Varieties in Topology & Differential Geometry, Spinors & Twistors — Part 1

0:00 What I want to say is that I'm going to tell you about my philosophy because it will be math and math. I'm going to tell you about math. I'm going to tell you about math. I'm going to tell you about math. What we need to know is that in all these questions, there are a number of ways to introduce yourself. And of course, it's important to show that these ways are equivalent to this. But in this seminar, I will choose a certain approach. There are others who are more favorables to some types of problems. But I will try to make an approach coherent. And then, when I think about it, I will invite what can be more important. So, the first part is on the pensar, so the pensar is something that lives in a particular space and more special than a variety. So, first, I would like to be interested in what is a variety, and I think it's not bad to come back to it, because, finally, it's lié to the notion of geométrie, it's an object fundamental in geométrie. It's, finally, what represents a geométrie in a very general sense. Today, all the space can not be represented as varieties, I think it's a lot of the way that we have to do it, especially in physical... So, the notion of space, in general, we think like an space of points, an ensemble of elements, these elements, we call points, and of course, for an ensemble of elements to be qualified of geometric, for those elements to be qualified of points, it is necessary to verify certain properties. So, to build a variety, we will define an ensemble of points and we will give them certain properties, which we will see. But it is also important that there are other points of view to consider, for example, an algebraic point of view, it is to say

2:30 a variety, we can be interested in the algebra of all the functions which are defined on this variety. This is an algebra, of course, because we add them, we multiply them by the numbers in an space vectorial, we multiply them by them, it's an algebra, it's an algebra commutative. And we can show that there is a correspondence 1-1-1 between the varieties and the algebras commutatives which verify certain properties. And the base of the geometries commutatives of Cohn is to generalize these objects non commutatives and finally, in doing this generalization, we will define new objects which are not different varieties, because the algebras are non commutatives, but which are defined by the algebras. So, we see that the point of view algebra on the varieties, rather than the geometry, allows to extend this concept to a new class of objects geometry, which would be in this case, these spaces non-computations, which have not the same proprieties, in which, for example, we don't know how to do it. And you know that, in physics, one of the problems of the physics, is that there are differences, the singularities, which are all liable to the fact that there are points, because there are some integrals that we can lead from 0 to 0, because there are points, and that diverge from what we call the divergence ultraviolet. So if we have a physics where there is no points, we think that it would be better. So, one of the tendencies of the theory today, is to try to work on spaces where there is no points. to build a theory of gravity quantic, and one of the ways to do the geometry correct in keeping the most of the symetrically or the symetrically correctness, etc., is precisely to pass to the geometry in the universe. So, all this is to indicate a point of view, already different from the geometry. It's not a point of view, but it exists, but it exists. So, today, I'm going to be interested So if I have an ensemble of elements, a priori, it doesn't have a geométric. So if I want to give you a character geométric, I need to define certain relations between the elements

5:00 which will allow me to call the points. If I have an ensemble point, The first level of structure that I can give you a topology in which I have a variety of topologies. What do I do? Since I'm talking about an ensemble of points, I'm going to define some sub-ensemble that I'm going to have to open, I'm going to verify some relations that I'm going to have here. and if I define this ensemble, this ensemble or the ensemble of points, it defines what we call the topology. These ensembles, these sous-ensembles, which are called the voisinage, which are called the ouvert, and if the points are in the same voisinage, they are, by definition, voisins. So, it means that we start to know what are the points that are voisins, So we have a first level of structure geométrica that allows us to do the geometry. And in fact, the topology is rich because we have spaces that are equivalents topologically and others that are not. And we can already do the topology. and if you can see the HSCM, and if you can install it, it will be better. Well, when we have an ensemble, when we have an ensemble of elements, we have to define a function. What is a function? It is something which, which means each element corresponds to a non-reel or complex. If we have defined a topology on the variety, it means that we can define the continuity of the function. It is something that is very important. Because now, when we have a function, we can decide if it is continuous or non-continue. So it means that we have already a number very interesting. But it is not sufficient. In general, we consider that it is not sufficient to do the physics,

7:30 at least the physics, classique parce qu'en physique, on veut faire plus que s'intéresser à la continuité des fonctions. On a des équations différentielles en général, on met des problèmes sur la forme d'équations différentielles, donc on peut être capable de relever les fonctions. Et ça, sur une variété topologique, il n'y a pas d'outil qui permet de le faire. Donc, si on veut pouvoir manier des équations différentielles, il faut définir un second So, by-dessus, the structure topology will be a different structure and will define what we call a different variety. So, it's a second level of structure. A topology can't exist without a different structure, but once you have a topology, we can put a different structure. And a few times, even through the variety of topologies, we can put several different structures. There are some theory that they say that they don't know, they don't know. After, we'll see that we'll continue to enfile the structures. Once we have a variety of topologies and different structures, we can continue. For example, we can put a connection or we can put a structure Riemannian, or a structure symplectique, and then we can define several things at the same time, which will have a relationship between them. But at the moment we are at the second level of structure, the variety of different structures, the different structures, and already it allows us to define a lot of things, namely the vectors, the forms, the product tensor, the vector, and that's how we will arrive at the tenseur. The anti-symmetry of vectors and forms which are called multivecteurs or multiformes, which we call them in general. The algebra... We will see that there are structures naturelles of fibres. I don't know what it is, but when we have a different structure, we have vectors, we have vectors and on a, in a manner canonique, des fibrés qui apparaissent, i.e. le fibré torsant, le fibré torsant, tous les fibrés tensoriels, mais aussi on a des algènes de glyphore qui apparaissent aussi de manière canonique, quelque chose qui est très puissant

10:00 pour faire des calculs. Donc déjà, si on a une variété différentielle, on a énormément de choses sur notre variété. Donc c'est ça qu'on va un petit peu étudier pour voir comment s'introduisent the different notions liées à la structure différentielle, et notamment les tenseurs. After, if we go more, especially if we want to define the spinners and the twisters, it will be necessary to define the levels supplémentaires of the structures on the variétés. Because from that, we will be able to define the tenseurs, for example, which is what we are interested in today, or a few other things also, but we can't define the spinners or the twisters. So we'll have to introduce... Pardon? You can do this. That's what I used to do. This is a strange statement. We'll define the additional levels. And we'll define two things which are different. They're one of the connections. because of the differential, we can define the differential, we can define the different infinities, and we can define the methi. Even if we give a differential, we can define the tables on the top. And in general, I think it's not only one, but it will have the probability of being, by example, Laurentienne, if we are interested in this instance, we can see what that means, but we can define the tables on the top. different. On peut définir arbitrairement des tas de métriques différentes, mais on va voir qu'il existe certaines relations, c'est-à-dire que quand on a une métrique par exemple, on va voir qu'il y a certaines connexions parmi l'infinité de connexions qui peuvent exister, qui sont liées à cette métrique. And from this curve, in the same time, we will define the spinners and the twist.

12:30 Thank you. Thank you very much. Since we define the sous-ensembles, these sous-ensembles are in fact the source of the variety and we have, by definition, the variety, we have a correspondence between each of the source with R plus N, which is, such as the R plus N, R2, R3, R4 can, in a certain way, play the role of the prototype. These are the spaces that you can do well, which are the spaces of the size of the number of dimensions generalised. and the same correspondance of the variety with the RN, which is what we call the coordonnées, which are some particular functions on the variety, which are not necessarily defined globally, but which are defined on each space ouvert, and then we revolve to reconstruct the variety.

15:00 So each point inside of the ouvert has its own coordonnées, or are all the same coordonnées? What do you mean? Each ouvert has a system of coordinates which is called a card and then when you take all the ouvert each ouvert has a card and of course the cards are broken so there are some functions that are defined and an ensemble of cards like that is called a matlas so on a matlas which is a system of coordinates for each variety in some cases you can define a global system It's global for the variety, but often you can go on a sphere, it's always missing a point, so it's a sphere, it's at least two quarts to define a system of coordonnées for the variety. And also the variety of topologies, of the space vectorial, how do you put in relation to each point with an ensemble of coordonnées in the region? It's the definition of a variety. The definition of a variety, it's that each sub-ensemble has to exist a gap, that is, a correspondance. And then it's very common because once you have these correspondances, these coordonnées, you can work with it. And you will see that it will be used to the next level. we're going to define what? We're going to define the different structure, since we've already defined the structure, the topology. Donc, qu'est-ce que c'est ? Comment on fait pour définir la structure différentielle ? On introduit des outils qui permettent de dériver des fonctions. Un outil qui dérive une fonction, on appelle ça une dérivation. Donc une dérivation, c'est quelque chose qui a une certaine fonction, fait correspondre une autre fonction, et on va imposer quoi ? On va imposer simplement que ça vérifie la règle de l'index sur le produit, la dérivation du produit. It's to say that FG' is equal to FG' plus FG' So, we have a whole ensemble of derivations and a derivation, we call it a vector and since we have already functions, it's practical, it can help us to define the derivations so, we have, we have defined the coordinates of x alpha, we have the evident derivations which are the derivations according to the condition and in fact what we can show

17:30 is that all the derivations which exist in a different variety we will be able to define from the derivations that we describe from the derivations according to the condition so in practice we can do it So, from a point of view theory, the ensemble of the derivations, which is what defines the structure differential, and the derivations, we call it a vector. So, we have an ensemble of derivations, we call it the ensemble of vectors, it's the same thing, we note in general phi, which is the structure of the space vectorial, since if we have a derivation plus one, we can add it, we can multiply it by a scalar, simply on va avoir dérivation a fois la dérivation appliquée à une fonction f va donner a fois la dérivation appliquée à f et on s'aperçoit que quand on a deux dérivations que j'appelle f et v, que j'appelle v et w, je peux appliquer deux fois la dérivation à une fonction f. So that's not a derivative, so that's not a derivative. In other words, if I apply FW-WV, we can see that it's a derivative. So when we have two derivatives V and W, their commutator, their crochet in which it's called, is also a function. So we have a multiplication of functions, a multiplication anti-symmetriques, and it gives for a function a structure of algebra algep de Lie. Alors on sait qu'une algep de Lie, en général, ça c'est la théorie des groupes de Lie, c'est associé à un certain groupe de transformation. Ici, cet algep de Lie est associé à un groupe de transformation, effectivement, et comme cet algep de Lie est énorme, elle est immense, le groupe va être immense aussi, c'est ce qu'on appelle le groupe des différolophismes, et en fait, le groupe des différolophismes sur une variété, these are all the transformations that conserve the structure of the differential. There are a lot of, it is to say that every time in physics or mathematics, we define the transformations, and we do not always explicitly, but it is almost always different. There are different types of different types of different types, like the isométries, the transformations conformes, and others, but the different types of different types are absolutely enormous.

20:00 And if there exists between two different varieties, Now I'm not talking about a different variety on the same, but a variety on the other. Well, we can consider that as a variety of differentials, these varieties are equivalents. The notion of equivalence and identity is quite subtle. And it's quite subtle, in general, in relativity. Because what says the relativity? c'est que l'espace-temps est une variété différentielle sur laquelle on a construit une certaine structure qui est une métrique lorintienne. Cela dit, si vous avez par exemple l'espace de Mikowski. L'espace de Mikowski, on peut le considérer comme un espace-temps qui est une solution dans la relativité restreinte. Qu'est-ce que c'est que l'espace de Mikowski ? C'est une certaine variété différentielle dans laquelle on a une métrique. Now we can look at another solution, which is the space-temps of De Siter, which is a solution different from the solution of Minkowski. In terms of different variety, Minkowski and De Siter are different. So in terms of different variety, it is the same variety. Only, it is true that the different morphism doesn't transform the matrix of Minkowski into the matrix Minkowski. So, as a variety, non pas differentielle, but métriques, they are not equal. So, a difference, you see, is something very, very, very vast, and of course, if we want to distinguish the varieties that we want to distinguish, we want to define the level of structure supplémentaire. Now, you see, the notion of a vector, we have defined before the notion of a vector. What is a vector? On va définir un vecteur en un point M d'une variété. Maintenant, chaque point M de la variété, c'est tout simplement la réduction du champ de vecteur en ce point. C'est-à-dire, le champ de vecteur, c'est quelque chose qui a une fonction f qui correspond à sa dérivée, qu'on va noter VF. Le vecteur V au point M qui correspond au champ de vecteur, qu'on va appeler V de M, It's something that, at the value of the function f in n, f of n, it will correspond to the value of the derivative of n. So, on définit, at a time of the vectors, on définit the vectors in each point.

22:30 And that is interesting, because that means that in each point, we have a certain ensemble of all the vectors. We call it the space tangent, and it is, in a way quite evident, a structure of vectorial space. So, the whole of the vectors of M, which we call the space tangent in M, which we note like that, tangent of M to the variety M, is a structure of the vectorial. It is to say that, finally, all the vectorial dimensions are in a sort of equivalent in terms of the vectorial. So, it's almost R to M. For the moment, we haven't defined the matrix. So, all the varieties of the same dimensions have the same vectorial space for each point. And we know that Rn, in the same vectorial space, and it's going to be any kind of vectorial dimension n, there is an action very natural in a certain group which is the linear group of n. What is that the linear group of n on a different vectorial rn? It's simply the group of transformations linéaires to conserve the structure of the vector. So this is something absolutely universal. You can see that already, we have all this from the different structure. That means that each point, we have a certain vectorial space which is defined by a group which is on the top. So this would be great for any variety. We don't have to talk about the effect or what that is. On the other hand, we have an algebraic element. Every time we have an vectorial And what is the dual ? It's what we call the dual form. It's, finally, all the applications linéaires on this space vectorial. So, every time we have an space vectorial E, we have our dual, and we know that it's also an space vectorial and that it has exactly the same structure that E. They are isomorphs, except that there is not an isomorphism point by point to be defined in an ergonomic, but we know that they are isomorphs. Here, in each point, On a un espace tangent qui est un espace vectoriel, donc en chaque point, on a son dual qui est aussi un espace vectoriel et on note comme ça et qu'on appelle l'espace cotangent en M. Donc on a déjà ces deux choses-là et c'est intéressant parce que chaque fois qu'on a un élément de taux étoile, c'est-à-dire ce qu'on va appeler une uniforme,

25:00 une uniforme c'est quelque chose qu'on peut appliquer à un vecteur et ça va nous donner un nombre. So, naturally, the space cotangent agit on the space tangent, and reciprocally, the space tangent agit on the space cotangent, because, in fact, the tangent is the dual of the cotangent. The dual dual is the space of the dual. So, we have this, and we can do very naturally the product sensoriel of these things like this. It is to say that the product sensoriel, which I think is what I do, but from our point of view, simply, when we have two spaces vectoriels, like E and E', we will consider the vectors of V and E', and then the sum of the vectors like this, and that will define an space sensoriel. So we will note that when we have two spaces vectoriels, We have a product sensoriel like this, and a element of this product sensoriel is like this, such as it could be... It's not necessarily... Every element is not like this, but every element is written as a sum of products sensoriels, of vectors elements. Okay? So, it's something that's very tangent. Well, there, we can do the product sensoriel, as much as we can, of the space tangent and of the space tangent. and the tensors, this is nothing else that these products tensorials. So, if we have a vector vector B which belongs to... What we need to know is that when we have a vector vector, we always have a base. It is to say that if the vector vector is dimension n, there exists an ensemble of m vectors tels that we can compose n'importe quel vector compared to these vectors of base and we will do this with the indices the vector is written as a sum of vectors of base with here the components which are the components of the vectors in this base but you have to pay attention because the vector itself is something which is really defined theoretically it is a geometry which is defined in the space tangent, and we say that these components, it must be defined as one base. There are bases, there are many. So we see that these components

27:30 are something that exists that relative to one base. So it's not really an object geometry defined, while the vector itself is an object geometry defined. I was told that we could do a product tensor. Every time we make a product tensor, we obtain what we call a tenseur. So if we make a product tensor of the space tangent, for the moment I see on a point, by itself, like this, and then we make a product of the space cotangent, also by itself, and then we have u x that, and b x that, on va dire qu'on a construit un tenseur de type P. QP, je ne me rappelle jamais les fourmets. Donc, en particulier, si on prend l'espace tangent lui-même, l'espace tangent lui-même, il n'y a que ça. Il n'y a pas d'espace tangent Q égale 0, P égale 1. Donc, les vecteurs ce sont des tenseurs de type 0. Donc là, j'ai pris la tenseur inverse. Et les formes, ce sont des tenseurs du type inverse. D'accord ? And remember what I've said, that's important for the following, we have the linear group which acts on him. What is the linear group? I've already said that it's simply a linear group of transformations and when we have a vector vector, where we write the vectors like the columns with their components in a base, the linear group is simply the matrix. Because the linear application on a vector vector is simply a matrix. So, the group linear, it's simply all the matrices. So, we say that a group has a representation when we specify how it acts on a space vector. So, the space of representation of the group linear GL, which is the number of n, which is the matrices n by n, l'espace de représentation le plus naturel qu'on va appeler fondamental c'est tout simplement l'espace des vecteurs un espace vectoriel de dimension n par exemple Rn mais comme l'espace tangent on a vu que c'était la même chose que Rn ça veut dire que l'espace tangent en chaque point l'espace tangent constitue une représentation

30:00 de, du groupe In each way, the group linear is a representation of the space of the vectors, and it acts in the same way also on the forms. So, the space of a form, that is to say the space co-longent, is another representation of the group linear. And in fact, if we have a tenseur like this, the group linear is going in there-dessus and in there-dessus, on a also n'importe quel espace de tenseur c'est aussi une représentation plus compliquée du même groupe linéaire c'est à dire que quand on a des espaces qui sont des représentations d'un groupe tous les produits tensoriels de ces espaces sont aussi des représentations ça c'est très important parce que ça va jouer un rôle dans l'esprit humain, dans l'esprit humain, etc. donc cette notion de représentation de groupe elle est vraiment importante We have a group G which is made of elements. Each element G agit on a vector, it belongs to the space of representation, in giving something which is simply the action of the element of the group G on the vector v, and we can do this as the product of a matrix by a vector, and the product of a matrix square by a vector, it's just a vector. So this notion of representation of the group is very important. And obviously if, for example, we do the product of the vector vector, the space tangent will be dimension n. the product of the space tangent by the human will be dimension 2n so when we have the representation of the linear on a product tensor which is a vector of dimension 2n we will not have a matrix of dimension n but a matrix of dimension double, etc. and if we have a product tensor with plenty of factors we will have something here which will be a matrix square of dimension So not N by N, but P plus Q fois N by P plus Q fois N.

32:30 It's always the same group. It's the same group? It's always the same group. It's always the same group. which has its natural representation in RN. We call it the fundamental representation. We have, for example, the group of rotations, the group of three dimensions, GL3, which will obviously act on R3. But if I make the product sensoriel R3 and that, it will be the matrix 3-3. If I make the product sensoriel R3 by human, I obtain a new space vectorial of dimension 6. L3 va agir dessus, je vais simplement tourner chacun des trucs, ce n'est pas une rotation, c'est plus générale, et donc je vais avoir une représentation de ce groupe dans un espace vectoriel du 306. Je ne peux pas descendre. A priori, je ne peux pas descendre. Je peux ou je ne peux pas, ça dépend des groupes. Mais par contre, il y a des groupes plus gros sur L3 pour L3 ? There are more groups that are bigger. Because all these transformations linéaires, which are on the right, they are not. There are no more. And there are no more transformations linéaires that are on the body sensor? Yes, of course. And all of that, when we go to the spinner, this question is going to be very important. So, on a defined in each point an space tangent. So, since each point of the variety has a variety, with points everywhere, and then we just see that at the top of each point, there is an space tangent. If we have a surface, for example, On voit bien qu'est-ce qu'on fasse sur une surface. Donc, prenons simplement le gris et tout ça. En chaque point, on a un espace tangent qui est de dimension 1. Donc, c'est la droite tangent. Alors, si on a une surface, en chaque point, on a un plan tangent. Si on a un espace, au-dessus de chaque point, on a un espace tangent à trois dimensions. Si on a un espace-temps, au-dessus de chaque point, on a un espace-temps tangent à quatre dimensions in definition, it's the space-temps of 1.2.

35:00 So when we have a variety, for example, in general, it means that at the top of each point of the space-temps and at the top of each event, associated with each event, we have an equal space vectorial of 1.2. So we can see that we have a kind of structure here where, at each point, we associate an space vectorial. This is what we call a structure of fiber. And like here, l'espace vectoriel qu'on associe à chaque point, c'est l'espace tangent, ce fibré, on l'appelle fibré tangent. Donc, du point de vue, c'est une variété. C'est une variété qui est beaucoup plus grande, puisqu'elle a comme dimension la dimension de l'espace de la variété du départ, plus la dimension de ceci, qui est égal à l. Donc c'est quelque chose qui va avoir une dimension 2n, une variété de dimension de l'est, 2n. and which has a particular structure since it is not any variety of dimensions that we use. Now, let's see the first thing. So let's finish. On a, here, our varieté of the beginning, with, here, I decide, exprès, on the top. When we have points there-dedans, we can see how they project everything here, on this thing. So, when we have a structure of space, we have something called the base on which all points are projected. And in fact, all the points that are in the space tangent, in the space tangent Tm, which is here, on the planet, are projected on the point M. Here it is on the point M2. So we have a projection in this sense, but we have no projection in this sense. That's very important. It's just for that it's not a product of space, it's not a cartesian space. If we had a cartesian space, we could project in the two senses, on the one like the other, so vertically or horizontally, here, we can't define a horizontal projection. So if we want to do that, we need to introduce structures supplémentaires. So for the moment, we don't know what vertically. And what is interesting, it's that, for us, we have this same group linear which is agile.

37:30 In each vector vectorial, the group linear is agile, And in fact, we can consider that the group linear, it acts on all the fibers tangents like that, but there is a particular action, that when it takes a point here, in the field, we call it the field, when it takes a point in the field, it goes into the field. It is to say that it will never take a point here to put it on the field. So it is in fact a transformation, we call it vertical, because it is not the structure of the field. And, in fact, this notion of action of the group is very important because since we have an space fibrous, like that, since the fibrous tangent is a particular case of the space fibrous, we have always an action group which is vertical. We call it the principal group associated to the fibrous. So, our fiber is something where we have the variety of the space base of the fiber. At the top of each point of the variety, we have a fiber, which is the space tangent of the point M. And what is interesting is that we can define what we call a section of the fiber. What is a section of the fiber? on va se dire, à chaque point M, j'associe un point de la fibre. C'est-à-dire que c'est une espèce de fonction. Donc à chaque point M, j'ai une section qui s'appelle S. À chaque point M, ça fait correspondre quelque chose qui est S de M qui habite dans la fibre. Et la fibre, en fait, c'est l'espace tangent. Et qu'est-ce que c'est qu'un point de l'espace tangent ? C'est un vecteur. So it means that a section of fiber tangents is something that, at each point of the base M, is associated with a vector to the point M. So nothing else than a vector. So a section of fiber tangents is a vector. So what we had to call the whole chamber of the vector, which we had noted earlier, and the whole chamber of the derivations, or if you want the whole chamber of the vector, it's the same thing, it's also the whole chamber of the sections of fiber tangents. Well, it's a language a little special, in this case it's pretty simple,

40:00 but this notion of fiber tangents is extremely important, because it is fundamental, obviously, in relativity, and in the structure of the variety of human beings, we can go here. But if we do the psychopontics, we can see that it is exactly the same structure that will grow, except that it will not be a fibrous tangent. We will have other fibrous, which are, for example, which are, for example, these fibrous, which are some of these groups, which are, for example, groups 1, groups SU2, groups SO3, And all the theory of quantum exchange, which we call the theory of jauge, is defined on the structure of fibroids constant. And this is interesting because it means that the relativity general, where, finally, we see that the structure is in the fibroids tangent, which is associated with the fibroids associated, but once we have introduced the metric, eh bien, cette structure de fibres et tangents associés à la relativité générale va devenir très voisine de la structure des théories de Jones, qui sont les théories en physique quantique et en théorie quantique des choses. So, on voit que par l'intermédiaire de ce formalisme, on va avoir un rapprochement très étroit entre la relativité générale et les théories de Joules. C'est-à-dire que ça permet presque, pas tout à fait parce qu'il y a des difficultés, ça permet presque de voir déjà la relativité générale comme une théorie de Joules analogues à celle de la théorie de Joules. The section is an application that at each point corresponds to an element of the surface. Exactly, it's an section. So there's no need to be a rule... So it's a continuous. And we know that when there's an opology, we know what it's about. continue. Alors, il se trouve qu'on ne peut pas toujours faire ça dans n'importe quel film, mais on peut le faire toujours, on peut toujours prendre des espaces ouverts et, dans un espace ouvert, définir une section continue. On ne peut pas toujours le faire de manière globale. Par exemple, sur une sphère, on ne peut pas le faire de manière globale. On dit que la sphère n'est pas parallélisable.

42:30 It's a unique one. It's a sphère... We can't figure it out. We can't figure it out. We can't figure it out. There's always a singularity. we have defined fibres tangents. We know what it is with the sections of fibres tangents. It's the same thing with the vectors. And we know that this group G acts on the space tangents that is to say that it is to say that if we make a product tensor, we will define the same way we can define the fibril cotangent. It is to say that instead of taking the reunion of all the spaces tangents, we take the reunion of all the spaces cotangents. It has the same structure, so it's about the same thing. And we call it the fibré cotangent. So it means that this fibré, on the top of each point M, has for fibres the space cotangent. This notation means that we have a fibré which is projected in M. M, it's the beginning, it's the base of the fibres. And a section of the fibres, it's M. It's not a vector, it's obviously the dual of a vector, it's something that acts linearly on the vector, so it's a different form. So you can see that all this is canon, it's to say that if at the beginning I define my vector, I have the whole number of vectors, all this is well defined. And I would have to, at the beginning of the vector, prendre les formes différentielles et tout construire par des champs de vecteurs à partir des formes différentielles. Bon, il se trouve que pour des tas de raisons, c'est souvent plus facile de travailler avec des formes différentielles qu'avec des vecteurs, donc c'est assez souvent qu'on doit présenter la structure différentielle à partir des formes différentielles plutôt qu'à partir des vecteurs, mais ça ne change rien, c'est complètement bien. So, what I also remember is very important, that the space tangent, well, in general,

45:00 a space vectorial and a dual have the same structure. So, a space tangent in one point and a dual have the same structure. they are isomorphs in as an espace atom. It's the same for the fibrous. They are isomorphs, but there is not an isomorphism canonic. If I give an element of an espace tangent, I will not know how to correspond to a particular form. If I do this, it will be a supplementary form. It will be, for example, an example. Well, so, we have the fibrous tangent. We have the fibrous tangent. Well, we can do the same way as the product tensor. The same way we can do the product tensor with the entire copy we want... ...that we have a lot of copies. It's a different discussion that we've had. This is an space tangent to an space tangent. I can do the same thing with the fibrous and I have a fibrous tangent tensor. which will be defined like this, by the products sensoriels of space cotangent, of fibrils cotangent and fibrils tangents, which one section is what? One section of fibrils like this, it will be, obviously, the product sensoriel of P copy of the space tangent in M and Q copy of the space cotangent in M. What is that? It is, obviously, what we call a chain sensoriel. The x-axis is also high as the x-axis. The x-axis is also high as the x-axis. The x-axis is also high as the x-axis. The x-axis is the x-axis. And there is a x-axis particular. When we can see a matrix a little bit later, we will define a x-axis particular which will be the x-axis particular. and which, in addition, will be synonymous in these two arguments. I have to talk about the repairs. In each vectorial, in each vectorial, we can define a repair. We can define an infinite, an infinite or, if you want, a base.

47:30 When we have, now, an espace tangent in each point, we can define an infinite in each point. What we call, an infinite frame in English, is something that, at each point M, makes correspond to an infinite base. of the space tangent. It's to say that E , it's, by definition, a vector vector of the space tangent in a variety of points M. And E, E, it's the real problem, which, at each point M, corresponds to this point M. So, we can do it in a continuous way. in general, we can't do it in a continuous way On le fait localement, couvert par couvert, et on travaille en général dans un couvert où ceci est bien défini. Alors, de la même manière qu'on avait défini le fibré tangent, on peut définir le fibré de l'opère. Ce n'est pas plus difficile. Qu'est-ce que c'est ? C'est qu'en chaque point, au-dessus de chaque point de ma variété, plus de chaque point M j'associe ici l'ensemble de tous les repères possibles à mon espace vectoriel qu'est-ce que c'est qu'un repère ? c'est simplement un ensemble de 4 vecteurs linéairement indépendants donc à peu de choses près puisqu'il faut vérifier que les vecteurs sont indépendants c'est ça ressemble au produit tensoriel de l'espace tangent 4 fois par une donc il n'y a pas de problème à définir ça ce qui est intéressant Again, once again, this is the ensemble of all the repairs, of all the bases possible in an space vectorial. In general, it's RN. How do we pass a vectorial of RN to another? What is a changement of base? What is a changement of base? It's in fact a matrix, always the same group, GLM. So, it means that, finally, in condition of having chosen one at the beginning, Therefore, we can identify the whole base of the space tangent in this group, because, finally, each group transforms into another group, and each group transforms into another group.

50:00 So, this is the same thing as this. which means that this fiber of all the repairs that we have to build, is the same thing that the fiber which is in the base of each point M, the group GF, the group GF, and the group GF. And as this fiber has for fiber the group itself, we call it a fiber principal. And it's very important, the fiber principal, So, what is the changement of section? What is the section of this repaire? You have already understood that a section is something that, at each point n, is associated with an element of the field, that is to say a repaire. So, a section is exactly what I call a repaire mobile. Changement of section is a changement of repaire mobile, that is to say that it is a changement of base in each point. as a change of base, it's an element of N, it's to say that it's an element of the group in R in each point. It's to say that a section or a loop of a loop, it's the same thing, and we can also interpret it as a an element of the group in R in each point. This is what we explain in saying that this vibrer of a loop which is for base N is a vibrer principal is the problem. All this, I remind you, it's just a different structure. We don't have a lot of finitions and we don't have a lot of finitions. Now, the connections, I'm going to make my pass because the time passes, I'm going to move on very quickly. Sachez qu'à la structure différentielle qu'on vient d'étudier, qui déjà s'était superposée à la structure technologique, on peut associer des autres niveaux de structure par exemple des connexions ou des métriques, tout ça de manière complètement indépendante. Sur une variété différentielle donnée, on peut construire une infinité de connexions différentes. We can also build a different infinities, which are defined independently. We can also define different types of things. If we want to define a simple structure, a simple structure. For the moment, we have to talk about connections.

52:30 I don't want to enter into details, but simply, we call it a linear connection. Because, again once again, it's liable to the nation. What I told you earlier about the fiber tangent, we are going to be interested in the fiber of the main fiber, which is the main fiber, as I told you. We have a variety that will serve as a base, with points, and on top of it we have the fibers, which, in the case of the fiber of the main fiber, each fiber is the all the way around N. Remember that, by definition of a fiber, there is always a projection that projects the fiber on N. But I said that there was no projection like that. Well, we can define one. We can say, by definition, I will say that this point, this project, I choose here, like that, I define a projection. I don't define two fibers in the same way, which is defined between two phi, which are infiniment coincides. I'm going to define it in a way infinitesimal, and this correspondence that we define is called a connection. This will allow to define, when I'm going to take the space tangent to this, I'm going to distinguish a vector which will be vertical and a vector which will be horizontal. I mean, I mean, the vector is horizontal if, in general, it lies an element to the corresponding element. But this horizontality, you see, it doesn't exist before. I was able to define it. The verticality, she, she existed, since the projection existed already in the vibrate. The horizontality, she doesn't exist. I was able to define it. For the definition, I introduced an additional level of structure which is called the connection. the connection, all the possible bases of the fibrer to one particular base of the fibrer? No, they are all the same base of the fibrer to each base of the fibrer. It's right. So, it has to be compatible with the action of the group. There are a lot of things to verify. I don't know about the details. So, on the definition of the impondence, it's called a connection. It's defined everywhere. It's not a sensor, it's an object of a different nature. different. What is interesting is that when we get a variety of differentials, there exists

55:00 an infinity of connections possible. How do we define it? In general, for writing them, we are going to see how they act on a base vector. We are going to ask how the vector which is here is going to be transformed, in which vector it is going to be transformed here. So when we do that, we define what we call coefficients of connections, place in a phase and on the transformation of the vector, it does appear that the coefficients are like this. And in general, in literature, the connections are defined like this. We define first the coefficients and then, if the book has an interest in mathematics, we show that these coefficients, finally, allow us to build something which is liable but the right definition is this, and it's better, it's better, because again, it's the same thing with the theory of George. And the theory of George, what we call a connection, we will define the same way the connections in the theory of George, which is something that makes one fiber to another. The fiber, it's not the fiber of the repairs, it's an other fiber vectorial, So, a connection in a theory of jones, what is it? It is exactly what we call a chain of jones. So, you can see the importance of connection. So here, in relativity, we are going to define a connection. Remember that there is an infinity, but we are going to define the way in which one is one among the other. And in this connection, we are going to perceive that we can with that, in relativity, the chain of gravitational, more than the metric. It is to say that to identify the range of gravitation and the relative general to a connection, it is something that is, how to say, that to identify the range of gravitation and the metric for certain reasons that are just liées to what we need. Now, once we have defined a connection, it is good because we know how to pass the vectors from one fibre to the other and, between, we know the derivative. So, each time we have a connection, it allows us to define what we call the derivative covariance. Normally, it's written with the Nabla, but, in the dollar point, I can't find the character Nabla, so I have written a T. So, here, we call Nabla x .

57:30 It's the derivative covariance of T in the direction of the vector x. You can see that one connection has two arguments and you don't know if it's not a tensor, but, despite all, on the first argument, it works tensorally, and on the second argument, it works differentially, it is to say that it's a rule of Leibniz, it is n'importe quel tensor, so the rule of Leibniz, in fact, it applies to the product tensor. If you have a derivative covariant, a product tensor of T1 by T2, it's T1 tensorier derivative covariant of T2 plus d'arrivée chorale d'Otéa en son réalité. Bon, je n'insiste pas là-dessus, c'est juste à voir qu'il y a des torsions, des connexions. Donc je vous ai dit qu'il existait une infinité de connexions possibles, mais que quand on a une connexion, on lui associe canoniquement deux entités qui sont ce qu'on appelle sa torsion et sa courbule. Alors je ne vais pas vous montrer comment ça se calcule, mais en gros, il y a des formules qui sont bien connues. if we have a torsion and a courbure which are linked to the combination of asymmetric and the covariance, whatever, what is interesting is to know that each connection is associated with torsion and courbure. So, technically... So, it's all for the connections. There is an infinite connections and that each connection has a torsion and a curve. For the moment, when I finish on the connection, I think about something which is the matrix. That is, on a variety, we need to introduce a matrix. It is something that allows to define the scale of two vectors in each space tangent. So, we want something, we want to build something that, in each space of the genre T, M, allows to define, finally, the product scalar of two vectors. This is defined by M, I have the 10 indices. So, we want a thing that I call GM, which is applied to two vectors like this. So this is a little gm which has two vectors which correspond, in a way, to their product scale. Well, we do that in each space tangent, but, of course, it will be

1:00:00 defined on the space tangent in general. So, finally, we have something which agit on two champs of vectors to give a little bit of something which agit on two champs of vectors to give, in general, their product, which will be, finally, a function, because it depends on the net, a function of the particle. Well, I think that everyone knows, at least. So, we define that with some, how do we say, some constraints. So if G is always positive, we will say that we have an electric Riemannian. And if G is positive, negative or null, we will say that we have an electric pseudo-Riemannian. and you can see that G acts twice on the terms in the way linear so it's another way to say that G is a terms of type 0,2 or 2,0 I don't know in any case it's something that lives in the space in the vibrator Tm, terms of Tm and, in fact, in the same part of the symmetry of this figure. Because it's a form of linear, I'm going to say, which is symmetric. And, as it's a form of linear, we can also consider it as a tenseur, and this tenseur, you know, it is diagonalised, and it has the values propres, and we can always find a base in which the values propres can always be 1 or 1. eventuelement des zeros si on était dans une variété générée, mais je n'ai pas ce cas. Et donc, suivant le nombre de 1, de moins 1, on va définir la signature. Alors, si c'est toujours positif, c'est-à-dire si c'est Riemannien, c'est que des 1. Sinon, si c'est pseudo-Riemannien, il y a des 1 et des moins 1, et le nombre de 1 et de moins 1 s'appelle la signature. Et quand on a un seul 1, ou bien un seul moins 1, on est dans le cas l'orentien. And of course, you know well that the space-temp of the relativity restreint and general, is a florentian.

1:02:30 It's to say, what is the signature? There are two conventions. There are one, one, one, one, one, one. There are one, one, one, one. It's a very interesting question to know which one is at least. If you make a statistic, I think you'll arrive at 52% or 52%, I don't know exactly what you mean. But in fact, it's not all that weird. What's interesting is that if you... I'm not talking about the subject of Glyphor, which is something extremely interesting with all of this, and which allows you to make a lien with the... You can see that these tenseurs, these tenseurs of a number of numbers, PQ, form a space vectorial. Now, I'm interested in particular the tenseurs of type P0 or 0Q. These are also spaces vectorials. and what's interesting is that I can define the part anti-symmetric skew in English I don't want to enter into the details but it's a little bit of parenthesis these are also these are also this is what we call the vectorial of the B letters and this is the Q form it's a structure of the space vectorial and what there is interesting is that if we have a G I can give them in addition to the structure of the algebra and if I I can call it algebra sorry I can call it I can call it I can call it algebra And then, I can, for example, make a direct sum of all these objects, all the values of Q. I don't know how to finish this thing, I'm sorry. Like that, I call it like that. That, it was all the Q-forms. That, it will be all the multiforms, quel que soit leur rang.

1:05:00 Et si j'ai une métrie, je peux donner à ça une structure d'abjet qui s'appelle l'abjet de pliforne. Bon, je ne veux pas rentrer plus loin dans ces détails, mais c'est extrêmement important. Et c'est canonique, mais tu peux avoir une structure différentielle et une métrie. Quand on a une structure différentielle et une métrie, on définit en chaque point l'abjet de pliforne, à partir de produits tensoriels antisymétrisés of the space tangents or tangents. It is an algebra of dimensions finies, dimensions 2.0, in fact, and it is extremely interesting because this, which is an algebra built from a tensor, makes in fact appear the spinners and the groups of spins who act on the top in a way extremely natural, so that you will see that earlier I was going to introduce another way which is quite laborious and here it is almost all alone and it is another... Well, in fact, not only one, you take the space tangent, you make all the products tensor and you take the symétries, you see that at some point it becomes zero, you make the sum direct, you have a natural product with the matrix and the anti-symmetric product, and it's an agent difforder. Well, it's necessary to do it, but it's not... There's nothing extraordinary to do here, and we can see the spin and the spinner, and it's a very elegant way of introducing them. Well, I have not chosen this way, but it is very important, very natural, and it is that it is generalizing in all dimensions, and there are plenty of other dimensions. Well, this is a parenthesis. So, here, I'm going to finish on this one for the next one. So now I define a variety of Riemannian or pseudo-Riemannian if the signature of Riemannian is not positive in a sort of formulae scalaire which can be interpreted as a symphony of 0.2 And now what is interesting is that if I have a connection,

1:07:30 if I have an infinite connection, I choose one, I know the derivative of n'importe quel tenseur, so I can derive in particular the metric. And the derivative covariance of the metric is a tenseur which is called the tenseur of non-meticity, which indicates how point the derivative covariance of the metric is non null. If, by chance, this is null, we are going to say that the connection is a metric. Theorem, given that it is a metric, there is an infinite connection. Among all these connections, there is only one metric metric and a torsion null. So, metric and symmetrics is null. This is called the connection of Levy-Chief. Rappelez-vous, chaque connexion a une torsion ou une corbure. Ici la torsion, j'ai dit qu'elle est nulle, mais il y a une corbure. Et bien la corbure de la connexion de l'Evichivita associée à G, c'est ce qu'on appelle la corbure de Riemann associée à la métrique G. Alors, bien souvent dans les livres, on introduit la corbure de Riemann de manière directe à partir de la métrique G. On se place dans une base et des répliques d'efficience par exemple. but it is important to know that the real definition of the curve is the curve of the connexion of the matrix associated with the matrix. We call it a curve of the human. It is a tensor. We can define the compositions of the tensor, the contractions of the tensor. If we make a contraction of two indices by two indices of the tensor of H. If you have a tensor of the chi, the electric multipliée by the trace, on obtient a tensor of Einstein, which is a tensor which intervient in a fundamental way in their activity general. If we do a contraction on the tensor of the chi, we obtain a corpure scalaire. And if we subtract the tensor of the corpure, the tensor of Riemann general, the component that is liable to the tensor of the chi, we obtain another tensor called the tensor of Weill, which is quite interesting, Notamment, il y a un lien conforme pour ceux qui connaissent, et ce sont des propositions intéressantes également qui jouent un rôle particulier dans les spinners et les twisters. Je ne vais pas être en ce moment en parler, mais voilà.

1:10:00 Pour tout ça, vous voyez que ça résulte uniquement de la définition d'une métrique. It is to say that if I define a metric, it defines a certain connection in a way univode and this connection defines a tensor of M.A. and the other tensor. If I am in the general, what do I do? I will try to find a metric, for example, because it is very common to have a metric to work. So what do I do? I will take the Einstein equation. The Einstein equation tells me what? Well, the Einstein tells us that the Einstein is equal to an Einstein that I'm supposed to know, which is an energy impulse, which is the material material of the universe. I solve my equations, because I solve my equations differentially, in putting the conditions of infinity to infinity, and I solve my Einstein. I solve the Einstein of Ritchie, I solve the Einstein of Ritchie, I solve the Einstein of Ritchie, and I solve the Einstein of Ritchie, and everything you do. Is it an inverse for each connection you can find an effect that the connection is emitted? Well, if you suppose a connection which is a tension... ...nude... Yes, that's right. I think yes. And... But it's more interesting than that. Malheureusement, I don't know how to talk about it. Because if you have a question... I don't know, I'm not sure. I don't know what I'm saying. What's interesting to know, I don't want to talk about it. I don't want to talk about the tetrable because it's something extremely important. just one word, that is, the metric is a tensor of type 0,2. In a base, we can define, when we have a vector in each point, there are four vectors, l'indice mu va prendre des valeurs 0, 1, 2, 3, 20, 60, pardon, des vecteurs en métier comme ça, mais je peux définir une base duale de manière canonique en demandant que l'action d'une forme e mu sur le vecteur e mu soit égale à delta mu.

1:12:30 Donc je sais définir une base duale et je peux montrer que, dans ce cas-là, We can compose the matrix in a simple way, like g mu nu is equal somme on the indices mu and nu that I do not. Pardon, g, the matrix, it's the somme on the indices g mu nu of these products consommés. We always have to work with that, so there is no problem. Now, in all the bases possible, there are some that are orthodorms. What does it mean that a base is orthodorms? that means that all the base vectors are of normal units 1 or 1, that's with the vector of 1, and they are orthogonal between them that means that the product scaler of one vector by another in this base is the column of the column I have here a chart for indicating that they are orthogonal and the column of the column is the matrix of Mitowski which is the column of the column of 1, 1, 1, 1, 1 so there exist these bases like this And if there is a base like this, there is also a base dual, orthonorme. So this base like this, we call a tetrable. How it is defined in each point, it's a group of mobile, a moving frame. And if we have a moving frame which remains orthonorme, we call a tetrable. And the base dual, we call a co-tetrable. So what is interesting is that if we have a base like this and we express the same matrix, of course, it will be ETA, U, U, E, U, E, U, D'accord? Here ETA, U, U, which is the function of the Newtonian. So if you want, if I know G, I can decide if a base of null is orthogonal or not. But what is interesting is that if I give you a base, I suppose that I don't have a metric defined in my reality. If I give you a type of metric and I say, I'm going to impose that they are orthogonal. Well, that defines the metric. How? Because I have given a base. Here, this is the same. And I built the matrix like this. And at this moment, this becomes a definition of the matrix. This means that instead of defining the matrix as a form 0,2, I can define the matrix as a tetra.

1:15:00 And I can do everything. You know, instead of saying that the gravitation is the matrix, I say that the gravitation is a tetra. Well, Mikoski, it's the only space that's pure. So, yes, Mikoski, it's Rn, it's R4. So, I thought it would be a bit different, but it would have been quite curious to choose... No, but the space tangent is built in a way to be flat. So, here, we are in the space tangent, and we want the most simple matrix. And we know that if we have a matrix flat, you can always be diagonalized, so you can place it in the base of the diagonal. So what is interesting is that this is the fundamental object that we call a tetrable, and what is also fundamental for this part, is that when we have two bases, E mu and epsilon alpha, remember what I explained, the changement of base is an element of the group. Now, if I have two bases orthogonal, I put the chateau on-dessus for the end of the day. If I make any change of base, at the end of the day, I will make a base which is not not not a problem. So if I want to preserve the nature of my problem, I have to make a transformation not from the group of GL, but from the subgroup of that, which is the group of the organization. The group of transformation is not a problem. Now, I will rest here, if I want to preserve the orientation or not. I will not enter into these details. but that means that, finally, everything that I've done, all the differential differential that I've done with the tensors, the vexors, the form, etc., which is applied with the group linear, now I can do the same thing, reduced, what we call a reduction of space vibrational, which is very rigorous, to the group SO, to the group orthogonal. SO is the group of transformation orthogonal So I can say I have what we call a reduction of the group of this to this. But if I have a reduction of the group, I can do what we call a reduction of the fibres. The fibres are based on the group. So the operation reduction of the fibres is completely defined. And it's interesting because the notion of repair mobile arbitraire becomes the notion of tetrade. And the tetrade, now, is the chamber arbitration.

1:17:30 The notion of connection is something that is something that has a value. The connection is something that has a value in the group, not in the group, but in the head of the group, this becomes a connection non plus linear, which is what we call a connection of spin, which has its value non plus in the head of the group, in the head of the group. And, in fact, we can reformulate all the relativities general in saying that the fundamental variables of the relativities general are the tetra and the connection of the structure. And even if there is a dynamic dynamic in relativities general, we can even remove a dimension and we can't put a tetra in the triad. Well, that's a good time. And what we can do is when we have a formulation like this, we can see that it's a formulation canonique in the sense of the dynamic systems, and we can identify it. And all of which we call today the quantum quantum quantum computing The spin is based on a dynamic vision of the reality of the general which uses these things, considered as the channel and its barrières. I will modify it a little bit. And this is very important, because when we want to introduce the spin and, of course, after the thrusters, it's from this that we start and we start. Because the spin is completely liable, not only to the HDR, but to the existence of the world. I have a question. In what sense do we say that this loop of quantum gravity is called background-dependent? What does that mean? I don't know. Yes, so what we call the background-indépendance, the relative general, in fact, is a theory background-indépendance.

1:20:00 What does that mean? All this is liable to the difference. I was going to talk about it at the beginning. Suppose two varieties, V and V, what is a difference between these varieties? It is a transformation, a card, a variety in one another, which conserve, which, rather, transform the different structure of the same thing into the different structure of the same one. From the point of view of a different variety, two varieties telles that a different morphism between them are two varieties different morphes, different morphes, different morphes, and so, in terms of different variety they are identically when they are to define an identity Well, now, one variety, we can do a different variety, and then on demand that it conserve the different structure. There are a lot of, as I say, that is a group absolutely enormous, which is the group of all the different morphisms of V. And finally, if we have the transformation of V by a different morphism, as a variety of differentials, it's the same thing. Now, when I'm in relativity general, I have a particular object, which is, for example, the chamber gravitational, which is defined as the structure. I have a level of plus. So what I'm interested, S, c'est là, c'est D, le V et le G, je reprends les étranges d'une chose quand je prends la même vie, est-ce que, est-ce que c'est la même chose que... my variété différentielle est transformée en dv g est transformée en dg ce que dit la relativité générale c'est que vg et dv sont la même chose physical, which is what we call the covariance. But you can see that if here I'm interested

1:22:30 at a certain point M, in my variety, the differentiomorphism transport this point M to another point. And yet, even if I change all the points, it's the same solution in general. That means that in a certain sense, the points are no longer. So there is no variety of points that is at the base of the relativity general. In fact, what counts are only the fields, So the sensoriel, the sensoriel, and also the material that exists in a sense, is related to the other. It's why the relativity general is a relational theory. This is a way to say background-indépendance, which is the relation of different champs, particularly the matrix, because there is necessarily the matrix of the relativity between the other champs, between them. That's to say that the idea is that if you want to define the champ, for example, the chance pinoriel between the electron, you don't define the chance pinaurial compared to the space-temps, but compared to the tenseur and the new tenseur. So it's a relational theory. And when we want to quantify the gravitation, the problem is to be able to quantify this covariance which is a covariance and, in fact, we don't really know. So the way that happens is that On va, d'une certaine manière, rompre la covariance, c'est-à-dire qu'on va se placer dans, non pas dans l'espace-temps, mais on va faire une espèce de feuilletage de l'espace-temps avec des variétés qu'on va appeler l'espace, devenir espace et temps, donc on a calculé la covariance. On va faire la quantification de ça en introduisant des conditions, des contraintes, si tu veux, qui disent qu'on doit quand même retrouver la covariance. And, for a long time, we thought that these constraints were absolutely impossible to solve. And what happened in the last decades is that, in changing variables, and this is what is happening in HTC, in taking not only the variables, just, which were metric directly and are dual, but in taking, on the contrary, the tetrable and the connection, with the coefficients of the length of the length,

1:25:00 and the length of the length of the length, we are able to express the constraints which were possible to solve. So we arrived at a theory quantic of all this, which is the graph of the length. And then the spin, we are able to formulate this in a way of quantic, but we are not able to define the dynamics with a different point. But the idea is that we have a theory which, even at the point of view, will conserve the background of the independence, which means that it is relational or that it is foreign. The spinners, that is the transformation of the introduction. We have at the beginning the space-stand Nikoski, yes, I will define the spinners and the twisters on the space-stand Nikoski, knowing that after, when we have the fibers on a variety, when we are talking about relativity general, we will have a fibrary where each space tangent is in the space of Minkowski so if I define the spinners there-dessus, the twisters there-dessus then we will have a fibrary, and it's there that will happen something interesting but there is so much work already that I understand Minkowski that if I would like to do that, it would be simple so I'm talking about Minkowski, which is R4, in a variety with a certain metric on it. L'espace-temps de Minoski, tantôt on le considère comme un espace de points, tantôt on le considère comme un espace vectoriel. C'est-à-dire tantôt l'espace affine, tantôt l'espace vectoriel. Normalement, il faudrait faire rigoureusement les distinctions parce que je parle des deux. J'ai un peu mélangé, donc de ce point de vue-là, c'est un peu, comment dire, confus. Mais si vous regardez, tout peut être restauré comme il faut. Well, what is interesting is that in the space-temps of Mitovsky, there is an action in the group of Lorentz, which is the rotation of Mitovsky. What are the rotations? These are the transformations, if we look at the space vectorial, which are the matrix. That is to say, in Mitovsky's space vectorial, all the transformations, almost all the transformations which are the matrix, which is also the symmetry, but it's the rotation, and it forms the group of Lorentz,

1:27:30 which is the rotation 1,3, it's because it acts in 4 with the signature 1,3. So it's something that this group of Lorentz, which is on the vectors of the space-temps, which are the vectors of 4 components. Well, it's true that this group of SO3 is a group, So, it's a group of leagues, you know that the groups of leagues are in the same time in the varieties, so they have structures topologiques. And it's a group of SO3, it's a group that is not simply connected. So, it has what we call a remetement universal, and in a certain way, we can plunge into a bigger group, which we call, in this case-là, SPIN3. So, in a very general, when we have a group orthogonal or something, we call spin the same thing, sombre vector universal. Classes. I have written for Minkowski. All that I can say about the spinners, it is generalised in a complete dimension, unless it is a pair, it is to say that the sombre is a pair, so that also the formular dimensions are different. And, as I have written for the case of Minkowski, and also for the theory of the difference. So here, I write it for the space of the mythoskip habitually, knowing that there is still a generation possible for other dimensions. So, it is important to know that this group has a universal development called Spin 1-3. It is to say that it is almost the same group, but it is a little more large. In fact, it is a little more large. We will see exactly how exactly it is. And it's true that this group spin-1-3, when we call spin-1-3, it's when we define it in terms of the universe that we call spin-1-3, and when we look at what it is, we realize that it has a structure connue, it's in fact the group linear of C2. It's to say that it's the group of all the matrices complex, two by two, who have a determinants unit. So that means that this group has a fundamental representation, which we know well, it's the group of all these matrices. But it's true that, as we can see, it's also a certain form, in this case, it's not a form of linear or logarithm, but it's a form of synchronicity, no matter. But you can see that, in the same way that we had,

1:30:00 for example the group orthogonal where the action is preserved in a quadratic form. Here we have another group where the action is preserved non plus on R4 but on C2 a form non plus symétrique but anti-symmétrique. Well, it's a little bit like this. So, when we write the action fondamentale of this group we are going to write in a form of vectorial which is C2. and an element of C2, that is really a vector of C2, that is a vector of two complex components, we call it a spinner. Well, it's the definition of a spinner. Sachant that there are over the transformations of this group and sachant that there is a certain form that will be preserved by these transformations. Alors, what is the link with Mikoski? what is interesting is that if we take this space, and we can take, if it's complex, we can conjugar, we can also transpose, which is interesting to do with the products. So when we conjugue and transpose, we have a reaction which is called the transconjugation, and we can also do the products. And the point absolutely fundamental which is at the base of the theory of Spinner, is that if we take this space of representation C2 and we make the product tensor of C2 by the same and transposed conjugated, we obtain the space-stander Mikoski. And so we produce this form antisymmetry by the form of antisymmetry transposed conjugated which is found by the same and found by the same thing. That means what? That means that a vector of Mikoski apparaît comme le carré d'un spinner. C'est plutôt dans l'autre sens aussi intéressant, c'est-à-dire qu'un spinner apparaît comme la racine carré d'un vecteur. Et ça c'est le problème. C'est l'intérêt de départ pour le spinner. Vous savez que Dirac, quand il a voulu introduire le spinner, il est parti d'une équation de Klein-Cordon, qui est une équation quadratique, et c'est bien dans les temps que j'ai pris une équation linéaire et une équation Schrodinger, So he tried to take the square root of this equation, that is to tromper...