, geometry, and the theory of the group, and there are others also, but it's those that I find the most important, to describe something, a theory, a model, or something. Here, the group that we are interested, it's the group of the rotation spatiales. The rotation spatiales, it's the same group SO3, it's the group of three dimensions,

10:00 the rotation in space, according to all axes, and then the group of the transmissions, It's the group of translations. There are three axes possible translations. So it's a group of dimensions 3. The translations, it's R3 and the rotations, it's SO3. And in fact, the space is homogenous, it is invariant, both sous the group of rotations and sous the group of translations. And in fact, it is invariant sous a group more large, which is the product of the two, and which we call the group of Euclide, because it is the group of symmetry, the group of invariance of the space of the grid. Of course, more an space is large, more there is a dimension, more the group of invariance will be large. So if we take the time that he has one dimension, it is also homogenous, it is to say that, at this point, it is a line and the only transformation on the line is the transformation. that the physics is invariant compared to the transformation temporal. On one dimension, the rotation is simply a symmetry. So, is there a symmetry in terms of the time? In the physics Newtonian, In the physics, as we consider today, in taking into account the reverse phenomena, it is something that we discuss today in this discussion. So, it is true that the group of translators, it is a very simple and a dimension. And it is also, finally, the right real. So, it is also an ensemble of the real. And we call RT to indicate that it is the group that we consider as H100 on the time. It is very important that the time is thermogène. It is to say that all the instant are equivalent. It means that there is not an instant first in the graph Newtonian. So, for example, there is no end of the end or the beginning of the world or something like that. There is no end of the end of the world. All the instances are equivalents. It does not mean that we could have at a point of time cyclical in a model like that. It is not the model of Newton, but it could still be compatible with the homogeneity of the time. And that means that the physicality is the same today, tomorrow, tomorrow, in a billion years or in a billion years. So, if we do an experience today, we can do it here, we can do it again, we can do it again, we can do it again, and we will always do the same result.

12:30 And you know that the foundation of the physical, it's that. It's to say that we can express a way that is universal, and then verify it at any point. It's the homogeneity of the time, but also at any point, it's the homogeneity of the space. And also, if you want to turn on your laboratory, it's the isotropy. homogénéité, cette symétrie de l'espace et du temps, on ne pourrait pas faire de la physique. Et dire qu'il y a un espace et un temps comme ça homogène avec cette symétrie, c'est dire qu'il y a un univers. Donc tout ça, on peut le résumer en disant que Newton introduit la notion d'un univers. Qu'est-ce que c'est que l'univers ? C'est finalement ce qui est garant des lois universelles. Et on ne peut faire de la physique que s'il y a des lois universelles. Par exemple, non pas certaines lois valables sur la Terre of the universe of the universe of the universe. So I would say that the important thing of Newton is to introduce this notion of universe, of universe, and he enonce the first of these words universe, which is, as you know, the law of the gravitation universe. It is very important the details of this law, but it is important also, it is also the fact that it is precisely universe. And I think that it is for that, because Newton is an universe, that we can say that he is the founder of the physical universe. Well, of course, these symmetries are very implicit. When we learn the physics Newtonian at the university, we do not say that the space is homogenous, that it is a group of symmetries particular. But I'm going to remind you, because when we look at the generation of the physics Newtonian, these groups can change, or change, or change, or change, can be modified and it's a way to see how the physics, the different theories of today, the physics quantic, the relativity, and other theories modern, can be considered as deformations in the physics newtonian. And when I employ my deformation, I put it here in a sense vague, but you will also see a little more later that it can be to be put into a mathematical sense, very defined and very precise. I have a question. Is this principle of universal universalism a-t-il a non-précis? A-t-il a non-précis, this principle of universalism, which is defined by the physicality,

15:00 is it a non? Is it why you have the word called without the physicality? No, but it's obvious that when we do physics, we always tell you that there is a method experimental And if we have a theory, we can verify it by an experience, and the experience we can refer to it at any point, at any point, at any point. But there is no more possible for this thing? I don't know. There is more or less the principle propernician, but it's not all at all. I don't know. I don't know. I don't know. So Newton, he expires his symétries geométriquement. In fact, when he uses, and when we use today, the coordinates to describe the space, finally, he does the algebra and, of course, he expires his symétries geométriquement. Today, as I told you, on les exprime par la théorie des groupes, qui est un produit très puissant. Alors évidemment, comme vous le savez, chez Newton, il y a un espace et un temps séparés. Dans quel sens ils sont séparés, on va le voir. Mais malgré tout, par commodité, on peut dire qu'il y a une sorte d'espace-temps newtonien qui est trivial, c'est simplement le produit. C'est-à-dire, on va dire, il y a une espèce d'espace-temps, c'est l'ensemble de toutes les positions à tous les instants. It's not an space-temps in the sense of relativity where there can be a mix between the spatial and temporal dimensions. At Newton, there is no one. But the fact of saying that it's a product like this, it means that, finally, we can't make a mix. And I would say, it's a bit abusive, of the space-temps Newtonian as this product. And finally, we can say that the symmetries that we just saw, Well, those of the space and those of the time are the symmetries of the space-temp newtonian. Of course, they are separately at the space and separately at the time. There is no one that combines the space and the time as there are in relativity, but I call them the symmetries of the space-temp for having a sort of language in general. But what is interesting is that these are not the only ones. It's to say that the symmetries of the space-time Newtonian, there are other than those that we are trying to see. There are the symmetries that concern the composition of the vitesse.

17:30 Because when we define the space Newtonian, the space and the time, we define the same time the cinematics. It's to say the movements, at least the movements that are free, in the sense that there is not a force, for example, an electric force or, at the time, a gravitation force, which comes to create, accelerate, freer, and modify the movements. So, what is the movement free? What is the movement free? We put a particle in space and we look at how it evolves. Since there is no force in which there is no force. The way in which the particle evolves depends on what? It depends only on the properties of the space and the time. Autrement dit, le mouvement libre des particules, c'est-à-dire la cinématique, c'est une caractéristique de l'espace-temps. C'est une caractéristique de l'espace et du temps, et des relations entre l'espace et le temps. Donc, quand on fait de la cinématique, on fait de la géométrie de l'espace-temps. Chez Newton, je me répète, c'est un peu amusant de parler de l'espace-temps, mais quand même, dans l'absurde, and their reports, to make the cinematics and to make the geometry of the space-temps. So, what is the vitesse? What is the vitesse? It's a length of length divided by a length of length, and that is something that is something that is what is the space and the time. So, the way the vitesse is composed, what does it mean to be composed? For example, if I'm in a train that goes to 100 km heure, If I launch a caillou that goes to 10 km heure, how much will the caillou? By the way, it goes to 110 km heure. So, the vitesse is composed in addition. It is something that looks intuitive, but when we explain it, we explain it by a certain law. This law is the law of transformation of the vitesse. These are also transformations. It is to say that we have a vitesse that, by the end of the other vitesse, will be transformed. the train transform the 10kmh of the caillou into a higher speed, 110kmh, compared to the road. So we have a transformation. This is called a transformation of inertia. In English it's called boost, as the word is more condensed, which we use quite often. Here it's called boost galilien, because the cinematic newtonian is the same thing as the galilien, since Galilien had already introduced

20:00 a large part of its characteristics. So, we call it the transformations of the galiléan or the boost galiléan. And you know very well that they are trivial. When I say the widths of the height of the index, it is actually something that is trivial. But it does not matter that it is part of the symmetry of the space-temps Newtonian. So, in fact, when we talk about the symmetry of the space-temps Newtonian, We have the first symetries of space and time that I have talked about, which form, as I said, the group of Pines, and we have to add the boosts galiléens. And it forms a bigger group. Remember, it was three... It forms a 10 dimensions, in fact. The group 2, we call it the group of Galilée. It completely caractérise the physics newtonian, and we call it the group cinematic, because it includes not only the symphony of space and the symphony of time, but also the symphony which contain the relationship between space and time. So, in fact, this group cinematic describes all the properties of the physics of the universe. The movement libre is a pure relation between space and time, no depends on the properties of space and time, and so of the group cinematics. Let's take an example, the principle of inertia. You know, what does the principle of inertia? It tells you that a particle, when she is all alone, she can't do anything, or be able to go in a uniform movement. If that was not the case, what would that mean? If she doesn't go in a uniform movement, it's that she accelerates. If she accelerates, it means that her movement changes, she changes, so she changes in a certain direction. If there is a certain direction, it means that this direction is privileged, because it can be the space that creates this movement. So it means that the space is not isotrope. If the space is isotrope, one particle cannot accelerate. So you see that the principle of inertia, which is something that we feel like a cinemathic of the matter, is a direct consequence of the isotropic of the space. which is something which is a consequence of the symmetry of the space-temps. I'm sorry, because it's something that we haven't always seen,

22:30 the passive inertia, it seems to be a physical property of the particles in the entire. No, it's a symmetry, it's an aspect particular, of the symmetry of the space-temps. And why the same argument doesn't affect the VPS? Pardon? Pourquoi les mêmes arguments ? On peut dire que si ça bouge, ça bouge en direction ? Non, parce que la particule conserve la vitesse qu'elle a déjà. Donc si elle a déjà une vitesse, c'est qu'on lui a primé cette vitesse déjà auparavant, donc elle la conserve. Et s'il y avait un changement de vitesse, ça voudrait dire qu'il y a une certaine direction privilégiée qui fait que la vitesse va changer dans cette direction. D'accord ? That's to say, I suppose, that the movement inertial is not due to the mass at infinity, to the mass at infinity. Attention, when I talk, yes, yes, of course, in this case-là... This action that you have mentioned earlier, is the principle of Marx. Attention, here, there is no interaction. What I say... No, no. La physique newtonienne, telle qu'est énoncée dans ce cas-là, elle n'est pas rackienne, puisque la symétrie de l'espace-temps, c'est une propriété de l'espace-temps qui est déterminée par rien d'autre que l'espace-temps lui-même. Si on était dans une conception rackienne, on dirait que ces propriétés de l'espace-temps sont en fait des propriétés, par exemple, de la matière à l'infini. Ça n'est pas le cas ici. Alors, si vous voulez, après, vous pouvez avoir derrière tout ça une autre théorie who will say that the symmetry of the space-temps are a consequence of something, I can perceive it, but as I said, no, it is not a consequence. I have not identified an interaction that would make the inertia of a particle, that is to say that the properties of symmetry of the space-temps, localement, are the consequence of something else. In relativity, we are going to arrive. and we are in physics Newtonian. So, I think, I think that we can give a formulation of the physics Newtonian which would be Mackenzie. I don't know if anyone has done this effort, but I think it's possible. But in fact, when we do that, we will get to the relativity. We can answer if it is Mackenzie or not and we will be able to do that by the next. So, I told you that the physics Newtonian was invariant compared to the 12th Galilean, I don't know if I can tell you, in all case, I can tell you, and that is a principle of relativity.

25:00 The principle of relativity is to say that, rappel you, I told you that the physics is invariant compared to the position in space, compared to the position in time, compared to the orientation in space, it is also invariant compared to the vitesse, if it is uniform. When you are in a train, you go to a constant speed, you do an experience, you have no way to know if the train is or not in déplacement. Why? Because the physicals are totally invariant compared to the uniform. And this is the principle of relativity, which has already been seen, seen by Galilée. And here, I call it the relative relative to Galilier, even though the adjective is superflu, I repeat that this relative relative tells you that the physical, all the laws of the physical are invariants compared to a change of speed uniform. It has absolutely nothing to do with the fact that the speed is composed in addition or not. So, in physics Newtonian, we have the relative relative, and we have, as I said, the composition of the vitesse by addition. When we go to the relative, we have always the relative relative, with the same enoncé, but what will change, is the law of the composition of the vitesse. As you know, it will not be a simple addition, it will not be something more complicated, it is called the transformation of the world. And we will talk about the principle of relativity Einstein, rather galiléen, but it is the same. I put Galiléen here to remind that this principle of relativity goes with the right composition of the galiléen, but the principle of relativity is always the variance compared to boost. In other words, here we have the addition of the stress. When we are in relativity, we will have the boosts which are more complicated, which are the transformations of Lorentz, but we will always have a relativity principle which is like the invariance of the physics compared to the boosts. Other than, you already know, to pass the physics Newtonian to the relativity, what do we do? We will simply change the group.

27:30 What do we change in the group? So, in the group, there are the rotations, the translations, and the boosts, and then we will change only the boosts. All the rest will remain, and it's like that we will retain it earlier. In 1905, the relativity will remain. Now, I insist on the importance of this Cinematic K, which is so constituted by translations, space-temps, rotations, spatiales and boosts. For the Cinematic Newtonian, it's the group of Egalith. Well, something interesting also, when we do the psichy plutonium, we have the habit of working with the space and the time that I have found in a sort of space-temps. But there is something interesting, and if you do a little bit of dynamic, you know what it is, it's the space-phases, what we call the space-phases, or more correctly, the space-phases. And in fact, all the cinematics and even the dynamics in the geometry, not in the space of movement, but in the space of phase. What is the space of phase? The space-temps is all the events. That you know well. What is an event? It's something that happens to an instant, to a position. It's to say, in this moment, at this place, it's an event. I don't want to give you the importance of what you need, but it's an event. Each thing that happens, my anniversary, my 30th anniversary, which is happened to a certain place, it's a certain event, etc. So you see what are the events? These are the points of the space-temps. In other words, the space phase is what? It's what? It's all the positions spatiales, I'm here in an environment, and the vitesses. And you know that a movement, a particle, for example, So we need to define, we need to specify, at an instant, the position of a particle and its speed. Three coordinates for the position and three coordinates also for the vitesse. So, gamma, it's the whole position and speed possible of particles. So it's an space that has 3 plus 3 dimensions.

30:00 And it is found that if you look at all the movements of the possible possible particles, there are a lot of, because it depends on their position initial and their position initial, there are more than two points in this space-to-face. It is to say that gamma, which is defined as space-to-face as an ensemble of positions and vitesses, is also the ensemble, all the movements possible of particles in the space-temps. And that, it's very general. It's why it's better to call the space of movement, because the space of phase is a term of ambiguous, while in all the theories that we can do, all, it's really large, not only in the city of Newton, but that it's not quantified, it's to say that it could be the relativity, it could be the movement of a sphere that turns, any system dynamic, there will be a certain space of movement, l'espace des phases, et ce mouvement, cet espace a toujours une structure particulière qui est celle de variété syntactique, je vous expliquerai tout à l'heure ce que c'est, mais c'est très important cette structure universelle parce que c'est la base de la quantification. Et quand on a un système dynamique, alors par exemple le système dynamique de la particule in the space moutonien. We will then go after to build a theory quantic of the same system. We will pass from a theory classique to a theory quantic. For example, from the mechanics classique to the mechanics quantic, from the theory of the champ classique to the theory of the champ quantic, perhaps from the gravitate classique to the gravitate quantic or other. In these cases, the first thing to do, is to find the space of the movement which caractérise in question, to be able to see what is the structure symplectique, there is always one, and then to apply some procedures which I will be able to identify, I will talk about it at the end. So, every system cinématique, and even after dynamique, is expressed by the structure symplectique of its space of movement gamma. So, what is this space of movement gamma? Well, what is interesting, What's interesting is that we can find it through the film. And why do I say that? Because the film film K, which I've already talked about, is so important that from him, we can find both the space-temps and the space of movements.

32:30 So, K is a group. It's a group of transformations, as you have seen, rotation and transformation. The group of transformations, it's a particular category called the group of Lee. The group of livres have a propriété, and in the same time that these are groups, there are also varieties, that is to say, in the language common, these spaces. So the group K is a group of 10 dimensions, which is very large, but it is also an space. And it is true that in the groups, we know how to make, we have often groups. For example, in the group K, which is very large, there is the group of rotations. There is the rotations spatiales. there is a group of translations. It is true that we can do what we call a quotient of a group by a group. It is a mathematical operation very well defined, I will not enter into the details, but the result is that if we do the quotient of a group by the group formed by rotations and boosts, you have seen what it is, what? On trouve quelque chose qui est une variété et qui est précisément l'espace-temps. Autrement dit, l'espace-temps, et ça, ça se généralisera, ça va rester vrai en relativité, l'espace-temps, c'est le quotient du groupe cinématique par un certain sous-roup. Ce qui veut dire que l'espace-temps et donc toutes ses propriétés sont totalement définies cinématiques. Mais il y a mieux parce que l'espace des phases, lui aussi, est donné à partir du groupe fondamental. C'est-à-dire que c'est le quotient du groupe fondamental par un autre sous-groupe, ce sous-groupe qui inclut les rotations spatiales et les transmissions temporaires. Alors, tel que je l'ai écrit là, il faut faire attention parce que comme il y a le temps, ça ne sera pas valable, cette formule, en relativité. Ça sera un petit peu plus compliqué pour arriver à sa relativité. But it is also true in relativity, and in a very general way, that the space of the phase can always be obtained from a group of cinematics. So, this is to insist on the role extremely important of this group of cinematics.

35:00 Here, we see for the group of Galilee. We also see for the relativity. But it is something that happens. You can invent a theory which will be your theory. probablement vous aurez dans votre théorie un espace-temps probablement, si vous voulez faire la cinématique de la dynamique vous aurez un espace des phases et bien, à peu près certainement, vous aurez un groupe cinématique et l'espace-temps et l'espace des phases vont se retrouver à partir du groupe cinématique par des relations de ce genre et donc, ce groupe cinématique, je répète, c'est encore une fois un groupe de symétrie, c'est le groupe des symétries de la théorie We have a little diagram. So you can see here a group of cinematics which, for the physics newtonian, is a dimension 10, in which there are the spatial rotations, the boosts, the constations temporelles. If I make the quotient of this group by the rotations spatiales and the boosts, as I said, I find the space-temps which, he himself, can be, if you make the quotient of the group cinematics by the rotations spatiales and the constations temporelles, which is the product of the space through the space by the space of the vitesses. This composition here is very clear for the physics newtonian, for the physics relativist. We can't do it in a way of a way. We can do it in infinities of ways. The problem is to know if when we're going to decompose the space standard in space and in time, we're going to do something that we really have the right to do. It's an envelope that's open. And I've also marked here the group because there is another group that is even more grand than the group of cinematics, which is the group that includes the dilation and the changement of unity. And it is found that a large part of the physical, not only the physical, is also invariant to the group of conforms. It is to say that some parts of the physics have a variance even more than the same of the cinemas. This is very interesting. It is something that is taken into account in certain ways of using physics, for example in the theory of twisters, or in what we call the theory of conformity, the theory of champ conformity, or something like that. So it is something that plays a role very important. but I would say that it is not something that is very clearly explained in the manual of physics,

37:30 so it is something that you have to keep in mind that this group conform exists, it is more grand than any cinematic, it is a group of cinematic, and it has a role to play, which I will talk about, but it is not totally clear. Now, I come back to the space of phases, or the space of movements. As I said, it's a variety. It's a variety, in the case that we interest, it has six dimensions. Three dimensions, which represent a position in space, and three dimensions, which represent a width and width. But we can simply consider it as an space. And we can define, on this variety, functions. C'est-à-dire une fonction sur l'espace des phases, c'est quelque chose que je vais écrire du genre F, on est en physique newtonienne, de R et de V. Vous en connaissez une par exemple, c'est de la grangien. On a la grangien qui est une fonction de R à vitesse ou l'énergie d'un corps. L'énergie d'un corps, c'est quelque chose qui dépend de sa position, par exemple par l'intermédiaire of the potential and its speed. So, it's something quite banal. And in fact, it's what we call an observable. It's something very general. A function of the position and the speed, it's what we call an observable system. So, the whole of the functions defined on the space of the phases, the movements, it's an algebra. Why is it an algebra? Because, simply, I can add the functions, I can multiply them. and it is found that this algebra, in plus, has a particular propriety, which is due to the fact, rappelez-vous, that Gamma has a particular structure, which is a particular structure. It means that this structure on the space of the phase will induce certain properties on the function. It's something very general. When you have an space, a variety, which has certain properties, these properties will always translate into the functions of this variety. So you see, we pass here from a notion geométrica, a notion of space, to a notion algebra, which is the algebra of functions on this space or this variety.

40:00 We'll be back here, this will be very important. Here we are in a particular case where the variety is considered as the space. the functions are called observable, they form an algebre and this algebre has a structure of poissons. That means that when we have two functions, F and G, we can define something which is the crochet of poissons. But it is important to know that this structure on the space of functions comes from the structure symplectique de GABA, et c'est elle qui est à la base des possibilités d'identification dont on parlait tout à l'heure. Bon, voilà pour la physique de tonienne. Vous savez que pour un tiers siècle, la physique de tonienne a donné la place à deux nouvelles qui s'appellent ça plus que des théories, c'est des espèces de corpus, d'un côté la relativité, parce qu'il y a la relativité restreinte, puis la relativité générale, puis la cosmologie relativiste qui construit la partie d'elle, d'un côté, et la physique quantique de l'autre. Mais, apparemment, ça diverge. C'est-à-dire que ce sont deux choses très différentes, qui utilisent des concepts très différents, qui utilisent un formalisme mathématique très différent, qui s'applique dans des conditions physiques différentes, c'est-à-dire la physique quantique plutôt qu'en microscopique, la relativité plutôt qu'en microscopique, et qu'on n'arrive pas à synthétiser, qu'on n'arrive même pas à harmoniser. What do we do with that? Is it really going to live with two corpus which are inflated? Or is it going to try to reconcile You know that it was already a great Einstein to find a theory that this theory is always pursued by the most theorists with the types of pieces which are theoretical in order to expand the supersymmetry, etc. What I would say is that, in any way, Quelle que soit la piste qu'on veut explorer, il faut essayer d'explorer les structures géométriques de ces deux théories, par exemple, en comparaison avec la structure géométrique de la physique de l'Union, pour voir au moins ce qu'elles ont en commun et comment on pourrait y trouver les points de ressemblance, les analogies qui permettraient d'aller plus loin dans l'unification.

42:30 Alors, relativité restreinte, là je vais aller très rapidement. You know that the space and the time of Newton are replaced by an space-temps, which is something more global, which, for the relative terrestrial, is called the space-temps of Minkowski. And what we call, I'll note it like that. It's a variety, and, of course, it's four dimensions. We can say it's three dimensions for space and one for the time. even though, as you know, in the space-temps, we can't say there is time and space. It's to say that we can't replace the space-temps in space-plus-temps or, more exactly, we can do it with an agility of a manner. I'll show you how to present this in a design. So, you have the space Newtonian, there's no problem. l'espace newtonien le voilà et le temps le voilà avec le temps qui s'écoule partout en tous les points de l'espace et à un autre moment l'espace il est là c'est l'espace demain l'espace après demain il n'y a pas d'ambiguïté l'espace temps de Mitovski c'est pas tout ça c'est à dire donc moi je suis dans l'espace temps de Mitovski voilà ce que j'appellerais ma ligne d'univers c'est à dire ma my first anniversary, my second anniversary, etc. So, there is something that is true, it's my time. And then for me, space is something that is going to be like this. But, I anticipate, someone else will have a different line of time. If she is inclined, it means that someone else has a speed compared to me. But for him, space is not at all that. Why, I say that space is that? Like that, for me, they are simultane. It is to say that this event, this event and this event, from my point of view, they are happening at the same moment. But from the point of view of this one, this three events will be happening at three moments different. T1 for him, T2, and T3. Autrement dit, pour lui, l'espace, c'est des choses comme ça, et le temps, c'est des choses comme ça, alors que pour moi, c'est des choses comme ça. Donc, les notions de temps et d'espace, vous le savez bien, dans la relativité, ne sont pas définies du tout de manière absolue, mais de manière distincte pour chaque observateur.

45:00 Donc, il y a une manière infinie, en relativité astrale, de découper l'espace-temps en espace plus temps. It's like that we can't say that there is an space and a time in relativity restreint. And in relativity general, it will be fine. In cosmology, we can show that there is a possibility of synchronization between... So, in relativity, no matter of observatory, no matter of object follows a certain line in the space-temps called the universe. For him, the time is well defined, since it is simply the length of the metric of his curve. It is to say that for me, between this moment, my birth and my 50th anniversary, there is no problem, I can measure this curve. The instrument I measure for this curve, and I will find, for example, 50 years here. In my time, but it has absolutely no value for others. 50 years of my time. Well, it's true that in cosmology, there exists an ensemble of objects, which are the galaxies, which have an universe, and we can show that there is a way, and this is a consequence of the homogeneity and the symphony of the universe, that there is a way to define There are sort of surfaces in time constant, which are, and we call it, the time cosmic. It does not mean that the other observers have not had a completely different time, it is to say that someone who is able to do this, he has a completely different time. But in any case, there is a kind of class universe of observers for which, for each one, the time proper is equal to the cosmic time. On peut démontrer, c'est vrai dans les modèles cosmologiques qu'on a, ça serait pas vrai dans les modèles cosmologiques, et c'est conforté par l'expérience dans la mesure où l'Univers se révèle effectivement compatible avec l'hypothèse dont le jeu d'été nous a rempli,

47:30 mais c'est une possibilité de rétablir maintenant cosmique ou cosmologie. Ce qui n'empêche pas, c'est pas parce que entre par exemple, disons, la formation des premières galaxies et aujourd'hui, on pense qu'il s'est déroulé 10 milliards d'années de temps possuible, ça n'empêche pas qu'il pourrait très bien y avoir un certain observateur pour lequel dans son temps de croix, il s'est déroulé un milliard d'années s'est déroulé. C'est tout à fait possible. C'est même quasiment certain qu'il doit y avoir certains observateurs, entre guillemets, ce ne sont pas des humains, mais des objets pour lesquels le temps qui s'est déroulé, c'est... Si vous prenez un photon, par exemple, pour le photon, il s'est déroulé because for the photon, the time doesn't occur. If you have a particle that goes to the speed of the light minus epsilon, it's going to be something that's going to be of an angle, I don't know if it's epsilon or epsilon 2 seconds, it's going to be almost nothing. So in 10 billion years of cosmic time, it can occur for some observators or a small time as you can choose your observator. So there is absolutely no contradiction between the existence of the cosmos and the impossibility of doing a decoupage of space-temps in space to the time, you can say that the impossibility is to find one unique. Sauf that in cosmology, just by the fact that there is this kind of class, the galaxy that seems to give a look at it, which means that there is a decoupage of space-temps. Alors, j'en reviens à la relativité astreinte, qui n'est pas encore générale et encore moins cosmologique. Alors, il y a des symétries. Je termine par ces symétries. Je suis censé parler combien d'heures ? En tant que SMIC, je vais vous raconter. Non, non, parce que là, j'arrive à peu près à la fin du premier quart. So, we're going to have four translations of space-temps. We're not going to be able to separate the translations of space and the transformations of time, but we're going to have four translations of space-temps. Three rotations spatiales, they're there. I don't know why, I don't know why. So it would be added here three rotations spatiales,

50:00 and also three transformations of inertia. There are also, but the transformations of inertia, they are not at all trivial. They are always called boosts, but not the boosts galiléans, they are called Einstein. At the point, we say boosts relativists, but the word relativists is not well chosen because there is already a relativity, as I said, in the physics galiléan or newtonian, So, 4 translations of space-temps, plus 3 rotations spatiales, plus 3 boosts, and their expressions mathématiques, which are the transformations of Lorentz. It's to say that these are the real rotations in the space-temps that mix the dimensions spatiales and temporales. That's why we have really an space-temps, and not an space plus-temps. So, if I add some 4 translations, plus 3 rotations, plus 3 transformations of inertia, This will give, first I add it separately, the three rotations spatiales and the three boosts. This gives a group of six dimensions called the Lorentz. So rotation spatiales and boosts. And if I add the four translations of space-temps, I get a group of 10 dimensions, which is the point square. This is the cinematique of the relativity restreint. So, for passing from the Newtonian to the relativity restreint, I replaced the three boosts galiléens, simple addition of the vitesse, by three boosts Einsteinians on the transformation of Lorentz. So I changed the group 2.4, I changed the group cinématique. And I could say that it's all. Because from the group cinématique, I will find the space of Antonovsky, I will find the space of the phase, I will find the cinématique relativist, I will find everything. So when I say that, I have all said. And you can see that it's simply changing a row. How do you call 4 rotations? Well, the rotations, there are 6. We are at 4 dimensions. One rotation to four dimensions is in a plane. So in general, how can you choose the plane of rotation? If you combine the three dimensions spatiales, you have three ways to do it. And then you can combine each of the three dimensions spatiales with the temporal dimension.

52:30 So it will give you three new ones. This is the three rotations spatiales, which are the same in R3. And this is the three new rotations which combine each dimension spatiale with the dimension temporelle, which are the three new ones. For example, if I call 1, 2, 3 these indications with I, if you want, 1, 2, 3, I call L1, 2, L1, 3 and L2, 3. And here, if I call 0 the dimension spatial, I have L01, L02, and L03. That's how we call these rotations spatially. So it's the same. Everything is in the cinematic. This cinematic is in the cinematic. Principle d'inertie, il est toujours là, mais il s'exprime encore de manière plus simple. C'est-à-dire que quand vous exprimez le principe d'inertie en physique newtonienne, qu'est-ce que vous dites ? Vous dites, un corps libre décrit forcément une droite à vitesse constante. Donc, principe d'inertie en physique newtonienne, une droite à vitesse constante. Principe d'inertie, le même, mais en physique relativiste, c'est une droite dans l'espace-temps. On n'a pas besoin de rajouter à vitesse constante, parce que quand vous dessinez une droite dans l'espace-temps, c'est forcément à vitesse constante dans l'espace-temps. Si la vitesse changeait, ce ne sera plus une droite. D'accord ? You can see, these are little details, but it shows how the passage of the space-temps simplifies a lot of the formulation of the things and the arrangement of the geometry. The principle of geometry, the principle of inertia, is a right. And when we start in relativity, it will be the same. Like the space-temps is curved, there is no right. What we replace is the right, it is the geometry. So, the principle of inertia and relativity general, it is a borderline. which describes a geodesy. It's not even more simple. On dit que change de vitesse dans l'espace-temps, it corresponds to a rotation, but it also corresponds to a rotation in the space of the momentum because the point of the right is not the same. Or is-t-ce que je dis des bénéfices ?

55:00 on peut le voir comme une rotation. Yes, it's a rotation, but a rotation completely trivial. So it's simply a group additive, because you have V1 plus V2, V1 composed with V2 equals V1 plus V2. And in Lorenz, you have V1 composed with V2, it's Lorenz, it's Lorenz. So in a certain way, we can say that it's a rotation. I was going to say no, but... It's that which is interesting. Yes, because... Well, we'll ask you a question all the time. Ok, ok. Ok, ok. Ok, ok. Ok, ok. what changed, as you have understood, it's the law of proposition of the vitesse, it's to say the details of the group. And in fact, you know that all this comes from the fact that the vitesse of the lumière is constant, that the experience of Michelson and Morley has confirmed that it was constant. What I want to say is that the only exigence of these constants It's to say that the only group, the only group cinematic that ensures the constant of C, it's the one, and there is no other. It's to say that if we chose C, then at the limit, yes, the only group cinematic possible, in fact, C doit contenir the group of Lorentz. or a kind of degenerate if the speed of the light was infinity and if the speed of the light was infinity the speed of the light degenerate in the group 2.2 the group 2.2 excuse me degenerate in the group 2.2 d'accord but there is no other solution than these two and if, for example you say I don't want to 2.2 and that would mean that you wouldn't want to have the constant of the weight of the weight of the weight. But as all this was done because of the crisis that we observed that it was constant,

57:30 which was not compatible with the addition of the weight, in fact, this is translated, and in fact, it's good to compare it, which has translated it into a group, and it's for that we have given it, its name to the group 2.4. Alors, même chose que tout à l'heure, l'espace-temps, c'est le quotient du groupe cinématique, qui cette fois est le groupe de Poincaré, par le groupe de Lorentz. Donc, on va trouver l'espace-temps de Mikulski. Là, le groupe de Lorentz, c'est les rotations spatiales de Vébousse. Et l'espace des phases, ça va être l'espace-temps, pardon, le groupe cinématique, divisé par les transmissions temporelles, divisé par les rotations spatiales. because I suppose there is a group of rotations temporally identified, so there is a time, so I have particularized the things. I have a little trick, but it's true. Well, I'm going to leave here for the relativity. Very quickly, the geometry, I wanted to do a little bit of the geometry, I was very happy to go back to the physics. It was simply to put in place the notation, the concept of the variety. One variety is what is generalizing the notion of space. One variety, what is it? We consider it as an ensemble of points. The points are the positions in space. We consider the variety of space. We consider the events if we consider the variety of space-temps. We consider other things. We have an ensemble of points munis of certain relations. Quelle relation ? That's what is important. Well, if there is no relation, an ensemble of points is not an space. These are points. We don't know if the points are proches. So, we need a certain structure to talk about the space, the geometry, the variety. The first level of structure, it's the topology. Structure minimally. For what we can do with the geometry, we need to talk about the topology. I'm going to tell you what it is. a little bit further, we need a different structure. So in fact, when we do the physics, in the very majority of cases, we suppose that we have a different structure. We suppose that we have a different variety. And then, on a different variety, we can still add different structures. Structures conformes, structures métriques, which is a variety riemannienne, pseudo-riemannienne,

1:00:00 structures d'espace libré, structures de variétés symplectiques, structures complexes, structures if we have a group of groups and a variety. All of this is a plus, eventuellement. So, we can also have a structure metric and a structure fibrée. In the same time, a structure metric and a structure symplectique, which will give a structure complex, etc. So, we can add a lot of things, but I would say that these two minimum levels that we can then enrich. It's not a relation to the science logic, it's not a medical condition. It's a bit complicated than that. What do you mean by relation between points? The relation between points depends. For example, the topology. The topology, you can see it as a topology. You can see it as a topology. To define a topology, you have to define what we call the ouvert, the voisinage. So, you're not going to enter into the details, the topology is defined as a correspondence So these are the pieces of the variety, which we will call the ouvert, and the pieces of an ecstasy that we know well. So, you will finally consider the variety as an assemblage of the open spaces that contain the points. And the fact that the points belong or not to an ouvert, that, we can consider that these are the predicates, and the whole of these predicates, which will, when we interpret in a way geométrica, define a topology, we can also interpret it in a way logically. It's a whole ensemble, and there is a total correspondence between the logics and the topology. What about the predicates? I don't know, I don't know, it's interesting to discuss it, but I know that it's So, this is to say that it is important to have a structure topology. It allows to describe the variety by coordinates, because these correspondances are nothing else than the definition of the coordinates. It allows to describe things like, for example, the number of dimensions of the variety. When I say, for example, that space is dimension 3, I am interested in the topology of space. And what's interesting is that if I have functions on the variety,

1:02:30 for me to ask if my function is continuous or not, it is necessary to define the topology. It's to say that the topology allows to define what is the continuity of the functions. Yes, just a précision, because I'm not sure. To be able to distinguish the open and open in topology, I have to work on infinities of elements or points. If they're finished, we can't distinguish the open, the fermes, the fermes. One open can be at the same open and the same? Yes, the two. Exactly. I don't think there's no clue. We can play with you. Yes, but if I consider three elements in my ensemble, an ouvert will be closed. Yes, but after that... That's what I want to say. Yes, it's important. Yes, it's important. It's important. It's important. It's important. It's important. It's important. It's important. Well, there are actions, but... Yes, I know well... I know the definition, but I try to understand a little bit. I think, I'm sure that even on an ensemble discreet, we can define what the difference is. Well, precisely, that's trivial. Not necessarily. And, just so, that's what I wanted to talk about when I was talking about. Now, one of the ways that is actually followed is the way of the quantification of the gravitation of the relativity general. The message of the relativity general is that the gravitation is the geometry. So if we want to quantify the gravitation, we want to quantify the geometry. If we want to quantify the geometry, we can't do it in a geometry given, because it's the geometry that we can even quantify. Alors, un moyen de quantifier la géométrie, c'est ce qu'on appelle les réseaux de spin. C'est-à-dire que lorsque vous avez un espace, en gros cet espace, vous pouvez le trianguler. C'est-à-dire que vous pouvez représenter, je dirais, l'essentiel de sa structure topologique, son nom saturn, si vous voulez, by a sort of triangulation like that

1:05:00 which will be an ensemble discret of points. I will be a lot simplifier. Well, when you consider the version pointy of this thing, you will consider an ensemble discret non plus of points geometry, but you will consider not the points, but rather the relations possible between all points and each of these things a certain opérateur quantique qui est par exemple un opérateur appartenant à un certain groupe, par exemple SU2 et il se trouve que SU2 c'est le spin donc ça va être un opérateur de spin et donc on va parler d'un réseau de spin qui est une sorte d'objet abstrait qui va représenter si vous voulez une sorte de pré-géométrie quantique. Bon, vous voyez bien que dans cette espèce de réseau abstrait il va y avoir un ensemble discret, je suppose not to call it a point, but something like that, a vertex. In this space, it can very well have a topology, and it is even very important because when you have a variety, the properties of causality, which is very important in this variety, are defined by the uniform structure of the variety, which is something that is a little bit less that the structure of the metric. And in the same way that the distance is defined by the topology, this uniform structure is also defined by the topology which is much less, which is the geometry, the cone of light, the causality, etc. Why do I say that? Yes, it is because we want to have a minimum in the geometry quantique, on cherche quand même à avoir une espèce de causalité, et c'est au moins la vue des choses minimum, donc on voudrait avoir une sorte de topologie, et il y a beaucoup de gens qui étudient d'une manière abstraite ces réseaux pour voir quels genres sont des choses discrètes, finies ou infinies, et voir quels genres de topologies on peut leur donner pour précisément, même dans quelque chose comme ça, qu'ils n'étaient pas encore géométriques, to define a sort of causality, or at least we can call it a pré-causality.

1:07:30 So, structure topologique, it's the first level. Once we have a structure topologique, it doesn't suffice, because, for example, here we have a lot of equations differentials, so if we have an equation differentials, it means that we can be able to derive a function, the topology doesn't suffice. To be able to derive a function, we need to have a higher level of structure which is called a different structure which is produced by the vectors what is a vector? it is the same thing as a derivative a vector, it is something which applied to a function gives a function derivative in the sense of the vector so vector equal derivation so to give a different structure to the variety it means to define the derivation the vector. Once we have the vector of the vector, it's a kind of typical vector. In terms of vector-s such as spiners. on can define what we call the form, by duality, and then, by product sensoriel, we define the tenseurs. So, when we have a different structure, we know what it is that the vectors, the forms, the tenseurs. And even, in some cases, what we call the spinners. It's the form linier and all that. Yes, the form linier, it's by duality, in a vector vectorial, in a point, then it becomes a champ, and then, we have the tenseurs, There are two forms of tension, which are the two forms, or the form linear, like the metric and other. There are different structures. And in the theory of physics, in general, we suppose there are different varieties, at least in the theory of physics, a little classical. Then, we can also add the levels of structure. But there, it depends on the theory. For example, we can define the connections. When the variety is a fiber space, I will tell you what it is in two words, we can define the connections. And the formalities of the space fiber and connections, which we will talk about in 15 days, is what is completely based on the theory of George. We can also put a structure conforme, or even more, the structure metric. structure métrica, so that's a role very important, because it's based on the relativity general. One variety rimanian or pseudo-rimanian, it's a different variety muni d'une structure

1:10:00 métrica. And it's true that when there is a structure métrica, it also allows to define certain types of connections. So, a particular connection, it's called connection of the Vichita, it's interesting, So structures fibrées, all these structures are interesting. The number of dimensions can also vary. So you can see that already, the geometry gives us all the materials possible. I would also have to put the structure of the volume here. There is also a structure possible for the variety. What is the metric? What is the metric? It's something very important. We have a different metric. So we know what the vectors are. a metric is a operator that associates at two vectors their products scalaires. It's all simple. We can also see how a tenseur of certain types. It allows to define the interval in the variety. If you use a curve in the variety, for example, it allows to define an interval. And when this variety is the space-temps, the interval can be a length or a duration but according to the course, it is orientated in a way or another, or even in an interval of length, that is to be a null length. What we need to know is that a different variety given is given to several metrics. It can be given to an infinity of metrics. There is an infinity of metrics also. Simulant. Simulant. And by the way, at a given a different variety, you can define the whole of all the metrics that could admit this different variety. It's an ensemble, it's a certain... Cet ensemble a lui-même une structure de variété. Et lui-même a une structure de variété métrique, je crois. Au moins dans certains cas. Si vous avez deux métriques sur une variété, G, G1 et G2, il se peut que vous ayez une relation entre les deux qui soit que l'une est égale à l'autre multipliée par une fonction. In this case-là, we see that the two metrics are conformes to the other. And we can define a class of equivalence of the metrics which is in fact the equivalence of all the metrics

1:12:30 related to this way. And when we have something like that, it's what we call a structure conform. So you can see that a structure conform, it's something a little less strong than a structure metric. Because a structure metric, by the theorem of the produce scalaire, it allows to calculate the angles and the length of the length. With a uniform structure, it allows only to calculate the angles. You know that the isométrie is a transformation, and it is important, but it is important, and when I talked about the symmetry of the newtonian space, I didn't have yet introduced this notion of the metric, but, in the space Newtonian, there is a metric, that is to say that we can measure the length, we can measure the duration of the time, and when I talked about rotations, which are the symmetries of the space Newtonian, it was, in the sense of, because they are the symmetries of the space and the time Newtonian. I did not say, but it was the isométries. And when we talk about the symmetry of a variety, The group of symétries is in fact the group of transformations which conserve the metric, which conserve the length, the distance, the duration, and it's the group of isométries. If, in lieu of a metric, we have, for example, a symplectique, it's different. We're not going to say that it's the isométries, we're going to say that it's the transformations which conserve the symplectique, On appelle ça des synplecto-comorphismes, ou d'une manière plus habituelle, des transformations canoniques en dynamique. Bon, peu importe si vous voulez, mais en tout cas c'est la même idée. C'est-à-dire qu'une variété peut avoir un hypostructure supplémentaire qui peut être une métrique, forme linéa-symétrique, puis linéa-symétrique, et des transformations, un groupe de transformation qui concerne la métrique, qui s'appelle les soumétriques. Ou bien la variété peut avoir une forme antisymmétrique, une forme symplectique, et des groupes de transformation qui conservent la forme symplectique, symplectomont. Tout ça, il y a des similitudes. Autre type de structure qu'on peut rajouter à une variété, indépendamment de la métrique, qu'il y ait une métrique ou pas, on peut définir des connexions.

1:15:00 differentials, which has a vector and a tensor associated with the derivative covariance of the tensor along the vector V. Because when we have a different structure differentials, it allows us to derive the functions, but it does not allow us to derive the vectors of the tensor. So there is a structure supplementary to derive the vectors of the tensor, which is what we call a connection or a derivative covariance. Sur a variety, there exists an infinity of connections possible. There are many. Each connection has a torsion and a curve. We can calculate I give you an infinity and I choose one and then you will calculate its torsion, its curve. Very good. if we have a metric, The metric is the case in general, because we suppose that we have a metric, we have a rhymanian, or rather pseudo-rhymanian, in the space-temps. When there is a metric, it allows to choose, among all the connections, a particular connection, which is called the connection of the legivita. How do we choose it? We ask that it doesn't have a torsion, and we ask that the derivative covariance, because a connection is the same thing as a derivative covariance, on the demand that the derivative covariance of the matrix is equal to 0. Well, it turns out that these two constraints define one single connection among the infinities of connections possible which is called the connection of the vitivitas. And when we do the general relativity, in all the manuals of general relativity, we have to talk about connection, it is this connection of the vitivitas which is a particular connection liée to the matrix. And sa courbure, toutes les connexions ont une courbure. La connexion Levit-Civita a une certaine courbure et comme cette connexion, je vous l'ai dit, est déterminée par la métrique, la courbure de cette connexion a quelque chose à voir avec la métrique. C'est ce qu'on appelle la courbure rimanienne de la variété. D'ailleurs, bien souvent, dans les manuels, on dit que c'est la courbure associée à la métrique, sans même parler de la connexion. Pourquoi I insist on the connections because it is a formalism that precisely can serve to have the same way the general relativity that I am talking about and the theory of George. Because the theory of George are completely based on the connections.

1:17:30 It is to say a potential of George, which is the potential of the champs which represent in physics what we call the bosons intermédiaires, which is the photon or the bosons or whatever, are very exactly connections on the varieties, which are the spaces fibrous. So, one connection is the same thing as the potential of the jauge. And the curve of this connection is what we call a champ jauge. It's a sort of derivative. When we pass from the connection to its On effectue une sorte de dérivation, donc on passe du potentiel de jauge au champ de jauge. Quand on dérive un potentiel, on obtient un champ. Et là, vous le savez ou peut-être que vous ne le savez pas, mais la courbure rimanienne, en relativité générale, c'est le champ gravitationnel. Enfin, c'est une manière de représenter le champ gravitationnel. Donc vous voyez qu'on retrouve ici, c'est la courbure d'une connexion, c'est le champ, And so, of course, the connection of the Vichita will be more or less the potential gravitation. Why, when we don't talk about it, just for what we're talking about? Well, it's a very good question. Because, for example, there are theories... Well, it's a question that is very complicated. I'm not sure of the answer to that. But... Well, it's a bit horrible. Well, I think that at the beginning it was a priori. Sauf qu'il y a une théorie de carton avec courbure nulle et torsion non nulle. La théorie d'Einstein avec torsion non nulle et courbure non nulle. Mais il se trouve que vous pouvez aussi, une fois que vous avez une métrique, je vous ai dit que la connexion de l'Evitch-Evita, elle était définie comme torsion nulle et compatible. You can define another connection, which will be a curve of 0, a curve of 0, compatible with the metric, and it will be a curve of 0. This will be a connection to Cartan, I think. And you can show that if you take a theory with these connections, you will be able to use a theory that is exactly identical to the algorithm. Now, the theory of Einstein-Karton is to take an arbitrary connection. You don't impose ni the curvature, ni the torsion nulle. You impose simply that it is compatible with

1:20:00 the metric, so it can be... At this point, it is not defined in a unique way. You have all the freedom. But in general, we do the theory of torsion nulle. On voit passer des papiers aujourd'hui où il y a des gens qui disent... Alors, soit des gens qui disent qu'on prend la version de la relativité générale qu'on appelle téléparallèle... Je pourrais vous expliquer pourquoi, mais ça prend un peu de temps. Téléparallèle, donc, où il y a une connexion de torsion non nulle et à courbure nulle, ou bien on prend des théories encore plus générales qui sont avec torsion et courbure non nulle. On l'appelle téléparallèles parce que, bon, il y a un format qui s'appelle Formalif des tétrades, je ne sais pas si vous connaissez, un tétrade c'est, en gros on peut dire qu'on a une forme, on peut décomposer une métrique G menu, comme ça, comme le produit tensoriel de deux formes, de différents mouvements, excusez-moi, on appelle ça une tétrade, and instead of asking for the connection to conserve the matrix, we can ask for it to conserve the tetrable. If they conserve the tetrable, it implies that they conserve the matrix, but it's not in any sense. So it's in this case that if we have a connection to conserve the tetrable, It's there that we can impose that they are in the middle and that they are in the middle. It's like that we can do it. There's an informalism that exists and that gives you to the work, but it doesn't seem to say anything. For example, the space physique is a variety of three dimensions. We have these isométries and transformations which concern the metric, translation and rotation. The intervals are called the length. The space that we have seen, the Newton, has a null curve, so it is euclidean, but there are three types of spaces where the curve is constant, that is the R3 that we are talking about, the sphère is a positive and the sphère hyperbolique is a negative. The first model cosmological of Einstein in 1917, it was a model where the space-temps had in the space spatial three sphères,

1:22:30 which is a positive. When we pass to the space-temps, a variety pseudo-rymanienne to four dimensions, which means that the metric is rather a pseudo-metric than a metric so I can't talk about the details here the intervals in the space-temps are either longer or less durées these intervals of the genre of light when it's finally the trajectory of a rayon lumineux because we know that the length of a rayon lumineux is always zero why? because the length is C2, dt2, minus dr2 but as the rayon is at the surface of the lumière, we have forcément dr equal c dt, so when you do c2 dt2 minus c2 dt2, it is always zero. So the length of the rayon is always zero, which is another way to say that for the lumière, the time is not. It is a way to speak, but it doesn't matter what it means. So, in the same way that there are three spaces There are three spaces to constant, there are three spaces to constant. There are three spaces to constant. There is one of the relativity restreint, where the constant is null, which is the time of Mikovsky. And there are two others, which are called the time of the Euciter and the time of the anti-Euciter, which are sort of hyperboloids, which are well known for the constant, and which are each of the solutions to the relativity general, but with a constant oscology positive or negative. For the conformity, I'll pass quickly. Simplement, I'll remind you that the conformity of the space-temps is the same thing that its probability of causality, so it's something that in physics is important. There are, for example, people who say that the matrix is something that we can perhaps see, and it's true, there are reasons to think about it, but in revanche, the structure conforme, before we could pass it, renouncing to the structure conforme, it's renouncing to the causality, and it seems to be something really fondamental. And in all the new theories, in all the theories, well, the causality remains preserved. So, remember all of a sudden, the film is the group that protects the structure metric, but it is also a group that protects the structure conforme.

1:25:00 And that's the famous group conforme, which was just a part of the diagram, and this group is interesting, which is related to the causality. It's the biggest group, if you want, who preserve the propriety and the causality of a zone. Just a point. Just a point. One thing that we form, that we conserve our own. Well, like we don't have a long length, classically, we define an angle by the length of the arc on the page, etc. What we call a long length, is it simply a function function or is it another thing? An angle. The angle, in general, if you have two vectors, the angle, when you have a trigonometry, it is defined as the trigonometry of the trigonometry of the trigonometry of the trigonometry of the trigonometry of the trigonometry. If you multiply the metric by a factor G, it means that here you multiply it by G, it by G, it by G, it by G, so finally it goes. So you can see that an angle is completely insensible to a transformation of the metric, a multiplication by a function, which is exactly the transformation of the form. the angles are necessarily a barrier of the form. Well, the group of Lee, as I mentioned earlier, there is something to say. As I said, a group of transformations, like all of those that we have seen, it's a particular class called the group of Lee, and it's true that a group of Lee is also a variety, which allows us to associate properties topology, like, for example, the dimensions. I have already said that the group of rotations is a dimension 3, so the Lorraine is a dimension 6, etc. Mais ce qui est très intéressant aussi, et qui va nous ouvrir les portes de l'algèbre, c'est que chaque fois que vous avez un groupe de lits, il existe une algèbre associée qui s'appelle l'algèbre de lits, qui a donc des propriétés algébriques. Et donc j'insiste, je vous l'ai déjà dit, sur cette correspondance, d'une part entre les variétés, les groupes et les algèbres. Et toutes les théories physiques qui existent aujourd'hui, Well, they are expressed rather than insisting on the aspect of the variety of groups or alges, but it can always be translated. And so, it's why we need to have a little bit of the three cords to the arc to be able to take each theory

1:27:30 and look at the bottom of the angle if we want to be able to compare it with other theories or if we want to be able to advance it. Well, the variety of symplecties, I don't have much time to talk about it. On peut l'avoir un peu comme une variété Riemannienne, sauf qu'au lieu d'avoir défini une métrique, on a défini autre chose qui a une forme symplectique. Ça ressemble un peu, c'est-à-dire que, rappelez-vous, une métrique, c'est quelque chose qui a deux vecteurs qui correspond à le produit scalaire. G de VW, on écrit comme ça. Bon, ben ici, on va dire qu'on a une forme symplectique qui a deux vecteurs qui fait correspondre aussi which is a sort of product of the two, but which is symmetric and anti-symmetric. It's interesting and, above all, it's a structure which is always present in the space of phase or in the space of movement and the symplectomorphism, for example, the transformation which conserve the form symplectiques It's exactly something that will take one possible movement and transform it into another possible movement. It's to say that it will take the trajectory possible of a particle and transform it into another trajectory. And all this comes from the optical optics. In general, if you look at all the rayons lumineux possible in a system, this ensemble of rayons lumineux, it's something that is possible in an space of movement, It's an space of phase, it's an ensemble, it's a structure symplectique, and all that is transposed to the dynamic of the particle, and also to the dynamic in general. It refers to what you had said at the beginning, that the cinematics are always a deformation, in a certain sense, of... Well, I think it's Newtonian. Yes, it's like a deformation elastic, etc., etc. the pressure that I have given, we are always in there. You will see earlier that the word deformation can also be taken in a sense much more precise than a sense mag. I will show you what I want. I will continue. The fibres, it is necessary that I don't have to talk about it. It is very important because there are connections, etc. Espaces group, Espaces RG, I have already talked about what I wanted to say.

1:30:00 but it's also that we can do the quotient of a group by a subgroup and obtain a variety. And these two examples that I have already cited, I will not go further. I just showed you that the space of the clit was the quotient of the group of the clit by the group of rotation. In the same way, the 3-sphere, it's the quotient of another group called SO4 by the same group of rotation. And the space hyperbolics, it's another quotient. And this, it's the three spaces possibles at the constant, Three spaces possibles, symétriques. Pour l'espace-temps, c'est pareil. Il existe trois espaces-temps symétriques. Celui de Newton, quotient du groupe de Gaillet par les rotations et les boosts. Celui de Minkowski, alors celui de Newton, il est un peu particulier et dégénéré. En relativité, il y en a trois possibles. Celui de Minkowski, qui est le quotient du groupe de Poincaré par le groupe de Lorentz. ou bien celui de dositaires ou d'antidocitaires, qui est le quotient du groupe, soit de dositaires, soit d'antidocitaires, par exemple de leur groupe. Ce n'est pas la peine de se rappeler tout ça, mais simplement pour vous dire que cette définition des espaces ou des espaces-temps des variétés symétriques comme quotient de groupe, elle est très générale, elle fonctionne toujours, même quand on va au-delà de la physique geniètre ou de la relativité restreinte. Yes, I will talk about algebra also, because when we have a variety, I have already said that we have associated algebra to these functions. Here I am going to talk about complex functions. When we can add and multiply the functions, it is an algebra. This algebra is commutative and associative. You know, it is a product and it is ordinary. And, as we considered the complex functions, there is the property of complex conjugation. When we have a function, we can take the complex conjugation, and if I take the complex conjugation, I return to the function, so we have an evolution. So we have an algorithm which is commutative, associative and involutive. An example of a function on a variety is just simply the coordinates. Or, what is important is that this algebra, its properties, completely define the variety. It is to say that instead of defining a variety by its geometry geometry, I can be able to describe its algebra. And theoretically, every algebra which is involutive, commutative and associative,

1:32:30 qui va s'appeler une étoile algèbre, définit une variété dont elle va être l'algèbre des fonctions. Ça, c'est important. Ça veut dire que je peux regarder une variété d'un point de vue uniquement algébrique. J'oublie la théorie des groupes, j'oublie la géométrie. Mais ce qui est très intéressant, c'est que je peux généraliser. C'est-à-dire que si dans cette définition j'enlève le mot commutatif, which is going to be associative and involutive, but eventuels non commutative, that will lead to something which is not a variety, but which is what we call an espace non commutative, a geométrie non commutative. It's something that I can no longer represent as a variety, it's like an espace of points muni with certain structures, but I can still represent it as an espace non commutative which will be made of points, which will be made of kind of structures, which is something that is not even a new, but it is something new, and which has been extremely interesting. So this one has been introduced, as you know, in the last decades, I would say, especially by the problem of Aliccone. And what is interesting is that, a posteriori, on can see all the physics quantic as a geometry non commutative. I'll explain. As I said, a dynamic system, at the moment classic, because we are not quantic, is always characterized by an space of phase. I prefer to call it an space of movement, which I note gamma, and which is a structure symphletic. Sur cet espace... Alors, oui, l'idée de la quantification, quand je quantifie un système, quel qu'il soit, si c'est un système dynamique, donc il a un espace des phases, faire une quantification de ce système, c'est remplacer la variété gamma par un espace, donc le mutatif. Faire une quantification, c'est rien d'autre. So, all the quantifications that exist, the mechanics of the theory of the champs, the quantification of the speed, etc., it's something like that. Well, that, it's written in two lines. But, of course, to do it, it's an other program. But in all cases, it's simple. What does it mean to replace gamma by an space non commutative?

1:35:00 Well, we do it by algebra. c'est-à-dire gamma, il y a l'espace des fonctions sur gamma, qui est une algèbre commutative. Je vais remplacer cette algèbre commutative par une autre algèbre non commutative. Comment ? C'est toute la question, on va le voir. Et ça, je vais considérer que c'est une algèbre non plus de nombre, non plus de fonctions, I call it the operators, and I call it operators because I suppose they act on a certain space, which is an space of universe, but in the conception that we have today of the physics quantic, it becomes almost secondaire. What counts is the structure algebraic, because we know that an algebra like that defines always a version non-commutative of an space of phase or an space of universe. So, the physique quantique, it's that. The quantification, it's that. It's based on an algème commutative to an algème non commutative. Sachant qu'on, par ce que j'ai dit avant, on sait qu'une algème non commutative, c'est l'algème qui représente une variété non commutative. Et aussi, on sait qu'on peut, en général, souvent on sait trouver quelquefois cette algème, but we don't know how to represent it in the form of an algebraic operator aging on a space of liberate well, sometimes we know how to do it in several ways and we don't know how to choose between these ways in any case, quantification, it's at least that well, now I'm going to go very quickly there are many ways to do it the quantification canonic the quantification by theory of the group quantification geométrica quantification by etat cohérent quantification by deformation I think Cocro will not have a lot of time Selon les systèmes, les unes ou les autres ne sont pas toujours applicables, par exemple la quantification geométrique, ça ne s'applique pas toujours, la quantification canonique, pas toujours non plus, et la quantification applicable, elle ne donne pas toujours des résultats équivalents, mais en tout cas ce sont des méthodes de quantification. Je vais vous parler un peu plus en détail puisqu'on va prononcer le mot déformation, je vais finir là-dessus. So I'm not going to do it. I'm not going to do it. And this will be my end. The idea is that there is a sort of natural process

1:37:30 which, from an algebra, which I will call this, not in giving a mathematical sense, but in algebra, which is not too complicated, to pass in a unique way to an algebra more complicated. That's to say, you give me an algebra, A, and I know this algebra, I will find another, I call it A, rather a family, based on a parameter. This process, which is completely defined mathematically, is called a deformation. If I take the Newtonian, the Newtonian is associated with a group, which is associated with an algebra. I take this algebra, I transform it purely mathématically, I obtain the relativity restreinte. I continue, I take the relativity restreinte, I take the algebra associé, I transform it, I obtain what? I obtain some solutions to the relativity general, which are the models of 2-6-3. I take the models of 2-6-3, I take the algebra associé, I transform it, and I can't do it anymore. This is what we call an algebra origine, so I can't go further in this way. in the physics newtonian, I can take another thing. Instead of deforming the function, I can deform the function on the space of the phase. In this case, I obtain what? I obtain a mechanical quantic. And finally, I perceive that every time there is a parameter that is introduced. Here, the parameter is the light of the light, or rather the light of the light. Here, the parameter that is introduced and here it is the constant of Planck. For me, these are three constantes fondamentales of the physics, and it is not a hasard, because, really, when we talk about the physics Newtonian, it is a kind of theory of base. The theory of the deformation tells us that we can arrive at 300 theories which are given by these three constants. These are the transformations, in general, minéaires? I will show you quickly. Physique Newtonienne, groupe cinématique de Galilée, au groupe de Galilée est associé, c'est un groupe de lits, donc il est associé à une algème de lits. Processus de déformation, l'algème de Lie du groupe de Galilée peut être déformé.

1:40:00 Donc, il existe une famille de nouvelles algènes qui dépendent d'un paramètre. Si je prends la valeur nulle du paramètre, je retrouve le groupe de Galilée. Là, je confonds ici le groupe et l'algème. Si je prends une valeur non nulle du paramètre, And if I take the value 1 over c, I find the group of points carrés, which is the group of the randomness. D'accord ? I'm going to continue. I do the same thing. I take the group of points carrés. There is only one way to reform it, in addition to a parameter. If I take the parameter, the 0 of the parameter, I remain on the same number of points carrés. if I take a number of values and I call lambda, I find that the new group is the group of 2-6 R, so it's the group that corresponds to the 2-6 R if the parameter is negative. And so what is it? It's a cosmological solution of the general relativity, in absence of material, but with a constant cosmological. So, the constant cosmological, in the same way as the speed of the lumière C, is naturally introduced in the theory of the deformation, which is the parameter of the unique deformation possible in my algebra. So, I have a way, finally, suppose that Newton had known the theory of the deformation without knowing whether the speed of the lumière was infinite, etc. he would have been able to build the relativity astral and the relativity general, in all of the solutions of Decyther, only from a point of view mathematical, since there is a mathematical process which is the same. How does that happen? I don't know. But, finally, it's a question that I ask you to ask your sagacity. The third information is interesting. Alors, le groupe de Citer, il est rigide, donc on ne peut pas aller plus loin de cette voie. Mais je reviens à la physique newtonienne. Alors, qu'est-ce que j'ai fait tout à l'heure ? J'ai pris le groupe cinématique de l'espace-temps, son algèbre de Lie, et je l'ai déformé. Mais il existe une autre algèbre associée à la physique newtonienne, qui est, c'est le gamma, l'algèbre des observables sur l'espace des phases.

1:42:30 Je peux déformer cette algèbre, et je trouve une nouvelle algèbre, Très exactement, l'algèbre d'opérateur agissant sur un espace de hyper, et c'est la mécanique. Et le paramètre de déformation, c'est la constante de Poirier. Bon, je vais m'arrêter là. C'était tout. D'ailleurs, j'ai dit qu'en ce moment, ça tombe très bien. Donc, théorie des déformations, ce qui est intéressant, vous avez bien compris, c'est là-dessus que j'insiste. I conclude, because for me it's a little bit emblematic the message that I want to make. If I look at the physics Newtonian, I look at the point of view mathématiques, which is at the same geometry, group, algebra, and all that. I can do a deformation in the function of a parameter 1 sur C, which gives relativity astral, and then relativity general and cosmological, I can do a other type of deformation with a parameter H which gives us the physical quantity. All of this, when we want to pass and enter the fields, there are the fibers, there are the connections. Here there are also the fibers and connections. We can do the link. When we look at the fiber of the fiber and we want to give a analog status to the fiber, in the theories of a large number of dimensions, such as the theory of Keynesotlain or the theory of Corn, etc. If we want to add some symmetries, we're going to add the supersymmetry, so we're going to pass on supercores. If we want to quantify, we're going to pass on the non-commutative of all this, so we're going to try to get rid of the non-commutative of Alain Cohn. You can see that there is a kind of scheme in which all this will take place. Well, this schéma, I would say, is not quite a point, it is to say, the theory of deformation, it is perhaps not the must, it is perhaps not the deformation of algebra that we can look at, but people are interested actuellement a lot of deformation. Well, there is something that we call a group pointy, on which people work a lot. Contrairement, well, no, it is not a group, it is just a deformation of algebra. I think it is a algebra of deforming the algemmes. That's a great role in all these new theories. And in fact, also, when we do the quantification,

1:45:00 we do the deformations, I would say, and a particular type of deformations which make a geometry which was commutative and which was non-commutative. You see, it's an expression of the way. Plombard concebera all this. Thank you. We talk about the variation of the deformation that we transform, what do we do, technically? The variation of the deformation? Can we respond quickly? Yes, when there is a matrix, or something else, or something else, or something else... Yes, there is a response simple. I know an element of the response, it is to say that when you have an algebra, you have a product AB, and in general, you have a anti-symmetric AB A, B, B, A. For example, I'll take a crochet of poissons or a crochet of lits for the algèbre of poissons and the algèbre of lits. This is your algèbre of the départ. When you go to the algèbre, you will have a new product which can be expressed in the ancient algèbre as well as a start product. A, B. It is true that And this star-produced, it's going to be decomposed as the old product, plus the fish, plus the fish, plus the other term. And so, finally, what's going to be preserved, it's this, this anti-symmetry, but here, it's not more like the first term of development. I don't know, it's not really a conservation, I don't respond well to your question, but I don't want to be able to pose it. There are surely things that are concerned. It's a question about the anthropologist. So, if we introduce it directly, naturally, the anthropologist, we expect the acceleration of the anthropologist. Of course. But it's perhaps that's what has incited people, inversement, to think about this kind of information. No, no, no, not at all, because this is the information that dates bien avant. Ah, no. The problem with the cosmology, today, we observe the acceleration of the expansion of the cosmology, you know. I would say that there are two kinds of responses that we find in the community of the cosmology.

1:47:30 There is a response that says that there is a form of substance of energy sondes, of quintessence, of energy divine, etc. who is the cause of this acceleration? And the second response is to say that in general, there is a term called the constant cosmological which is a constant fundamental and which has for effect to cause an acceleration. So there are two ways to respond. I chose the second one, but most of the physicists, especially the physicists of particles, say that it is the energy divine. The problem is that I don't know what is the energy divine, Well, for me it does not exist. So, I am part of the people who seek to comfort the point of view that the astrological status of the physics, like G, C, H, and so it is legitimate to say that it is something that is in the physics, perhaps at the origin of an acceleration cosmic. And this theory of deformation, it seems to me that it is something that comes to an argument supplémentaire to say that it is naturally a constant of nature. But if it is directly linked to the record of the book, it would mean that it is fixed and that it is fixed, it is fixed, it is fixed in the book. No, no, no, no, no, no, no, no, no, no, no, no, no, no. If you have a connection direct between the constant costrology and the rayon of Hubble, it would mean that it is fixed, so there is no one? The constant costrology, it is very exactly the rayon of the curve of this famous space-temps of the Citer, which is an space-temps a little more constant, with the time like this, and the space space which are the three spheres. it's an hyperbolic of three dimensions that we can consider as long as R4 with a metric plus or moins. So, it's an ideal model cosmological model. It's the model cosmological of Decyther, which corresponds to the solution of the general relativity in the void with a constant cosmological positive. If we had a constant cosmological negative, that would be anti-deucytaire, which is completely different. Well, there are two ways of concealing the models

1:50:00 there-dedans. There is a first way, which is to say that the space-temps is purely the model geodeucytaire that it is there, so that it is excluded because there is first a contraction and then an expansion. The second way, which is to say... Big Bang ! Non ! Laisse-moi finir. Deuxième manière qui est de dire l'espace dense avec un morceau de ça, avec ici le Big Bang et puis tout ça. Bon, moi je n'aime pas beaucoup ça, c'est pas beau pour des tas de raisons, parce que c'est pas complet géodésiquement des choses comme ça. Et il y a une troisième manière qui peut être la bonne, qui est de toute façon, il y a une observation fondamentale dans l'univers, c'est qu'il existe de la matière. So, this is the site of the material, it is not the site of the material. So, the real model of the cosmological model is not the site of the site, but it is a solution of the reality of general, with lambda positive, yes, but not with the material, and with the material. So, it is something that is very exactly a model of the universe with a big bang and an expansion of the cell. It works very well with all the observations. It's very good. For three minutes of analysis dimensionality, the rayon of Hubble and Hubble is at a point of 1%. It's to say that the universe, with the constant of the physics, H-bar, G, is produced by the three gases of the particle. On obtient 113 billion years of the year. I have to say that it will be three minutes of analysis dimensionality. So, it would mean that... It would be 15 billion years. Non, mais c'est réputation. Réputation, 3 minutes, 10 minutes. Je suis désolé. Donc là, c'est-à-dire qu'il y a tout le monde collectif que tu nous dises, puisque tu n'aurait introduit la consente que toi le dis. Ah, je n'aurait introduit pas. Non, parce que tu n'as pas jeté dans les 3 consentes que tu dis. Si, si, j'ai mis où. Là, je ne l'ai pas mis, mais il est en... Non, non, attends, il n'est pas dans le consente de la déformation, mais moi, je le prends dans le consente de la montagne. Comment ? D'accord, mais quand tu indiques là-bas, tu signales qu'on ne peut pas prendre la formule, The mass is not as much as the principal. Well, we'll see, that it's not as much as possible. Ah, I don't know. Well, in question, what's the mass? What's the mass? I can tell you.

1:52:30 It's the indices that represent the representations unitaires and reductives of the total scale. I can tell you. The mass is the speed. I can tell you. question. Mais on peut lui donner un sens de différence entre l'éducation à gauche ou à droite du signe égal. Et j'ai lu quelque part que lorsqu'on écrit une expression de je ne sais plus quel tenseur de Ritchie ou autre, je ne sais pas en fait de la famille, eh bien l'éducation la plus générale fait intervenir une constante qu'on appelle lambda entre autres choses. Est-ce que c'est purement mathématique ce truc-là ou est-ce qu'on peut lui trouver un sens physique ? Well, the physical sense is the constant chronological. It's to say, there are two categories of genre, I would say. There are those who say that it has no sense of physical, so we don't want it, which comes to lambda is equal to zero. And there are those who say that it is constant like the other, so it is there. So the equation of the relativity general, you have to write R menu, G menu tensor Einstein, so equal T menu, so equal t menu plus lambda g menu, for those who croient. There are two versions. The first version of Einstein was that, then he was passed to that because he needed for his collection, for his solution cosmological. Then he was returned to that, for the second reason. Well, the maître has always said that. There are people who have always said that. I have always said that. There are people who have explained it. Thank you. The other thing, for me, is to qualify as a great-puissance diagrammative structurel, the way to demonstrate things, which is very rare, because you work, in a certain way, with an espace of the Q, with the dispositif and the concepts,

1:55:00 and that, it's a very rare thing, but in the final, it's not an artificial intelligence, but it's not rare. We know. and also not only on the subject of the concept, but also on the passages, to say, on the aspect traductive of the traduction. When you insist on the book of Lee, in fact, there were the traductions between varieté, group algèbre, or an enroi, like that, and there is no idea of the algebraic. I think that it's an aspect philosophic, quite important. I think there is an adresse metaphysic. And the metaphysic, in the two senses, is the metaphysic and the metaphysic. M.I.S. metaphysic. The structure of the concepts is all about that and then the metaphysic. And then there is something that I have noted that you are also referring to a certain moment, In particular, the proposition of the relative terrestreint, is a sort of an insistence that I call the principle of contision or simplification, or simplification. on the fact that when we pass from the symphony galile to the relative terrestre, there is a simplification, for example, fundamental to the conceptually, and that we could be able to relate also to a compactification notation, Par ailleurs, oui, c'est-à-dire que, pour moi, quand on dit que la relativité est une belle théorie, pour moi, c'est dans ce sens-là. C'est-à-dire qu'écrire un problème en relativité, c'est souvent beaucoup plus simple en physique de tonnerre. Et peut-être que ça va après la résolution de l'équation, c'est un peu compliqué, But, for example, if we write an equation, if we wanted to write an equation, if we wanted to write an equation, if I wanted to write an equation in the form of Newtonian, I would like to write an equation, etc. I would like to have 10 components and everything, that would be horrible. And this formalism, which is called covariance, is very simple, it is extremely simple. And this simplicity can say something.

1:57:30 When I talk about the principle of inertia, everything is there. When we say that it's a right, and it's going to say that it's a geodesic. When you say that the particles are in the geodesic, you have to do it. That means that the orbit of the planet Mars, for example, that Kepler had 3 years ago, so the answer was that it's a geodesic. Thank you very much. This is the same question, but in a way more complicated. If, for example, we talk about this passage of interest in general, of course there is, let's say, an aspect purely algebraic, but it's not clear, or is it reduced to that? The passage of interest in general... but no, no, I haven't talked about it practically. I've talked about the passage newtonian to the relative terrestre, and then in theory of deformation, I've talked about the passage of the relative terrestre to the relative terrestre of Citer, which is a particular solution to the relative terrestre which, for me, represents the vide. Because today, when people talk about the relative terrestre, they say that there is a kind of vide of reference, which is the relative terrestre of Mitovsky, and we add the material and I say no, but with the reference of the cyther, I add the material cyther, and it's what they want to do with the cyther. It's not the space of Mikovsky who excite to give the material to all, but it's the space cyther. And I see it like that. What you have noticed in the history of cyther, is that if we combine too space and time, Poincaré has found an space-time, and he has already said that it would be a tier. It's not the distance. It's not the distance. There's a lot of difference between time and space. So when we make it like this, it's not the difference. When I did my analysis, I removed the C because I considered it was too long, and I calculated a distance. Because the importance of it, it's not the length of the length of the length, but we only said that. So I calculated a distance, and I found it in three minutes. the distance, today, it is a good goal. Yes, but it is so important. It's really important to me. It's really important to me. It's not me who can say it.

2:00:00 It's not me who can say it. It's not me who can say it. It was grave, not it was grave. I want to say it's not. I want to say it's not. But I will say my remarks as well, if you want to go. You said at the beginning of your exposé that these principles unificateurs were not taught at the university. So I don't know why. But I was a teacher at the university, when I did my studies, I didn't have to have any speak of these principles to be able to speak at the point of the isotropy and translation. But it was not even further. Even if, at the time, I made an episode of the relationship which was perhaps a fact. But I regret it. I would like to say that. Why is it not said, I would say, not at the university, but after the bac? Well... Well, maybe not at the bac, but... Well, not at the bac, but... I don't know what you said, but... I don't know... I'm afraid of the... I'm afraid of the manuals of relativity. For us, we see the terms of the analysis... It's incredible! Well, I've learned the relativity like that, but we have a lot of merit, because, in fact, I don't know, it's not G mu nu, it's G. G, it's an object geometry which has a sense on which we can work. Well, if we describe a multilineer, for me, it's clear. Well, if we describe the Tijikar in parallel, covariant, countervariant, and all that we want, it's true that it's difficult. It's not that it's difficult, it's not that it's a big problem. I mean, the real difficulties are not there. And for example, the connections which are defined, which are not as tense, but which are not tense. Well, when we define what is really a connection in a way, there's no problem. So I think there is really, again once, a big progress to make. I don't want to monopolize the word, but to respond a little bit to your opinion, we read that right after, in the big bang, at the moment of an enormous unification, the time and the three coordinates of space could be unified in four coordinates which are the time and the space. But what could that be? I didn't have understood that. No, I didn't understand that. That's my question. But for you, it was not different.

2:02:30 No, no, no, no, no. My question is, what could that be, this kind of nature, a little different, but which unifies the time and the three coordinates of space? If we can have an idea. Well, there are several ways to answer, but, of all, the responses possibles are evident today not in the case of the theory established. It's evident in the case of the theory speculative, like geometry quantic, or gravity quantic, or cosmological quantic. The idea, one of these ideas, which is the Hawking, is that we can have, well, if you dessine, you can have a piece of paper, if there is a piece of paper, you can have a variety like this, well, it's not even a piece of paper, but it's fine, I cut and I recollect another variety, which is like this, well, you can see that if I do it with I'm sorry, but all this is in the case of some theories, which are the theories quantic of the champs applied to the cosmology in the version euclidienne and non plus laurentienne. I'll explain what that means. But it means that here we have something which is a line of the genre of time, which, you see, will become a line of the genre of space. And that, it can happen. It happens, for example, in the horizon of the noir or something like that. the notion of the genre of time or the genre of space is not necessarily well defined. Why do we arrive at these conclusions like that? I would say that it's because in theory quantique, the champs have the habit of doing a prolongement analytically, which works very well. It's to say that, in a certain way, that we will consider the functions...