La Structure Géométrique des Théories de Yang-Mills

0:00 Yes, that's fine. Yes, it's fine. Yes, it's fine. Yes, it's fine. You can start already? I can't wait. Thank you. Is that okay to meet you for what I have? Um, yeah. Thank you.

2:30 It's for Michael, I don't know, I don't know, I don't know, I don't know. We are all ready. Yes, we are ready. Yes, we are ready. Yes, we are ready. Yes, we are ready. Yes, we are ready. Yes, so... Well, we are ready. Thank you very much for being here. And we will begin. Ok. Bon, d'abord, je veux remercier les organisateurs. Bon, il y a un représentant ici, Guido, Michel et Alexei, je vais m'avoir invité. Alexei est au Canada. Pardon? Alexei est au Canada, il est exclusé. Ok. Bon, je parlerai de la structure géométrique de théorie de Ian Mills. as you know, the importance of the general theory of relativity has not only been to be a theory of relativity, but also been the beginning of a general program of geometry, of interaction fundamental. This sort of generalization of the general relativity general has taken form of what we call the theory of Young-Mills. In the same way that the general interaction gravitation in terms of the geometry of the Spartan, the theory of Young-Mills describes the rest of the interactions, the interactions electromagnetic weak and weak in terms of the geometry of certain abstracts and internal and that we will describe. The main problems conceptually in this story are the problems of the relativity movement, the problems of the space, of the time and of the space interne,

5:00 the problems of clarifying the role of symmetry in physics, and the problems of clarifying the role of coordonnées and the principles of covariance in physics. Cette histoire n'a pas commencé avec Einstein, mais il a une longue histoire, et les personnages les plus importants de la même sont Descartes, Newton, Leibniz, Kant, Marx, Poincaré, Einstein, Weich, Schrödinger, et maintenant, je dirais, Barwood, Julien Barwood. Well, there are some canonics, what we call the Gage Arguments, which have the virtue of putting in evidence the problem of the theory of Young Mills or the theory of Schoch in general. We start with a theory of Champs-Livre, with a global symmetry, vis-à-vis the change of repercussions, that we can change the repercussions, but we need to do it in the same way, in all the points of the difference. Then, we invoke a principle of locality, which is not very precise, that we require that the theory is invariant in the case of the repercussions, because the global change is not a sort of relativistic sort of locality. The arguments are never very clear, like what is happening in the paper of Janssen-Niels. And to guarantee that the theory is invariant, vis-à-vis a symmetry, a local transformation, there is a technique technique which is very simple and very mysterious at the point. which is the following. What we need to do is take the Lagrangian of the theory and substitute the derivative ordinary by the derivative covariance. If, for example, we do this for the Lagrangian of the electron, we find this Lagrangian which is naturally invariant vis-à-vis the symmetry which is a local variance. I just want to compare it. Is it correct that this principle of symmetry, global, that comes from the kind of principle, the theory of interaction, other than gravitation,

7:30 so that the principle of locality comes from the relativity general? The principle of locality, we say that they come from the relativity restrain, It means that we can't do the same thing in the same time, but we need to do it locally. And the quantum mechanics doesn't enter here. In fact, it won't be there in the entire exposé. I will not talk about quantum mechanics. I will not explain why. Well, as I said, the garantia is naturally invariant vis-à-vis des changements locaux de repères, mais il faut payer un prix. Il faut introduire de nouvelles variables, un nouveau champ. Bon, la chose étonnante de ce principe de chose est qu'on peut identifier ces nouveaux champs avec une interaction, dans le cas de la garantie de Dirac, avec une interaction électromagnétique. C'est-à-dire qu'on obtient une théorie avec une symétrie locale and there are new degrees of liberty that describe the interactions, what we call the champs of charge. It's really a very mysterious argument because we start with what we call an epistemological challenge, that in theory, we don't have a system of coordinates that we use to prepare the objects in question, but we obtain a new physical content. new. On obtient les champs électromagnétiques et on obtient les termes exacts d'interaction entre les champs électromagnétiques et la matière. C'est-à-dire on obtient trop, on obtient beaucoup. Cette façon de poser le problème est effectivement ça, une façon de poser le problème. Ça veut dire qu'elle n'explique rien, à mon avis. Ce que j'essayerai to follow or to try to decipher the concept of this argument, and in my opinion, in order to understand what happens here, it doesn't need to be a lot of origin, it needs to be very close to the fundamental concepts that are in the game. And that's what I will try to do. I will start with an example trigonometry trivial that, in my opinion, is the key to

10:00 comprehension of this type of theory in general. Well, I said, I don't have an example of the geometry trivial which, in my opinion, is the key for understanding what is happening in this type of theory. Let's take a look at V as an object geometry which belongs to the space vector. As an object geometry, the vector V does not depend of the system of the coordinates that we use to re-reparate. It is to say that we can identify the vector V at the equivalence of the reference and the coordinates of the vector in each reference. We can think that if we have seen this, we can trace the line of marking between what we call the object geometry absolute system that we use, that we have the geometry geometry in itself and then we have all the quantities of derivatives that are liées to the particular repercussions that we have chosen. In my opinion, this is not the line of marking correct because if there is a fundamental technique if it is observable, that is to say if the variables objectives of the geometry geometry are a variant vis-à-vis a transformation of the repair, what we call a passive transformation, they are also a variant vis-à-vis a transformation active of the object geometry. This is a fundamental point of what we call the equivalence of the AIMIS. Let me give you an example. If we take, for example, a vector V and we choose a repair S, we can change the repair, c'est-à-dire faire une transformation passive et utiliser, par exemple, ce repère esprime. Oui, par exemple, les coordonnées de sélecteur sont 1, 1, ici, on aura...

12:30 Mais on peut aussi faire ce qu'on appelle une transformation active. C'est-à-dire qu'on n'échange pas de réflexe, mais on tourne le vecteur, qui aura aussi les mêmes coordonnées. Bon, l'équivalent de Leibniz, ce qu'elle dit, est que ces deux situations, il n'est pas possible de distinguer entre ces deux situations. C'est-à-dire entre une transformation passive et une transformation active. Bien sûr, ici on peut la distinguer. Ici c'est clair qu'on n'a pas touché le vecteur. Et ici c'est clair qu'on a tourné le vecteur. Mais ça c'est parce qu'on a un repère, on a le tableau, on a la chambre, on a un repère de fond, un repère absolu qui nous permet de discerner entre les deux situations. Mais si on plonge l'espace vectoriel dans les vides, il n'y a plus une façon de to distinguish between this situation and this situation. But what does that mean? That means that, that of a certain way, the vector v is not an object geométric. What we thought was that it was an object geométric in itself is not an object geométric in itself or an object geométric absolu. But if you look at the repair, it's exactly the same situation, because you don't have this... Yes, it's exactly that. It's exactly that. We need to find a repair in this situation because we don't have a repair. We don't have a repair. We don't have a repair, so we need to find a repair. that means choosing another vector. And that is completely... After, once we have chosen another vector, the situation is completely relative and symetrically. I can turn one, I can turn the other. Okay. So the conclusion is that the vector V, as an object geometry, cannot be identified in an absolute way. It is to say that the problem is not a problem of coordinates. The problem is a problem of identification of an object geometry. We can't identify it in a way canon. In a certain way, I would say that the equivalence of the line is the point which allows us to pass into consideration epistemological, that is, the invariant objective

15:00 doesn't depend on the system that we use, has a property of the space after in question. That is, that these points can't be identified in a way absolute. It is a property of the notion of vector vector. A point, a vector, a vector vector, has not a number proper. Absolute. There is not a vector absolute. Of course, we could say that, for example, in this situation, let's imagine that we have a vector absolute and that this point is the point V and this point is the point A, where A and V are the number propres of these vectors and these vectors. that the point B is here and the point A is here, and here the point B is here and the point A is here. That means that we can now distinguish between this situation and this situation. Here we have the vector A and here we have the vector B. Okay? like we can't distinguish this type of name, but we could say that there is a difference physical or geometric between these states liés by a transformation active, but that this difference is not observable. It is to say that we could think that there is a sort of space in with an absolute position, but we don't have access to this absolute dimension of each object geometry. This is a very valid position. In fact, if we support this position, after developing the theory of George, we get to the conclusion that if we support that the states liées by an active transformation are physically equivalent, physically different, that this vector is objectively different from this vector, the theory that it is not determined is not deterministic. So it is not a question of interpretation. If we think that there is a sort of space absolute, we need to explain what we do with this determinism that we result. Here, for example, you have a citation of John Norton. Excuse me for my English and French. There are two grounds for accepting legumes equivalents.

17:30 These grounds depend on physical considerations and cannot be proved. We admit legumes equivalents in order to minimize instances of taking physical state of affairs that cannot be distinguished by any possible observation. Of course, we would not be guilty of incoherence if we deny lavenous equivalence and allow the possibility of observationally undistinguishable by different space-times. If we deny lavenous equivalence, we force undeterminism in many space-time theories. Ce qui m'intéresse dans cette édition est un premier lieu le fait que But Norton says that this equivalence of Leibniz depends on the physicality. This is not an argument completely priori. We are deciding that these two situations are not distinct because they are the same situation. As I just said, we could say that they are different situations, but we can't test them in an experiment. This would be a theory, a variable cache. at which we have no access, and we have a phenomenon phenomenon that results. Well, as I said at the beginning, in the case of the theory of John Mills, On va décrire les interactions en termes de la géométrie de certains espaces internes qu'on va attacher à l'espace-temps. Il y a un formalisme géométrique général pour décrire ce type de situation, qui est le formalisme de l'espace-surveil. Je vais décrire maintenant comment on comprend la matière dans les quatre de ces formalismes. On a l'espace. Okay? At the top of each space, we attach a fiber which, in our case, will be a copy of an space vectorial V. And in the theory of the champs, the material is written as a champ. It is to say, in terms of technique, like a section of an space vectorial. What does a section of an space vectorial vectorial? It means that we are doing a choice, an election of an objector in each fiber. For every point of space, we are going to choose an objector in the corresponding fiber.

20:00 This is a section of an objector vectorial space. Before we get into the situation, we could say that we have a continuous distribution of atoms where each atom has a position F in the space and a position V, in an internal space. In a certain way, we could say, okay, a section, for each point of the space, I'm choosing a vector in an internal space. So why call it a section and not a function, a function between the base M and the vector V? In fact, there is a important difference between a section and a function which is the following. Dans le cas d'une fonction, l'image de chaque élément peut être choisie parmi les éléments d'un même espace vectoriel V. Dans le cas d'une section, l'image de chaque élément peut être choisie parmi les éléments d'une copie différente de l'espace vectoriel V. C'est-à-dire, pour chaque point X, on a une copie différente et l'image X sera choisie dans cette copie. Bien sûr, cette distinction pourrait sembler une distinction scolartique étant donné that all the copies are identical, they are all identical copies of the same space vectorial. But in fact, when we make the copies identical in an object symmetry, there is a certain way, there is an irreversibility. For example, the case of an sphere, when we make the copies of the sphere, all the copies are identical. But if I take a point in an sphere, there is no way to say correspondant dans une autre sphère. C'est-à-dire que j'ai des objets qui sont identiques, mais je ne peux pas les identifier d'une façon naturelle ou canonique. Si au lieu d'avoir une sphère, j'avais un map-monde, maintenant je peux prendre un point d'une map-monde et décider quel est le point correspondant dans l'autre map-monde. Ça c'est parce que le map-monde a une structure interne, elle a brisé la symétrie vis-à-vis des rotations. But in the case of a sphere, there is no way to identify them in a way of nature.

22:30 And that comes from what we have said earlier in the space vectorial. Each point of a sphere has no own, there is no real. In our case, we are in a completely different situation. The different copies of the sphere vectorial V are identical, i.e. they are identical, but they are unidentifiable, i.e. there is no isomorphism canon. it is not possible to identify in a way of nature the vectors belonging to different fibers. If there is something that doesn't matter, maybe we can stop it instead of leaving it to the end. Another representation of this situation which is, in my opinion, the most important aspect is the following. Au lieu de penser qu'au-dessus de chaque point on a une fibre, on peut penser que les points ex de l'espace, au lieu d'être des entités simples, sans une structure interne, ont une structure interne. Et la structure interne est dans l'espace vectoriel, c'est comme si on avait gonflé les points de l'espace. This situation, this representation is, in my opinion, more appropriate because we are trying to highlight the fact between the different fibers. There is no connection between one fiber and the other. However, if you represent like this, in the case, I can say that between fibers different, you are in condition of continuality. Yes, exactly. Yes, yes, that's why I said that I was mentioning another aspect, but here, in this case, there is a total space, there is a topology space, there is a continuity, etc. So here, we can believe that there is a natural relationship between one fiber and the other. It's not a question of the space ambient, I believe, you can't have that, but even you have a sculpture defined locally, It's a question of collage between spaces vectoriais, defined locally, you see? But it's a, yes, it's a requirement that doesn't allow me to identify the vector in one fiber with a vector of the other. Okay, so that's what I'll put in evidence with this representation.

25:00 If I have a vector here, I don't have any way to know if it corresponds to the same vector, or if it's a different vector than another fiber. And the material is a continuous choice of vectors in each space interne. I'm following here the terminology of the field chattelette which is called monade to each space, to each point gonfler. Well, I'll resume a little bit what we've done just now. the norm. First, we can say that the space is like a principle of differentiation between copies identically and identifiable in the space vectorial. As said, each fiber sees around them other fibers, certainly identically as they are as copies straight, but affirming curiously their heterogeneity. Les fibres sont monadiques, il n'y a pas de rapport naturel entre elles. On ne peut pas établir une identification canonique. La matière en tant qu'indicée intermonalique est une pure dispersion sans connexion. Chaque atom n'a pas de lien avec les autres. Et l'espace-elme, c'est comme bien appelé une pure extériorité déliée, c'est-à-dire une paramétration continue de monade isolée. Ça c'est une terminologie réglienne, non-reciproque, I found an impression assez convenable. Now we are in this instance of pure dissemination, of pure dispersion, where we don't have a rapport naturel between each fiber. To advance and give a connection to this structure, we have to define another geometry geometry which is what we call the principal space. principale. Dans un espace principale, au-dessus de chaque point, on va attacher, non pas un espace vectoriel, mais l'espace de tous les repères dans l'espace vectoriel en question. C'est-à-dire que chaque point de la fibre sera un repère. Quelle est maintenant la structure d'un tel ensemble, l'ensemble des repères? Afin de répondre à cette question, il faut tenir compte du fait que si on fixe un repère d'une façon arbitraire, on induit un isomorphisme non-canonique entre l'espace de repère

27:30 et les groupes, ce qu'on appelle les groupes structurels, qui est les groupes de rotation. Pourquoi ? Si on fixe un repère, on peut identifier n'importe quel autre repère à partir de la rotation qu'il faut exercer sur le premier pour y arriver. Donc on peut identifier S0 avec l'élément identité du groupe, la rotation identité, et on peut identify le repère S avec la rotation qui nous permet d'y arriver. Cette structure dans laquelle si on fixe un élément, on induit un isomorphisme non canonique s'appelle un torsor. Comme dit Jean Baez, a torsor is like a group that has forgotten its identity. Et si on lui fait rappeler son identité, on le trouve un groupe. but as to choose an identity is an arbitrary choice, the histomorphism is not canon. It is very important to have the fact that it is not what I have at the top of each point. It is an space of reference. After, I can choose a reference in each file and this is how I define what we call a reference mobile. That's to say I'm trying to choose a repair in each fiber, in a way of continuing. Now, we're going to suppose that we have a repair, we're not going to choose a repair in each fiber because there is a repair absolu in each fiber. That's to say, we're going to suppose that we're in a situation in which we can distinguish this situation from this situation, If we do this, we create a canonical identification, this time, because the repair is a repair privileged, between the space of repair and the structure structure. We say that the ensemble of repairs absolutes is defined as a mobile absolu, and the existence of a mobile absolu is a vectorial space in terms of absolutes. That's why I call the theory of Schoch-Mütonian, this structure. and each space in terms is an absolute space. Now, we can, in consequence, compare the vectors who belong to different fibers. Because I can compare the coordinates. If I have a vector privileged in each fiber,

30:00 I can compare the coordinates in a fiber with the coordinates in another fiber. And this is a comparison privileged because the vectors are privileged. We can say that each vector can be identified canonically. that a vector annexe has a signification physical absolute. This structure is invariant vis-à-vis the changements rigid or globals. If I change the reference here, I have to change also here, or I lose the identification that I defined before. If I keep the identification in comparing the coordinates, I have to change the reference On peut dire que l'identique est devenu canoniquement identifiable. Chaque torseur a trouvé son identité, et que l'interogénéité introduite par l'espace M entre les différentes monades a été réduite, parce qu'on a une façon canonique d'identifier les fibres. Techniquement, on dit que l'espace fibré est un peu trivial, It is to say that now, at the top of each point X-DM, I don't have a report, but I have a group structure in question. I have always the same thing in every point. I have lost this heterogeneity between the fibers. Dans cette transparence, j'ai essayé de mettre en évidence quelle est la notion qui subchase à une théorie invariante vis-à-vis des changements de références globaux. Ce que je suis en train de supposer, si j'ai une théorie invariante vis-à-vis des changements de références globaux, est que chaque espace interne est un espace absolu, où chaque So, each of these things is the case of the Newtonian mechanism for the space-time. I don't know if I don't understand, because there are two senses in this context. There are two different senses of what we can call the absolute absolute, or what we can call the absolute absolute space, like the Newtonian. Yes, but now the absolute is not the space, but the space interne. Or, you can choose, in each field, you can choose to have a fixed reference. And then you can have a sort of relationship, but even more weak, which you can't really

32:30 reconstruct. Yes, absolutely, like connection. You see, it's not necessary to choose in each field. it's not necessary that you have a global structure. No, I haven't said that. What I've said is that if we fix a mobile retard in a way of doing it, and that's the only thing we can do in fact, we're not authorized to compare the vectors in comparing our coordonnées. That's in fact, I can't. There's a relationship between the fibers. Yes, but it's a relationship that I can't use to tell if the material changes or not. But if we suppose that there is an absolute repair, that this is not the case, I could do it. But that is not the situation that we have. In fact, unfortunately, there is no absolute repair. But if I fix the repair of a person arbitrarily, I can't compare it, I can't compare it. I wanted to highlight that with this excentrician situation, where I think I have a reference as God given as we say in English, I obtain a theory that is invariant vis-à-vis the global change. That's what I wanted to highlight. And the global variance occurs from a supposition on the nature of the space in question, in order to know that these spaces are the absolute spaces. Okay? But this is not the case that we found because there is no reperts. If we had a theory like this, where we suppose that there is a reperts in each fiber, there are several interpretations that we could make. In the first place, we could say that this internal space is an absolute structure that is incontourable to the theory. It is to justify the need to use this type of repair. Unfortunately, we have no repair but we have to imagine that we have a repair because that is a condition of the theory. Otherwise, we can't do the physics. And another interpretation is the one that says that the presence of these theories

35:00 are observable, but evident, is a weak theory. It is to say that it is not to justify the need of these spaces, but it is to eliminate this notion of an absolute space. It is to substitute the transcendental critique for an effective critique. It is to say that when he did not critique the Newtonian mechanism, he wanted to fund it. For Leibniz, the mechanics of Newton had a problem, so it had to really make a critique effective. So what Marx and Leibniz said was that it had to eliminate these structures absolues by working only with the relative quantité. Finally, Einstein showed that, in fact, we are not obligated to eliminate the notion of the absolute space, we only need to relate. And to relate these structures absolues, we need to introduce new degrees of relational freedom, namely the chamber gravitational It is very interesting because what he is trying to say is that it is not necessary to eliminate the space absolu and construct a relative with the level of liberty that remains. It is necessary to relativize the notion of the space absolu and add something, add some interactions. After, we have constructed the relational theory, like the theory that Marx, Leibniz, Hullet, and it is not evident that the relationship between the relational theory and the general relativity. Well, in the case, as I just said, in the case of the general relativity, we have to introduce the chamber gravitational. In the case of the theory of Young-Mills, we have to introduce a mathematical entity called connection, which is, in my opinion, the physical and mathematical entity the most important of the physics of the XXe siècle, I would say. What does a connection allow us to do? If we have a network in a fiber and we have a path in the space between the fibers, the connection allows us to transport these vectors along the path. It is to say that the connection allows us to choose a path in the total space which is what we call the horizontal movement of the chamber in the base.

37:30 After, at the technical level, the connection is a different one. If I take the boulevard by a section of this connection, and I find a chamber in the base of the chamber that can be identified with the chamber of the church that I included in the first transparency. This connection represents effectively the interactions. I think that's the connection with the torsion or without torsion? I think that's the endorpsion. In fact, I've never seen the problem with the torsion. In the math, it's more large than the torsion. I've never seen the connection with the Yang-Mills with the torsion, but I don't know. Bon, quels sont les points importants ici ? D'abord, que ce rapport intermonadique ne se fait plus localement. C'est-à-dire que pour mettre un rapport avec cette fibre, il faut suivre un chemin dans la base. Ça c'est un point important. Dans l'espace fibril nutanien, ce n'était pas comme ça. J'avais un report absolu qui induisait directement un rapport qui ne dépendait pas des chemins dans la base. Et après, un autre point fondamental est que les transports parallèles dépendent en général du chemin suivi dans la base. On pourrait dire que l'information transmise par un chemin dépend du canal de transition choisi. Ça c'est un point fondamental parce qu'il montre qu'on ne peut pas identifier les fibres. the fibres without identifying them. Because if I take a chemin, I obtain an element. And if I take another chemin, I obtain another element. So I can't identify this repaire with this repaire or this repaire. A connection does not have an identification, it is to respect the heterogeneity between the fibres. After, instead of following I can follow a loop between the fibers, I can follow a loop between the fibers, which is a point and return to the same point. That is a loop. And a loop is defined in the same way that a loop between the fibers is defined in morphism between the fibers. Now, we define an automorphism, that is, a rapport intramonalite. We can say that each way to get out of itself is defined in the fibers.

40:00 This loop allows us to define what we call the loop of anonomic, and it's an observable Here I say that the parallel transport depends, in general, of the chemin. What does that mean, in general? It means that there is a quantity that we can deduce from the connection, which is what we call the courbure. And what is the importance of the courbure? If the courbure is null, we can show that the parallel transport doesn't depend on the chemin. Techniquement, on dit que la connexion est intégrable. Et il est possible donc de définir ce qu'on appelle un repère mobile naturel, en intégrant la connexion, et identifier naturellement toutes les monades. C'est-à-dire, si le transport parallèle ne dépend pas du chemin, je peux prendre n'importe quel chemin et j'arrive toujours au même point. Donc je peux identifier ces deux repères. Donc je peux tracer ce qu'on appelle une section, qui est une section ennuyée naturellement par la connexion, par une connexion à coupure nulle. If, by the way, the curve is not null, which is the general case, the parallel transport depends on the chain, the connection is not integrable, and the different fibers cannot be identified. We can say that the curve is obstacle to the possibility of reducing the heterogeneity introduced by the space M between the monades identiques Vx. If the connection is null, I can identify all the fibers, so I'm reducing the heterogeneity which produces the space between these fibers. Here, there is a description in his style passionnée of Gilles Châtelet on this situation. He said that by the existence of a cobweb, the monad experimented the rebellion of space is captured by a cartridge cartesian. It is then forbidden all the global decompositions to assume the complexity of all the decompositions instantanées by the infinitesimal connection. The cobweb permet to appreciate in some sort the flux of exteriority which is in each monad. We are also trying to appreciate positively the connection as a foundation of the spatial

42:30 As an aptitude to assemble the different incarnations of the same structure Each monade holds the space En esquissant simply autour d'elle un system of coordonnées La polvure concept derivée fait obstacle à l'aboutissement d'un tel projet Elle introduit une espèce de négativité en géométrie In fact, we have here a sort of, what can we call a dialectic, in the sense that the space introduces a difference between the identity. There are copies identically who are no longer identifiable. The connection makes a rapport to what has been so differentiated. And the curve does obstacle that this connection retombe on the identity. I have a principle of dispersion, I have a principle of assemblyment, and I have something that prevents this assemblyment to retombe on an identity. Well, as I said earlier, in an space that we have a connection, there is a intermonautique qui n'efface pas pourtant l'hétérogénéité entre les monades. Je ne suis pas en train d'identifier ce que l'espace a différencié. L'espace M n'est plus seulement un principe de différenciation, sinon qu'il devient le support de parcours qui véhicule le rapport entre les monades. Ce que je dis auparavant, pour connecter des monades, il faut suivre un chemin. Je ne peux pas le faire en abstrait. Comme le rapport entre les monades ne dépend pas d'une identification des coordonnées, comme dans l'espace fibré newtonien, maintenant je peux changer la référence localement. C'est-à-dire que la théorie est invariante naturellement maintenant vis-à-vis des transformations de choses locales, définies par un groupe qu'on appelle le groupe de choses. Techniquement, c'est le groupe d'automorphisme verticaux du fibré. What is important is that this invariance locale is a consequence of the fact that the space in terre is not an absolute space. It is to say that I am now in order to identify the fiber in a connection, not an identification. So each fiber is heterogeneous to the other. So now it is legitimate to choose the port of the terre in each fiber. But this symétrie occurs from the fact that these spaces are not absolute.

45:00 It is to say that I do not have absolute absolute in each fibre. Is it clear? I think when you say that I do not have absolute absolute in each fibre, it is a bit ambiguous. Because what you want to say is that you do not have absolute absolute global. No, I mean that in each file I have no repair absolutes. What do you call an absolute? You have a repair local in each file. Yes, but it's a repair that I chose. It's not a repair that is made with the file. Well, we thought that the repair was made with the space, apparently. So it's not obvious. You mean that there is no choice canonics in each file? Yes, but the fact that there is not a canonic comes from the fact that there is no repair. And that is not obvious. But it's absolutely the same way that there is no repair canonic in the global space. That's what I said before. I'm trying to interpret the theory of Yang-Mix exactly the same way that the general relativity. I'm trying to say exactly the same thing. In each paper there is no repair absolute. Exactly the same way that in the space there is no repair absolute. was not so. In fact, we believed that there was an absolute error. We believed that the fixed star was an absolute error. We believed that there was a fixed point in the world. And so we believed that each thing could be an absolute error. But this is not obvious. In fact, we had to suppose that we had an absolute error to develop the mechanism of Newton. It's incredible that all those who critiqued Newton did not have been able to develop a technique like the Newtonian. It had to be wrong, not to be wrong because Newton knew very well what was the weak point of his theory. But it had to be left aside. It had to continue. Pardon ? Pas de réflexe absolu, mais classe absolu. Pourquoi classe ? Merci. Hein ? Quand même, il a eu une relativité galilienne, il n'y a pas... Oui, mais le problème de la relativité, chez Newton, ce n'était pas le problème de la relativité galilienne, mais c'était la relativité de l'accélération. L'accélération... Bien sûr qu'on avait une relativité, mais qu'il n'y avait plus une relativité restreinte, plus restreinte que la relativité restreinte.

47:30 Well, the last part that we will do in terms of the geometry structure that we will consider, is the following, we have defined the connections. The connections are dynamic, they are physical entities, so we have to define the space of configuration of connections, L'action du groupe de choses sur chaque connexion fait de cet espace un espace vibrer, c'est-à-dire j'ai l'espace A de toutes les connexions, et j'ai une action du groupe de choses sur cet espace qui le projette sur l'espace quotient, même ici CH, c'est le groupe de choses. Et c'est maintenant, en théorie de Ian Mills, qu'on fixe le repère. And how do we do that? We do... What is this structure geométrique? There is an space quotient where each point is an equivalent of connections which are equivalent, which are physically equivalent. And each fiber is the effect of all connections which are equivalent in the action of the group of charges. So, for choosing a reference of a person's abstract, we need to trace a section of this space. For each fiber, we need to choose a representant. Of course, once we have chosen a reference, we need to establish a method to enter the objectives of the theory. This is not trivial, we need to introduce a formalism, which is quite complex, called the formalism of RCT. It is a formalized homology and, in general, it is a formalized which allows us to obtain the homology of the Alchèvres de Lille group of Schoch. And the fundamental point of this formalized is that the homology of the zero can be identified with the ensemble observable of the theory. It is to say that it is a formalized homology which allows us to draw d'une façon algébrique quels sont les invariants objectifs en ce qui concerne l'action du groupe de chauche ça ça semble très compliqué mais en fait c'est ce qu'on a dit au début c'est à dire on a on fixe un repère après il faut décider quels sont les invariants vis-à-vis le changement

50:00 de repère et maintenant les groupes de symétrie qui changent de repère et les groupes de chauche donc il faut établir quels sont les invariants vis-à-vis l'action du groupe de chauche et ça Well, I will start my conclusion. First of all, we can say that here I will summarize what are the three fundamental points in the notion of space that comes from the theory of Young-Mills and of the general relativity. general. En premier lieu, les points des espaces en question, soit l'espace externe, soit les espaces internes, n'ont pas de signification physique, les points. Autrement dit, ces espaces ne sont pas des espaces absolus, il n'y a pas un retard privilégié, c'est ce qu'on peut appeler la égalité de l'espace. Bon, comme l'a dit Einstein, the requirement of General Cobain takes a way from space on time, develops women's physical perspectives. Ça c'est une affirmation qui n'est pas évidente du tout, parce que moi, au moins, quand j'ai lu ça pour la première fois, je me suis dit, mais comment, dans la physique newtonienne, l'espace était un truc absolu qui était là, et d'après Einstein, l'espace devient une ambiguïté physique, with a dynamic, with a deviant object, but, pourtant, he is saying that the relativity general takes away from space and time from the assignment of physical objectivity. So, this is not an affirmation all at all, no ? But, well, I believe that it is to read this phrase in the sense of the point of space, and not the significance of physical Yes, that's right. But it's not obvious, because we have the beam gravitation that we ajoute, it's a structure defined on the space. So we are facing the support, but we remain with the structure. It's like if we had a smile without a cat, right? We are facing the cat and we remain with the smile. — Par contre, dans Internet, tu peux dire que exactement dans le cas instanien, c'est une covariance qui presque signifie une objectivité, c'est-à-dire tu as… — Oui, mais c'est une objectivité qui n'est pas attachée à l'expertence, oui, c'est-à-dire… — C'est covariance, le sens exact, l'objectivité dans le…

52:30 — Oui, oui, dans le sens que je viens de dire dans le transparent précédent. There is a group of theory that determines which is the invariant of this theory. But it is interesting. In a certain way, it is necessary to add a structure of the beam gravitational to efface the space-temps. It is paradoxical, in a certain way. It is necessary to add a structure to the space-temps to make sure that the space-temps disparate as an object object. You don't add a structure, you identify. I'm trying to add a connection, a matrix, it's a structure, yes, I'm trying to complexify the structure of the variety, I'm trying to add something, but, I insist, no, it's necessary to add something to the variety because the point of the variety has no signification, absolutely. Bon, j'avance. Ça, c'est le premier moment important dans la conception de la notion d'espace qui l'écoule de théories des jauges en général. En deuxième lieu, l'espace M, en tant que principe de différenciation, introduit une hétérogénéité entre les monades. Ça, c'est un point fondamental aussi. On peut dire, bon, l'espace-temps but he has introduced this space. And in the same time, the space, as a condition of possibility of a rassemblement, is the horizon of a parkour that is known as the monar. This is the same thing that I have already said before. The space introduces a heterogeneity, but in the same time, the horizon to recollect what we have separated. Voilà une citation de Ghegel qui, je crois que peut-être renforce la chose, mais je crois qu'il souligne déjà ces trois aspects de la notion d'espace. Il dit « La détermination première ou immédiate de la nature est la très universalité de son être hors de soi, l'indifférence non médiatisée de celui-ci, l'espace. is called the ideality of the one beside the other because it is the one out of the other. When we do abstractions of those who, in the concept of Kantian, belong to the idealism subject and to its determination,

55:00 there is this exact determination that space is a simple form, i.e. an abstraction, and, to say, the one that constitutes the immodality. Speaking of space points, as if they were the positive element of space, is inadmissible, given that because of its indifference, the space is only the possibility, and not the possessor, of the posteriority reciprocal. Why do I say that I have these three moments? Because in the first place, in the reason of the indifference of the space, that is, we have here an argument of the symmetry, which was already present at Leibniz, the points of space don't have a signification of physical. That is what he calls, after Kant, the idealism of space. At the same time, being one next to the other as an extériority in different and immediate because there is no mediations that connect what we have separated, and the condition of possibility of an extériority reciprocal, i.e. the space with condition of possibility of a assemblyment connective. Je vais souligner avec tout ça que ce n'est pas suffisant de rester dans le moment de l'idéalité, dans le moment où on a effacé cette objectivité de l'espace. Je suis en train de résumer tout ce qu'on a dit, donc ici je vais souligner trois moments. The first moment was when we had a real space without connection. At this moment, the space M operates only as a principle of differentiation, i.e. it produces a pure dissemination without cohesion. We were in an instance of the total dispersion, where there was no relationship between the filters. After, we have a unique space and connection. Now, the space M operates as a principle of differentiation which is a condition of possibility of a rassemblement. It is what Hegel is after, a reciprocal relationship, which is located locally, because the connection establishes this rassemblement in a local way. And in the other extreme, we have a space of space without connection, but which has been canonically trivialized, which I call a space of space newtonian.

57:30 Now we are in the context of an identity pure, n'introduit plus d'hétérogénéité entre les différentes monades. On peut dire que l'espace n'y passe pas. Ici c'est intéressant, on a les deux extrêmes. L'extrême où l'espace n'introduit pas aucune hétérogénéité, on a l'extrême où l'espace produit une dissémination extrême et on a les termes intermédiaires où il y a une connexion qui permet de relier sans identifier what has been discarded, what has been differentiated. In terms of the relativity of the movement, I would say that the character non-absolu of the space in question, so the space external, so the space internal, has as a consequence of the relativity general of the movement, but in an ensemble of the hierarchies of the freedom. Dans le cas de la relativité générale, on peut dire que l'accélération n'est pas relative à l'espace absolu, mais à une autre entité physique, à savoir les chambres habitationnelles. Pardon. Je pourrais remonter à la question précédente. Alors, là, peut-être c'est utile de retourner à l'analogie avec théorie newtonienne, théorie relationnelle et relativité générale à ce point-là. La théorie newtonienne, elle vit dans l'identité pure, et le cas de l'espace vibré muni de la connexion est le cas de la relativité générale. But the case of the space vibrational connection doesn't correspond to any physical theory. The case of the relational theory is something that we take the case of the pure identity

1:00:00 but it begins to operate with the idea that there is no equivalent to the other. So it's the same thing, it's really identical without being differentiated, but it's not something that we can... But it's not absolute. It's the same space, but it's not absolute. It's a different critique. Yes, it's not absolute and it's not identifiable. Because it's not absolute. That's why we are in the case of what I call a pure dissemination. Because I can't identify what the space has. And I can't do that because each term has not an absolute reference. It's a term with a symmetry in a strong sense, in a absolute sense. It's not a symmetry of a change of reference. It's not a symmetry epistemological, let's say. It's a property of the sphere that has not an absolute reference. absolute, that I can't choose a point in a canon, to identify a point in a canon. What I wanted to suggest is that, in fact, the two possibilities that we have to use in It's the case of the functions, in fact. It's really the same space. It's not different. And the case of the vibration-connection canonically trivialized, in a certain sense, est retournée à l'idée des fonctions plutôt que de la déception. Exactement. Ah, et là, on pourrait ajouter la critique que cet espace où les fonctions prennent leur valeur n'a pas de repère absolu, et on dirait une théorie relationnelle.

1:02:30 Pardon, dans le troisième cas? Oui, dans le troisième cas. Mais j'ai un repère absolu, je n'ai compris. Si justement je sais partout la même chose quand il y a un repère absolu, je peux l'indiquer tout. Ah, mais… Je ne sais pas si j'ai… On y entre. Si au lieu d'avoir une section, j'avais une fonction, ça veut dire qu'au-dessus de chaque point, j'ai la même chose. Oui, bien sûr, mais tu as un espace vectoriel, mais… On peut appliquer la critique que tu as faite au début, que si on a un espace vectoriel et il faut penser à l'équivalence Leibniz, en effet, si on a un seul espace vectoriel, il ne faudrait pas introduire un repère absolu. On peut dire que c'est la même chose ou une chose différente, parce qu'on a l'identification des espaces vectoriels, mais c'est la seule chose qu'on peut dire, c'est quelque chose de relationnel, mais on ne prend pas le repère absolu comme la chose qui permet l'identification. Oui. C'est peut-être plus naturel de voir des choses comme ça que par le détour de l'espace UV et puis l'identification en disant un préfet absolu. Je ne sais pas si je comprends le fond. I don't know if it's the same thing. Mathematiquement, of course, you can always distinguish your space vectorial just by the point. It depends on your position to the fact that you live in the space vectorial. Because if you just take your unit, you have some ways to distinguish between the two?

1:05:00 Well, of course. I suppose that the mathematical structure that you have to use is an space vectorial. But I could use an space vectorial with a reference. It's the same thing as an space affine to an space vectorial, where I have I could think that I could use another mathematical structure in the space vectorial. No, no, no, it's not that. I mean, if you look at things like mathematics, you just say that you have a sphere, you have vibrations with all the space vectorial tangent, Very good. You have all the problems with the NCT because you have some repairs on your sphere, for example. So you can introduce space ambiance or not necessarily you can have a sort of code that we use in the geography. You have an indexation of points. And then, all these spaces are indexed by points of sphere. Oui, mais c'est exactement ce que je suis en train de dire. Il y a des possibilités. Soit on a un espace ambiant qui fonctionne comme un repère absolu ou quelque chose comme ça qui nous permet d'identifier les points, soit on peut faire un choix arbitraire. Je ne vois pas une troisième sortie. Soit on a quelque chose qui nous permet d'identifier les points d'une façon canonique, soit il n'y a pas une façon d'identifier ce qui est quand même identique. but it's like a non-propre, it's just like a non-propre. Yes, so each point has a non-propre, so each point has not a non-propre. That's what I'm trying to demonstrate with the example here. And in fact, apparently what happens is that each point has not a non-propre. It's to say that there is no doubt about it. But this is not, I insist, an argument a priori. I can think that in place of a sphere, I have a mark of the world. At the top of each point, I can think that I have a mark of the world which would naturally allow me to identify one point with the other. Because the space interne could be a structure with a structure in concept. But this is not the case of the sphere vector.

1:07:30 Also, in mathematics, you know, it's not, normally, we look at these things in a more constructive way. You are part of this idea, which is maybe interesting as an approach. I don't want to critique it, but just other approaches, a little more, I don't know, more standard, if you want to just construct these things, tranquillement, c'est-à-dire juste tu dis bon je peux avoir une idée, j'attache à chaque point mon fibre et tu vois, c'est-à-dire je ne parle pas de cette idée que c'est pas nécessaire que tous les, disons, toutes ces fibres soient mises au nom, c'est ça que tu issues quand je dis c'est tout identique et pas identique, c'est copie, tu vois, c'est It's not necessary to take a bit more constructive, it's just to say, there is a fiber, and it's what we call it, I don't know, the pre-fessor, for example, we have a dissimulation, everything like that, but then you add a sort of condition of collage, and, of course, if you start in the different case, like you do, which is for the physical, Dans quel sens tu as déjà ton connexion dedans de début, tu vois ? Mais je ne vois pas quelle est la troisième possibilité entre ces deux situations. Non, non, je ne sais pas. Je ne comprends pas le point. Je veux juste dire que ton approche, bien sûr, c'est très important, intéressant. Je ne veux pas objecter quelque chose quand je dis. but just from a point of view mathematical or standard, does not necessarily mean all these sorts of problems. I don't know. I don't know. Because you built your... Yes, yes, yes, yes. But of course, what you're saying is that I can build a vectorial space where each point has a name. Of course, I can do that from a point of view mathematical. But apparently, it's not the best way to describe the interactions. Bien-sûr, c'est ce que j'ai dit auparavant quand j'ai dit au lieu d'avoir une sphère on pourrait avoir une map-monde. Dans ces cas-là, bien-sûr les choses pourraient être comme ça, ce n'est pas une déduction qu'il faut avoir une sphère avec une symétrie, on ne peut pas la déduire, c'est un fait physique.

1:10:00 Et peut-être il y a une structure, comme je l'ai dit au début, il y a une structure, which is what we call an space in itself, but it's another position. To say that each space is an absolute space, but I can't access it from a physical measure. But what I said at the beginning is that if we assume this way, we have to explain things that are a bit complicated. For example, the theory that occurs is indeterminable. But it's a very valuable possibility. I'm not sure that it would be absolute. If you build vibrations or the cables in general, the faisceau, I don't see where you are necessarily. You see, I think it's a little different. It's just that you don't talk about the beginning, the idea, that you have something identifiable, an authentic space, and then you distinguish it, and then you compare it. It's just for the other logic, the more constructive. But it's not to say that you have something... But the logic constructive is the same thing. If you have an ester, if you have an enchantment to each point, there is no way to decide if a vector here is the same vector or not. But why... Of course, it's not the same. But I can say that every vector is very well identified. It's not the same. But why... You see, you can't talk about the beginning, but you can't talk about the beginning. But if I'm in an entire space vectorial, I'll be in this situation. I can make a transformation. If I don't plunge the space vectorial in an ambient space, if I don't plunge the space vectorial in real life, there's no way to distinguish between this situation and this situation. I don't understand. Except if we assume that each point has a own own. But if I don't assume that each point has an own own and a certain space vectorial plonged in the living, there's no way to distinguish between this situation and this situation. If I say that it's not like that, I don't understand. But it's important, right? But I don't know how to distinguish between this situation and this situation if we don't have an external or internal structure to the space vectorial Well, in the case of the theory of Yang-Milz and the relativity, it's important to distinguish because I have a chamber of material and so I want to see if the chamber of material changes or does not change.

1:12:30 And to know if the chamber changes or does not change, we need to compare a vector here and a vector here. You can't distinguish between the two images, but you can distinguish between the two transformations, because they will be dual, it's not exactly the same transformation. And you define it in a general way, that you have any coordinate system, when you say coordinate free, you don't have the coordinate system fixed. on l'a accentué au début, même si on travaille coordonnées frites, sans coordonnées, même comme ça l'objet géométrique ne peut pas être identifié. Ça c'est justement le point que je veux souligner, mais ce n'est pas une question des coordonnées. Même si on laisse toutes les coordonnées à côté, ce que j'appelle, l'objet géométrique absolu dans la séparation vectorielle n'a pas une nature absolue, parce qu'il n'y a pas un repère qui pourrait me permettre de l'identifier. C'est un point très très important, c'est pour ça que je vais comprendre ce que tu For the mathematician, this question of identity, I always do anything with identity, it's not a problem. For who? For the mathematician. To the point of view of mathematics, it's a problem that doesn't exist like that. For me, it's really the point of view of the theory, It is not a problem of reference. It is a problem that even an object, as I said earlier, the line of marking is not passed between the coordinates and the object geometry. The line of marking produces a clivage in the object geometry itself. The object geometry itself is not an object geometry itself. There is no object geometry itself. except if we have an absolute respect, an ambient space that allows us to fix things in an absolute way. Of course, it could be the case, but it's not the case in physics. In fact, it's not in contradiction with the fact that the two operations are the one one to the other. It's like an element. You look at the circle, you look at the circle and then you move and ask if it's the same circle and the circle. And of course you move

1:15:00 But it's not the same, but it's the same circle. Justement, these are the copies identiques. That's what I also said. The fundamental point is that I have the copies identiques, but I can't identify what is identique. Of course, it's the same circle. But if I take a point in a circle, I can't identify it with a point in another circle. Sauf if I have the symmetry of the circle. C'est-à-dire si j'ai une structure interne au cercle qui me permet de repérer les points d'une façon absolue. Mais si je ne peux pas, si le cercle est à une symétrie, je ne peux pas identifier un point dans un cercle avec un point dans un autre cercle qui est complètement identique. C'est vraiment incroyable, non ? C'est des choses identiques que je ne peux pas les identifier. Bon, après on continue. I will end with transparency. I talked about the relativity of the movement, which was another important point. I said that the character non-absolu of this space in question has as consequences the relativity of the movement, but this is the fundamental point which is in the ensemble of the archives of the degree of liberty. This is the point which was introduced by Einstein. I can't build a relational theory only in effacing the absolute space, but without anything ajouter. There is an interaction. In the case of the general relativity, we can say that the acceleration is not relative to the absolute space, but to another physical entity, namely the chamber habitational. It's a curiously familiar to express the general relativity. Because in general, what we can say is that there is no difference possible between an effect of inertia and an effect of fragmentation. So it's not to say that there are two different entities, but in fact, there is no one single process. What I want to say is that, let's say we have the matter. Newton has the matter and the absolute space. absolute. And the matter can be reparated by the absolute person by this absolute space. Now we have the absolute absolute space. So we obtain a theory relational, but in the effect of having a theory relational, we have to add a new degree of freedom gravitation. So now we have a theory completely relational, where we have the correlation

1:17:30 between the degree of freedom material and the degree of freedom geometry. But we need to add the degree of freedom dynamic and effective. That's what I wanted to mention here. I don't know if we are trying to say the same thing. I'm sure, but it's the word entity that I'm talking about. Entity, for me, it's a synonyme of the degree of freedom. It's to say that we need to add a new degree of freedom with a dynamic, with an agrarian, etc. En même temps, avant l'identification, effectivement, on identifie le champ gravitationnel et l'espace-temps. Mais en même temps, on peut aller à l'étape d'avant qui consiste au fait qu'auparavant, ce n'était pas identifié. Et le choix, et c'est ce que fait la revue que tu cites Rodébillis, Madeline H. Técard, c'est que finalement, ils disent que soit on vient de le champ gravitationnel, soit on vient de l'espace-temps. Oui, c'est ce que je dis quand la réveil citation d'Einstein. Absolutely. On peut finalement se débarrasser de l'espace-temps, et c'est pour ça qu'il n'y a plus de conception relationnelle, de champs sur des champs. Soit on garde les chats, soit on garde la sourire. Bon, je lis la citation de Rovelli, il dit, « Reality is not made by particles and fields on the space-time, it is made by particles and fields, including the gravitational field, that can only be localized with respect to one another. No more fields on the space-time, just fields on fields. That's in his book on the quantification. Well, to finish, I wanted to highlight another thing, which is the following. Kant, I think a very important point of view of the Newtonian, which is the following. The Newtonian physics begins with the notion of an absolute space, which is called a background absolute. It is a priori condition, the theory. And for that, the theory Newtonian remains a relative. I start with the notion of absolute, so the theory that I obtain is a relative, which is called a background dependence. Par contre, ce qu'on a compris après Einstein, c'est qu'il faut faire exactement le contraire.

1:20:00 Il faut commencer avec un principe de relativité qui dépendisse de la théorie de tout type de background. Ici, j'appelle ça une suspension de la théorie. C'est-à-dire que chez la théorie, je suis en train d'éliminer les backgrounds sur lesquels la théorie est appuyée. So we need to suspend the theory, it is to make it independent of the absolute absolutes externes, which are not described by the level of the effect. After a theory, we can extract the absolute quantity, it is to make an objective. If we start with a bad absolute, we start with a good relativity, we can then obtain the absolute correlations, because these are the absolute which depends on an interaction. Here I take of course the term absolute in the sense of absolute for the group of symmetry and the theory. In our case, it's for the relativity general, it's the group of different morphisms and different parts. And in our case, it's the group of George. I represent this equation in the following way. For example, I start with a theory where I have two degrees of freedom which are, in a certain way, to pull up a background absolute. And what we need to do with a theory is to suspend it. It's to say, to relativize the background. It's not to eliminate it. It's only to add a group of symmetry which remove all the physical characteristics of this background. but we also have to add an interaction between the degree of liberty. This is the chat, this is the sourire. This is called a suspension relation in theory. We start by a relativization of absolute balance, that is to say that we impose an invariance in terms of a transformation active by a group of symmetry chosen. And then we introduce these interactions in a way that the different degrees of liberty, in a certain way, support them mutually. Because we have effaced what allowed us to repair it in a way absolute way. Thank you very much for your patience and attention.

1:22:30 J'aime beaucoup l'idée que la justification des symétries locales doit être trouvée dans une notion de connexion. So, we need to have the idea of a non-trivial connection to justify the idea that there is no absolue dans les fibres. Mais, en continuant d'un discussion d'avant, je ne suis pas sûr que c'est la seule façon pour critiquer la notion d'un espace absolu. I think that the critique of Leibniz and the theories related to the Bavre are a different way. Yes, I am completely agree. It is possible, and that's why I separated the Bavre of Einstein from the Bavre. Einstein had given a possibility that I was going to describe. in fact, who worked on the definition of relational theory, which does not depend on... It is to say, what he did Barburg, he worked on a relational theory of the Leibniz, and then he showed that the canonic Hamiltonian of the general relativity has exactly the same shape as the relational theory that he built. But it is actually a different way. It is to say that it is possible to build a relational theory without anything ajouter. If it was that the point, yes, I completely agree. Just, that's a very good thing. I can see things more mathematiques. And it's, to my opinion, very interesting to compare, because maybe in mathematics there are certain things that are perdues, which are the physical intuition. But, on the other hand, when you talk about symmetry,

1:25:00 which structure is mathematique? Is it a group of symmetry? Yes, but you know, even a solution absolutely natural, rather than talking about groups of different people, look at what it's called, Propoide. It's to say that you don't have an intensity of one thing, but you have several, and they are all in the same way. Yes, that's all the nature. And the second thing, of course, if you have something more general than the hypoids, and that's the category, I don't know if in physics... I've heard Alain Pond and he talked about something about the theory of the jauge, something that we discovered. I don't know how to build it, but I don't know how to build it. But for all these issues with the space of the fiber, you can build a little bit in a way, apparently transparent. Maybe it's a sort of a false, I don't know, simplicity is a little bit a little bit and can't avoid certain problems that you pose, but the simple way to build a sort of thing is what we call the vessel, You just take part of the topological space, because in fact, all these local stories, of course, it doesn't depend on the properties, let's say, differentiable, or even vectorial. It's not possible to see something about vectorial, in general. But it's all the topological properties that are different from what is global and local. Then you can add something like a free fiber You can open it and put it together And this also shows that this idea with Monade I don't know, I'm not a famous name But there is something that I'm a little gêner In this idea of Châtelet To look at this fiber like Monade And this was only the terminology that I had in Châtelet I didn't want to make it I was not sure that this was a very good expectation. Yes, it was not the intention. Yes, yes. And plus, you see this situation, like you said, you disperse absolutely, you have your fibres, what we call a bandeau in English, and then you add conditions of collage between fibres

1:27:30 who respect the continuity of your base space, which is called FISO, finally. And here it is not necessarily that it isomorphism, it is to say that you can have a sort of connection, it is to say that it works in one direction and not in the other direction. But in this approach, of course, you don't have any problems in this formula, because if you want, it's maybe naive. There is a formulation that you would have made in this structure. At the beginning, you have different fibers, you don't have questions that are just different, and then you call them, and they are also different. And when you call them, you have something like connection, or mobile, but you see, it's a little bit different from the other side. And, well, I don't want to say that it's a real point of view, but it's just to sign that we can see things on the other side, you know. Yes, of course, maybe we can use other structures. In fact, I think it was Erichmann who introduced the group concept, the group OIL, just to substitute the notion of connection and to make a connection and all that. Effectivement, il y a d'autres structures possibles, mais bon, je n'ai jamais vu au moins une formulation, mais peut-être qu'il faut l'affaire, de la théorie de l'Amus sur la relativité générale ou de ce type d'outils mathématiques. Mais bien sûr, ici, je travaille dans le cadre de l'espace vibré et de connexion. Après, on peut substituer la notion de l'espace vibré par une autre notion peut-être plus générale et la notion de connexion par une autre notion. Si, si, si, cette question maintenant je comprends bien. Si tu prends une variété différenciable, comme dans le cas de l'activité générale, et tu as des espaces vectoriels que tu définis, alors tu as connexion déjà par défaut. C'est-à-dire que tu n'as pas besoin d'ajouter cette... Quand, pardon? Dans quelle situation? You already have connection. If you have a structure of differentiability, you have a different form of the exterior, and you already have connection. It's a bit like connection. In our case? No, not at all, no?

1:30:00 No, not at all. Si tu peux isoler tes espaces fibrés, c'est par sorte de manière artificielle, parce qu'il y a connexion naturelle, tu peux dire, même dans les structures mathématiques, parce que tu prends les cas spécials quand beaucoup de structures sont déjà là. Dans les cas que je viens de décrire, il n'y a pas de connexion naturelle. C'est un point fondamental, parce que justement, en plus, la connexion devient une entité dynamique. Donc elle change, il y a tout un espace de configuration de connexion possible. So, if we have a metric, we can determine the corresponding connection, but this is not what we do here. The connection is a dynamic entity, and we have an space of configuration, we have an agrarian which, in which the equations describe how the connection changes. So there is no natural connection. naturel. Pour obtenir une connexion naturel, ça ne sert à rien. On a une connexion naturel dans le cas de fibres et d'un champ, si il y a la connexion du poids. Question philosophique. Au sujet de ce que vous avez montré à la fin, de la conclusion which consists of an absolute background, I think that the pair of all of this, despite what it seems to be, is Kant. Because Kant, what did he do? He removed the idea of an absolute space, and put in the form of the general in relation to terms that are plongered in space. And what did he do exactly the operation numero 2 that you described, that is to suppress the space as the things, the substances, etc. Et à la place d'être une structure générale de relation qui est la structure légale des catégories. Je pense que c'est… Oui, je suis complètement d'accord. Je suis complètement d'accord. C'était forcément du bravo, c'était Kant qui l'a… Mais le point que je voulais souligner, c'est que pour lui, il ne fallait rien faire à They could only interpret it in a way transcendental, but in fact, it's... Yes, but in the same time, yes, I can't agree with you.

1:32:30 So, he was a bit rigidified by his interpretation transcendental. In the same time, who knew that it would work, he liberated all the possibilities of reading the theories and of the reconstructing on everything else. That's why the quantism is very active in the interprétation of the things that we do. But it's not at the level of the physical space, it's at the level of the transcendental of the physical space. It's not at all the same level. But it's relational, but it's... What do you mean by the physical space? The physical space is the one that we know. I don't know any other space than the physical space. It's the one you know. It's the one you know. It's the one you know. Einstein said something amazing, which is more transcendental than Kant. He said that the physics describes the reality, but what is the reality if it is what describes the physics? To say that the space is a particular space, and the space, let's say, that relates to a form of sensibility, it's another thing. It's exactly the same thing. it's a way to describe the same thing, that is, essentially, the recognition of the possibility of the existence. I am completely agree. That's why when I summarized the three moments in the constitution of the space, the first moment was what I called the identity of the space, which is the effacement. Le passage de cet espace à un espace qui est là, quand même, mais qui n'est pas un espace absolu, qui n'est pas en fond d'absolu sur lequel on peut repérer les choses d'une façon absolue, mais qui, quand même, il est là d'une certaine façon. Il est là. Il est là. Même dans la relativité générale et dans toutes les théories contemporaines où on utilise... C'est pour ça que j'ai dit que Einstein n'alimine pas la notion d'espace, il la relativise seulement. Moi, j'ajoute très simplement que ce qui est là, ce n'est pas l'espace physique à ouvrir, etc., c'est les conditions de la mise en relation qui sont constitutives d'un espace. Après, l'autre point fondamental, c'est si cette condition de mise en relation découle d'une subjectivité transcendantale.

1:35:00 Hegel, bon, il ne se signe pas à ça. I would very like to say that, because there are all sorts of other interpellations that are made out of the ancient, which are completely based on the real conclusion of the subject. So that's not it? Yes, yes. It's not a problem, no? Is it possible to give a different response to the idea of subjectivity without that the whole world can maintain? It's a big problem, in fact. Yes, yes, there is a way to do it. It's a way to do it. It's a way to do it. that wouldn't you be following, which are certainly not at all very strange transformation. One question on the transformation passive and the transformation active. Dans le cas des transformations spatio-temporelles, on peut faire une distinction entre transformation passive et active quand on regarde un sous-système. Oui, bien sûr. When I take a system, I do the function active of this system, and it's something different. Why? Because there are different relations between this system and the rest, probably. And in the case of storage, there is no possibility there is this possibility? Yes, it's just the same thing. What you said about the subsystem is exactly that. Here, we can actually distinguish between these situations because it's in the tableau, it's in the subsystem of this salle and there is a global gap that allows us to differentiate between these situations. That's why the ideal situation was the place of space in the city, where there is no environment that allows us to repair things. When it comes to the theory of things, there is an equivalent between the active and passive transformation. The active transformation is the automorphism of the fiber.

1:37:30 And the active transformation, what we do is trivialize the fiber with a section. and instead of doing an automorphism in the fiber, we change the section and this is the version, let's say, passive of the transformation of Shosh. But it is exactly the same equivalent. There are two ways to do a transformation of Shosh, one passive and there is no difference there is no difference between the two? No, there is no difference, while in the case of the temporal transformation, there is this particular case of the sub-systems where, in fact, there is a possibility to make a distinction. but I do not see the same possibility in the case of the transformation of the globe. Is it correct? I don't know if we can imagine a situation where we consider it... I imagine that yes, we can consider it a physical situation with an electromagnetic chamber localisées, et peut-être on pourrait trouver un environnement qui nous permet de distinguer entre les types de transformation, si on prend un subsystème. J'imagine qu'on peut le faire, oui, je ne vois pas aucune raison pour laquelle on ne pourrait pas le faire. Si on considère un subsystème, il y a un rapport qui nous permet de distinguer entre les deux. But I don't know, in theory, I don't see any obstructions. The problem that I imagine is the problem of Einstein. If we do a local transformation, if we interpret it as an active transformation, it's problematic. It's a consideration like this that I'm trying to find analogies to this system.

1:40:00 This situation is exactly the same as Einstein's argument. Yes, of course. C'est d'ailleurs le seul type d'événements physiques qu'on peut repérer, ce sont les coïncidences entre les lignes de monde de particules, ici ça serait la même chose mais interprétée dans le cadre d'un champ électromagnétique, mais c'est exactement le même type d'interprétation. J'ai essayé d'ici de homogénécer vraiment la théorie de Yang Mi, c'est la théorie de la relativité générale. The title would have to be the future geometry, the theory of Schoen, in general. In general, perhaps, when we work on the philosophy philosophy of the theory of Herm-Mills, we immediately take the theory of Schoen, and we interpret it from this side. For me, it's dangerous because we are dealing with a lot of things. We are dealing with the theory of Schoen. That's why I tried to stay as close as possible to the general relativity. Is it in the physical, it's really simple or not? Because in mathematics, the modern way of doing this sort of thing is more simple, it's much more simple than the future. Yes, that's a question. But maybe in the physical, it's something absolutely... No, not at all, not at all. In fact, I would say that there are two ways. There is the first one that says that it should absolutely eliminate the coordinates, it should absolutely work. And, for example, in all of which I have described, the double de Wilson, the solennomy, allows to define the quantities that do not depend on the coordinates. On the other side, it would be like the theory Leibnizian, instance, the theory Markiano-Leibnizian, where I eliminate really the coordonnées and I rest with the relative quantité. The other one is the one that says, well, we need to use the coordonnées, but we need to add a couple of symmetry that will remove the physical physical coordonnées. In fact, I have a situation for this type of question. Here I am going to show you two arguments that defend the position in which we need to work with the coordinates and with the symmetry instead of working without coordinates and without symmetry which is exactly another position completely valid working without coordinates and without symmetry. You don't have three symmetry

1:42:30 if you don't have to work without symmetry? Tu ne pars pas. Non, je ne pars pas. Je travaille sans coordonnées et sans symétrie. Mais pourquoi ? Pourquoi tu ne peux pas travailler sans coordonnées mais avec symétrie ? Parce que justement la symétrie, les symétries sont les symétries… Tu peux garder symétrie sans… Dans les… bon, dans les cas de… dans les cas de… Dans les cas de… Bien sûr que non. Well, that's a nice day, everyone, what they say. I don't want to say anything about it. And after the situation of Einstein, where he defends the position after which he has to guard the coordinates. because it's a bit of the idea that, say, you need to identify everything, you need to identify everything, and you don't need to know anything, or you need to know everything. In fact, I think it's for, at the point of view, mathematics. Of course, it's just how it works in physics. But in mathematics, it's just to not identify anything. or you can even look at something more dispersal, but you can look at all these transformations, in a way more global. It's a bit like a choice, mathematically not justifiable. I think in general, that's why these situations are in the same direction. It's technically, I think, more easy to work with the coordinates and with the symétries that are in the simulation of physical coordinates. It's the way, let's say, real, no? But that doesn't mean that we can follow the other path. So, I can change? Dans le... Dans ce... Ici, il y a des possibilités. So on peut travailler dans l'espace quotient, où on a éliminé toute l'ambiguïté relative aux coordonnées.

1:45:00 Ou soit on peut travailler dans l'espace total. Si on travaille dans l'espace total, d'abord il faut fixer un repère, casser une section, et après il faut introduire un formalisme qui nous va permettre d'intirer des invariants d'offrir. Mais bien sûr, si je travaille dans la base du chiffré, je n'ai pas tous ces problèmes-là. but working here, at least in the theory of Young Mills, it's really torduous. It's a space where the sea quotient is not trivial at all. So that's why we work in total space, working here and working on the observable space. But of course, it's a choice. I can force things to work here. Here it is clear that if I work here, I'm trying to eliminate the group of symmetries. I'm trying to cosyenter everything by the group of symmetries. But effectively, it's another possibility. I'm not obligated to work in the total space. You can't accept this? Pardon? How do you say this? These are the four people who created the formalism. It's B, K, R, S, T, T, and T, with the pronunciation corrects in each case. But these are the four people. In fact, it's the BRS that have published a work from one side and the AT that has published a work from the other. What is the idea? Do you calculate the group of homology? I calculate the homology of the group of George. In realising the thing very simple, I work with an algebra differential on the fiber. I define the different forms on the fiber, which are what we call the fantasmas, which can be identified with the form of Moray-Kartan in the group of things. I define a vertical differential on the fiber, and I define a cohomology. I have a cohomology of the type of RAM, with the difference here,

1:47:30 it's an infinite dimension, so it's much more complicated. And then I move to the cohomology, and everything is controlled in a way telles que la homologie au niveau 0 coincide avec les fonctions qui n'échangent pas quand je bouge dans la direction de la fibre. C'est-à-dire que je change le repère, mais les observables, les quantités physiquement observables ne changent pas. C'est fondamentalement ça, et on peut identifier ça avec la homologie dans le chèvre et l'hygiène du groupe de choses. Mais, si je travaille ici en bas, je peux laisser tout ça à côté, non ? Mais c'est tellement difficile de travailler ici en bas, que je préfère la complexité en haut que la force implicitité en bas. Thank you very much. The next week, there is no seminar on Monday. Par contre, le samedi, toute la journée, il y a le connoisse, le connoisseur, le connoisseur, le connoisseur, le connoisseur, le connoisseur, le connoisseur, le connoisseur, le connoisseur et moi-même, et si vous êtes sur ma liste des solutions, vous allez recevoir tous les détails. It's just got a piece of ice cream. I'm going to go out on the bottom and see how these are. Thank you.

1:50:00 Thank you.