Invariance & covariance, geometrical spacetime (part) / discussion / lunchtime conversations

0:00 It's a trend in mathematical development if we consider in a very general way the idea of understanding the Galois theory as the theory of groups of indistinctions. And so one of the questions to be asked is whether we can consider the group of indistinctions that is present here as a particular case or a case of a group of indistinctions of some kind to try, once again, to make a sort of Galois theory of physics, of physical theories. All the prejudices that mathematicians and physicists suffer from the elaboration of the equations. In particular, these are the theories developed by Carquet. We should know if there is a group of Galois that would allow us to consider the constants of physics in an even more general way. So he calls this, it makes you smile, so knowing that in this case, without going into too much territory, we have to deal with extensions that are not necessarily algebraic, extensions of transcendence, it would also mean that we need to know if we can work on extensions of transcendence, knowing that we don't know much except on the calculator. There is a movement that works in this direction and is engaged in the aspects that are very important in this work. In the independent formulation of less-principal coordinates, we consider objects as different types of abstract applications.

2:30 The vector-conjunction is the application of real-time values ​​in the real world. The infine connection is the application of vector fields in vector fields. The matrix parser is the application of vector pairs, vector couples in the set of real numbers. In this formulation, a theory is not a reference to a coordinate system. So, we have four varieties, and then we have the geometry theory, which is constituted by the theory of T, and we have a tangent vector, T in sigma, a course class for the variety, and the set satisfies the laws of movement of T. So, the class of models extracted by theory is independent of the coordinate system. It depends on what we mean by the old one. We don't have the components directly in the database, but we borrow them in the system of given coordinates. And it can be very useful, even if it is still very difficult. The components of a tangent vector form a square, as shown here, and the components of an infinite solution form a square, as shown here, and the components of a geometric transformation form a square, as shown here, and the same for the other geometrical components. A geometric object can therefore be seen as an application that has an input, a coordinate system, and an output, a set of real-value functions, state, alpha, and so on, defined on the X system. So the equations of space-time theory become a differential equation system for the components of the geometric object of T in the component system that we have chosen. So we have the intrinsic form, without coordinates, of a dynamic equation of the movement, given in this way, and in terms of coordinates, it becomes... This is what is given here. D2, A, I, B, U on the gamma of J, K.

5:00 A, I, A, A, I, A, J. A, I, A, A, J, A, K equals 0. The A's are the functions of the vector K in the coordinate system. And the gamma of J, K are the functions of the vector B in the coordinate system. So, in general, an independent equation of coordinates becomes... So the rules that describe this change are based, as we all know, on the laws of transformation for the types of geometric objects in question. So, when we change the coordinate system to another, the equation of our theory changes, we move from a linear equation system to a second system. I repeat again, so either the genetic equation we have given, if we have a flat variation, we have a linear coordinate system with gamma minus gamma. And in the X-ray, the genetic equation becomes this, dA by dAx times u equals 0, and in the non-linear coordinate system, the same, non-linear, the same genetic equation is given by the HPS equation. All of this, I remind you, because I have the impression that when we work on these notions of covariance, invariance, etc. This is the concept of coordinates, and there are several ways to get rid of coordinates. So independent coordinates, and there is a question of knowing if independence in relation to coordinates has a physical meaning as such, or if it has a mathematical meaning as such.

7:30 When we do relativity, we can say in a certain way that it is a mathematical tool necessary to do relativity, we work with coordinates, but it has a meaning that goes beyond. It goes beyond, it's not just the fact that we work without coordinates, we have to bring ourselves back. And on the other hand, historically, that's why we have to spend a quarter of a chair on coordinates. We think that the notion of an observer is mathematically or physically linked to the notion of a relationship with the fixation of coordinates. So, the intrinsic genetic equation has two arguments, the vector t is now a function of infinity. In the coordinate system, one of the arguments disappears, the function of t is null, and the differential equation system only has the argument of the tangent vector as an argument. And when we move on to Y-axis, the components of D reappear. Exactly. The second argument, but they are not at all the same, is that the geodesic equation for a given connection is simply to say that the tangent vector is parallelly transported by a vector. So the argument is true. The unknown is the tangent vector. The connection is supposed to be given at the same time. If we start from a specific point of view, a particular system of coordinates, a differential equation system in the case of the degrees that are given in such a system, so we select a class of the geometry over it, but the selected class depends on the particular system of coordinates that we have. If we start from a differential system, we select a class of degrees that are different from the class selected by the equation given in the case of the degree at the base. So, in relation to the different choices of the coordinate system, the same equation will be used to select the distinct classes of the model.

10:00 If we have a predictive system, a system selects a class of the model of the form m of the standard equation m of the state, so the system is covariant in relation to a change of the coordinate in the form of a new system in the case of the same class of the model that is selected in relation to the new system of the coordinate. So here is the first option. Now, the covariance group of a differential equation system is the largest group of coordinate changes such that each group transformation satisfies the specific condition. In the second way, we can see the transformations of our group as the transformations of the variety of automorphisms. So, these are automorphism applications. And so on and so forth. So, sufficiently continuous, that means that we can use displacements, and finally different terms. That means that we can use polarity in general for these two, but ... And, under the reality of the H-transformation, the human object changes from θ to Hθ, whose components in relation to xi in Ht are equal to the components in relation to θ in relation to y in qi in qi.

12:30 And m is just the point, it's a double-edged sword, as I told you earlier. No, no, no, after, when I told you I was going to say that the transformations are more or less the same. In the second approach, we can see the transformation of the group, that is to say, it is more or less continuous, we use the neighborhood. The M, the big M, that's the variety. And what do you mean by the neighborhood of M? The appartement, yes, the neighborhood. Locally, that is to say, you look at all that? Yes, of course. Locally? Yes, yes. Locally, it is less strong than locally. But the problem is that, well, maybe I... Well, I want to interrupt because it will allow me to say things. Do you want me to ask questions now or after, Jean-Jacques? No, but I will continue to talk a little bit because... So the objective is still to try to find a distinction, to arrive at the point of distinction, so by focusing on one of the different works, to make a distinction between covariance... And invariance. Exactly. And invariance. And diffeomorph invariance. Again. And to give you a definition of what it is. What it can mean to you. And to introduce you to the problem it poses afterwards. Because to do that, we have to create a concept that is the notion of absolute object. And in the constructions that have been made by different...

15:00 In this way, we can show that we do not have in the system that we have this absolute object that we have just forged, which will then allow us to define what we hear by theorem of invariance. On the other hand, what we also call, in the same case of theory, is independent, background independence. So independence in relation to a basis, a value, a reference. And it is through these two concepts that we are trying to forge a satisfying definition of the notion of invariance groups, or even of subgroups, which allows us to narrow down what we wanted to look for in general relativity studies. There are many problems posed in the context of this notion, on the one hand, of absolutes, and on the other hand, of independent equations, and objections that are not completely finished and far from it. So, in fact, this is the style, the sequence of a transformation, a large H of reality, which in turn changes from a large Theta to a large H of Theta, including the components, which is important, the components in relation to X in HP, the components of Theta in relation to Y in P. So we write what we want to write in P. So that means that the object is not real. It means that it is not real either. There is a connection, for example, that is transformed differently. No, it doesn't only mean that it's two years old. The relation up there, I don't know if you can see it. Well, we put it back. Because H is the... Yes, I think it's what you're saying.

17:30 The J is missing a term there. It's for a scalar, then. It goes for not only one. Because the port of H is a scalar or not? Well, the port comes back higher. No, H is for a sphere. H is the difference between them, that's it. No, but... When the difference between them, it's just a... So, take a coordinate change, which gives you H. The coordinate change is XY. Then there is a H-conformation, which belongs to the group of conformations of the automatic system. So, X applied to HP equals Y applied to P. All types of coordinates induce a transformation to H, and vice versa, it is also true that all transformations to H induce a transformation to H, given in this way, so we will see. And now, in the components of a geometric object, here is what it looks like. Or finally, we will come back to it if you want. So the two approaches, either you can consider a coordinate system as a window through which we can see space-time and its geometric structure. And when we move on to a second coordinate system, we see space-time through another window. The word window is intentionally chosen, knowing that the monads of the unit are without a door or window. So in fact, the way in which they communicate is by using coordinates. Consider a set of equations of R4 as a square or a circle, which can be displaced from one window to another. And the only geometric terms are those that satisfy the equations related to the coordinate system in question.

20:00 An equation system, from this point of view, is invariant, under a coordinate change, if the same objects are visible when we move our frame from window to window. I have a window, I go to another window and I see my geometric object, but in addition I have a frame that also moves from window to window. So we can give this definition, a system of equations is covariant for a random change, if the same objects are visible when we move our frame to know it. It does not depend. When we deal with a transformation of a variety called H, X is the frame and the window, and we change the geometric objects themselves from Φ, objects based on Φ, HΦ, a system of equations is covariant by H if All states, great and small states, and great and small states, are divided by the same window. The point of talking about an isomorphism is precisely that you don't need to talk about windows. You don't even need to fix a frame. Rather, what you call a frame is a given system. An isomorphism is defined independently of any given system. Okay, okay, but then, what does that mean for the variants? And what does that mean for the derivatives? You're talking about making a difference between a variance and a variance, and even in the two approaches, if I replace covariance by a variance... No. So, according to the definition of H, a frame is covariant for a concentration H of the variety in the case where it is covariant by the coordination of coordinates which is composed by H.

22:30 What is the meaning of the covariance group? What is the meaning in particular of what is called general covariance or general covariance? We consider what is and what is not covariant. So the equation of the geodelic given here is not covariant in general, it is for linear or linear proportions. On the other hand, the equation where I just introduced the Gamma ijk is covariant. So the ordinary derivative of a vector field is not covariant, but the application of an operator of derivation to a vector field is. The components of the operator of derivation are corrective terms that make up the covariance differentiation. So the connection to P is called covariance derivative, which is important. I'm sorry, maybe I don't quite understand. We transform an equation that is not covariant in general into a Keeley equation by replacing the original derivatives with covariant derivatives. The star-star equation is the general coordinate representation of an intrinseq equation, which is not star-Large. Simply because the ordinary derivative of a tensor is not a tensor, it is a covariant derivative. Yes. Now we can say, of course, that a covariant equation in general, or generally covariant, if we can form it independently of the coordinates. Only the covariant theory of the general equation gives us a correct description of the space-time theories. Theories that are not generally covariant, such as the traditional Newtonian theory, or the terrestrial relativity, are developed differently. We are in a flat space, and beyond the distance of the Cartesian coordinate system, or the binary coordinate system, in which the components of certain geometric objects become constant. They are called temporal-temporal geometric objects. The terrestrial human being is covariant. It is a geomorphism of a certain geomorphism. It is written in a non-verbal way, so it is a variant of all the changes that exist in the world. We can put a physicist only in relation to the law, not in relation to science.

25:00 Megabase has the same... Yes, yes, wait, wait. It is a variant like the Rhinoceros. I can continue a little. The equation system D2Ai over D1U contains only the components T indecisively, T indecisively, T indecisively, T indecisively, T indecisively, T indecisively, T indecisively, T indecisively, T indecisively, T indecisively, T indecisively, T indecisively, T indecisively, T indecisively, T indecisively, T indecisively, T indecisively. For example, the shape of Gamma GK equals zero for the D-convection or the shape of Gij with this signature of the diagonal for the matrix. That's what I'm talking about. In general, the covariance group will be a subgroup of the linear group. Traditional formulations lack covariance in the general sense by not including all the other geometries that are actually supposed to be. We can then, for example, bring the equation to a general variation by saying that there is a system of linear force in which there is a magnetic field equal to zero and the equation d of ai over du is valid. What does it really mean? That is to say, one we can do in the case of Nielsen-Kontz which is the other in the case of Eugénie-Magnon? No, in which we ... To say that it really presupposes the case of Eugénie-Magnon? Well, really, really, it means general, it means generalized. Jean-Jacques, the covariance group is the change of coordinates. It's not a linear group, but it's the group. Vertical automorphisms on the variety of the principal figures. The covariance group has infinite dimensions. There is the linear group and the two-dimensional group. But, be careful, I'm not talking about the covariance group. The covariance group, in general, is the differentiating group or a subgroup. For the moment, I see it.

27:30 The dichomorphism group or the group of differential coordinate changes is the same, in general, in general relativity, the group of variance is the dichomorphism group, but which is, in fact, the linear group, it's the gauge theory, it's the same thing, that is to say, the linear group is the global gauge group and the dichomorphism group is the local gauge group, that is to say, when you transform the gauge symmetry, in this case, the global symmetry, that is to say, rigid, The question is whether you preserve the properties of geometric objects. The definition of a geometric object. Geography is co-variant compared to geomorphism. If you give me a definition that I don't have yet, a definition of co-variance in the sense of completely generalized with... The traditional principles of relativity can only be understood as requisites of covariance. For example, the general principle of relativity is almost always accompanied by the general principle of covariance, the requisite that the group of covariance of our theory is the group of indecisive transformations. From the simple requirement that the theory is formulated in an independent style of coordinates, this requirement can be satisfied by any of these theories, which can be formulated in this kind of formulation, with this kind of formulation. And we will come to Kretschmann. So Kretschmann, by pushing this idea further... So here we have articles by Hubert of the Physicist Institute for the Relativity of Postulats, Einstein. Here we have a sponge that tells us about the relativity of this theory in physics analysis. The general principle of coherence has no physical signification, but it has much less. It is simply a certain type of formulation of physical theories.

30:00 The simple space-time, where you have the line E3 3R, In which all the notions of absolute motion and their signification in all possible senses can acquire an independent description of the coordinates as easily as general relativity's space-time. So the question is to know, in this case, which form of covariance will manage to capture the signification of general relativity. So the notion of covariance group, according to this theory by Ketchman, which has been discussed by Einstein... So the notion of relativity is not the right notion to interpret a traditional part of relativity. So all of our space-time theories are covariant by the group of coordinates transformation, but they are trivially covariant by all subgroups of coordinate transformation. So there is a need for another notion, which may be understood, provided it has a refined meaning, the notion of covariance, if we have elaborated it. There is no other notion than that of covariance groups, and he proposes to give the notion of invariance groups or symmetry groups at this moment. He says that this is the right notion. So I come back in a few moments to the statement on which the general principle of covariance is based on what I have said. The opinion was to share the characteristic of ARG and its independence in relation to the background, in relation to a base. They were almost immediately accepted by Einstein. We find this in volume 7 of the complete edition of The Station, which makes it difficult to define, to give a sense of the ERG by PGC, the generalized principle of covariance. A better definition would therefore be the notion of dichromorphism invariant. These are very clear, including general relativity and general covariance principles. And he says that the general covariance principle involves the general relativity principle, but it is not the key.

32:30 So the situation is more complicated because Kretschmann says that TGC has no physical meaning. So Kretschmann says that any principle can be rewritten in a form equivalent to the general covariance. So Einstein, on the other hand... It has been maintained that the general principle of covariance, which is simply historical, could express a conflict if we elaborated in a more general way a concept of formal simplicity. It is a concept that people who work in terms of analytics, people like Northam, etc., will be able to elaborate. Just to confirm the terms, that is to say, a variance, what is it about? It's not about a variance, but we still have to be precise. Okay, but what's the difference between a variance and a covariance? A covariance, as Kretschmann says, if it's a covariance, it doesn't work, because any theory can be a covariance. So if we want to understand what a theory is, it's not the concept that allows us to grasp it. So we need something more. What's more, it's invariance, but rather invariance group or symmetry group. And when you say invariance, what is invariance? Yes, that's it, that's it. Equations, that's invariance. No, but if I understand well... Listen, if I understand well, but maybe it's wrong. When we speak of invariance, it must be a constant value. It must be a value. That is, it's not just covariance, equation... No, it's the symmetry of the equation. No, but in general relativity there is no symmetry of equations, there is no invariance. No, but when you write symmetrical equations in equations, let me see, I'll give you the... Excuse me, but... I'll try to give you the conclusion. No, but what definition do you give of the covariance? Is it the same thing as the differential of the invariance of equations or not? Yes, but the difference is the preservation of the curves, so that it's like that, and of course...

35:00 We don't know how to give everything that is given there. So, the definition of covariance by g is the following equation. It means covariance. So it's for example. For example, it's an equation. So, gamma is... So that's exactly it. It's the invariance of the equation that differs from each other. That's it. And invariance by g is not the same thing. It's that the given system does not move. That's the invariance of the quantity sigma by g. And that's what... There is no invariance in relativity, for example, or diffeomorphism. There is invariance of equations. It is covariance. Here there is a kind of constant. You keep the same sigma. Diplomorphism transforms the metric into another metric, which is not equivalent, but the variety transformed with the metric transformed is equivalent to the previous one, which is the covariance. But in general, the metric transformed is not the metric at the start, except for some particular diplomorphisms which are called diplometrics. The invariance, that is to say, is to transform the solution of the equation into the solution of the same equation, that is to say, it can be a symmetry in the equation. That, by default, except in very particular cases. What I have learned is that the principle of invariance of the general relativity is the principle of equivalence, and that it is absolutely not a matter of the general relativity of the movement, even if it is a mathematical theory, but it is not a mathematical theory.

37:30 You have the tangent space, so in the tangent space you have the Minkowskyian physics. When you work on a variety, you mean that there exists in each point a tangent space, if it remains Minkowskyian, you have the idea that it is not on the part of the player. Exactly, in fact, on a variety, you can not use the metric. Yes, yes, yes, in space-time, in space-time, in space-time, in space-time, in space-time, in space-time, in space-time, in space-time, in space-time, in space-time, in space-time, in space-time, in space-time, in space-time, in space-time, in space-time, in space-time, in space-time, in space-time, in space-time, in space-time, in space-time, in space-time, in space-time, No, now you take the theory of Kalou-Zaclin. There is a fifth dimension, but which is predicted, which does not have the same status quo, so you are not covariant in the five-dimension variety. You are not a variant, you can not make a dichromorphism in the third dimension. But here more precisely, there is something constant in the absolute sense, that's it. That's the deeper notion and we come back to it. What do you mean by invariance? Yes, invariance is that. Yes, you have values that are just in a simple sense. What I don't understand is precisely, in Galois geometry, what are the symmetries of equations. And here, it's the symmetry. Invariance is the transformation of the equation into the transformation of the same equation. And I wrote here, for example, to give you this example, if you take Maxwell's equation that I gave you here with Gerard de Roche, Thank you for your attention and see you in the next lecture.

40:00 We find a similar equation. It is true, it is not true in the second case. Can you show on this equation what this means for the variant? Well, that's what's written there. I'll show you how to rewrite it. But it's not the equation, it's a variant. It's F, the human tensor. I don't have an invariance in the sense that the solution of equations is that we have equations. Again, I don't want to refute Kachman, but that's Kachman's argument. But when you talk about solution of equations, what precision is it? Well, it's precise. I think I know the solution. But is it homomorphism or something similar? That's the difference between the two. Or it's just numerical values, it's very... that's what can make a difference if you understand. So, if I rephrase the argument of Kashmar, it is to say that the covariance of the different m covariance... So it means that your equation, as you see it, is geometric. An equation that is written in a special coordinate system is also written in a covariant system by imposing the coordinate system as your basic equation. In other words, it means that realizing invariance by covariance is not the same thing, and it is more complicated than simply realizing covariance by covariance.

42:30 This is the second case. Excuse me, but I don't see which invariance you're talking about, which is not the covariance. In relativity, nothing is invariant other than in the sense of covariance. To realize an invariance by a scalar, it turns into phi prime. Your phi prime of x prime equals phi. Your phi prime of x equals phi of x. It's also invariant. Yes, but phi is invariant. So, if we look at, for example, the quote by Pauline Misa, to realize an invariant of quantum mathematics, it is non-criminal and involves a general order, I think, in general, and so it is a new edition of the book commented by the professor of mathematics there. I continue, then, if you want, we will resume. A proposition that was made by Anderson in his recipe manual in which he explains that in reality, the components of any transformation in reality are of the same value in rt which is phi. This transformation is called Messe phi, a symmetric invariant of phi. Here we have h phi equal to phi. Anderson thus defines the symmetry group or invariant group of a theory in the largest subgroup, the group of both normals, leaving some objects of the invariant theory. He calls these objects of the absolute theory. We must distinguish between what we mean by absolute objects and dynamic objects.

45:00 The absolute objects of a theory are those that are not affected by the interactions defined by the theory. Absolute is the metric of restricted relativity and not absolute is Newton's mechanics. Examples of dynamic objects are the metric of general relativity, the electromagnetic field. So there is this distinction that is still being made by Anderson, which was very discussed. An absolute object is determined by the field equations of a T-theory. Friedman discusses this. He defines an equivalence relation such that the absolute objects are determined by equivalence made by this relation. So we have a model of the T-field equation which is given. Since T is covariant in general, we know that this model is also a model by the H-automorphism. The general covariance is given here. Again, he says that if we have this, then we have this. But he says that if we have this, if we have the general reality, we do not necessarily have this. So once again, the point, but maybe once again it's a question of vocabulary, is that general covariance is not the same as symmetry, And to obtain a theory that captures once again the essence of the general reality, we must give ourselves the notion of symmetry group. So I will perhaps go to the link between symmetry group, like Anderson, and covariance group.

47:30 What happens is that there is a link between absolute objects and theories that describe flat spaces in which there are inertial coordinates. Such systems are defined by the components of certain objects that become constant. The symmetry group of a theory is a group of the variety that preserves the absolute objects. These objects, in turn, serve to specify a class of privileged coordinate systems in which the equations of the theory become equations for, this time, the only dynamic objects. Distinguished between absolute objects and dynamic objects, the absolute objects become constant and disappear. So, the standard formulation of theory. If we refine the vocabulary, the group of symmetries that Anderson wanted to look for is actually the covariance group of the standard formulation of theory. We can consider a theory in which the models contain two objects, Φ and T. We suppose that Fischer defined an inertial coordinate system of T, so absolute, and that θ is dynamic. If M is a model of T, then M is possible. If M is a T-model, then M is a T-model for the standard formulation of T relative to the inertial system, which is not restricted. So what Michael Friedman says is that if H is an element of the T-symmetry group, then we have H applied to Φ equals Φ. The inertial system of HΦ equals the inertial system of Φ. So M is a model for the standard formulation compared to the inertial system.

50:00 In the first place, we have distinguished covariance groups and symmetry groups by distinguishing which was the lack of the generalized covariance definition and now we show that the symmetry group allows us to give a more satisfying meaning to the covariance group. I will go directly to the end to give you ... So I summed it up in 8 definitions, I start with 3, so an equation is invariant by hypermorphism if and only if it admits diff as a group of invariants. Definition 2, any field that is either non-dynamic or in the space of solutions consisting of a diff in orbit is called absolute structure. So in fact we need to give a notion of what we mean by absolute structure. This means that we are building an orbit under the action of diff, which is given by the orbit in the space of solutions. So, definition 3, a theory is called independent of the bottom, if these equations are diff invariant and if these fields do not include an absolute structure. Two fields, T1 and T2, are locally said to be equivalently different if and only if, for a point P, the same, there is a neighbor of P and a different one that acts in this way. A field that is either non-dynamic or whose solutions are all locally said to be equivalently different is called an absolute structure.

52:30 Is it the same that is said to be linear? Yes. I concentrate the definitions of the article I read there on linearity. All the solutions are locally different from all the solutions in the dynamic field. Yes, but a field that is either a dynamic field or in which the solutions are all locally different, equivalent, we call it the absolute sphere. But the solution 2 says the same thing. Yes, you're right. No, 4, sorry. The field 1, excuse me, yes, it's the same. Okay. Yes, okay. A solution that is independent of the background, if it increases, these equations are invariant in the sense of 3 and these fields do not contain the absolute structure in the sense given previously. So this is the state of the definition. So once again, it is a question of trying to give a definition. We proposed what we call covariance, a sufficiently elaborated definition of covariance so that it can reflect the physical properties we are looking for and what we proposed on the basis of Kretschmann's critique, then on Anderson's propositions, then the critiques we made to Anderson.

55:00 There is a lot of difficulty in defining a group of symmetries. A group of symmetries, once again, is what I mean by the symmetry of the equation that defines it. And so it gives the definition of 8, knowing that there are some critics on this notion. Because the question is to define. This will allow us to define the invariant symmetry. The equations are invariant, but the definition of Newton's program means that there is no invariant structure. How does that work? The equations are invariant, but the differences are invariant. Do you have theories? It is to have the notion of dito m invariant that we have built these absolute structures. We do not have the absolute structure if and then we have this invariance there. You try not to have it and then you hold it. No, I'm trying not to have it. That's what I don't want. So, an independent theory, if these equations are invariant to each other, these fields, the fields that exist in this theory, in this data, do not have the structure of... The principle of definition 5 and 8 is a simple one. When we talk about the independent balloon, we are trying to mimic it, that is to say that we are looking for the green line...

57:30 In which the states of the matrix are elements and the states of the hyperboles of the banknotes of the matrix. It's not a matrix, it's a question of terminology, but when we talk about quantum gravitation, it's not about the matrix itself, it's really not about the matrix itself. So maybe it's a couple that everyone who speaks about quantum mechanics can modify as a coefficient of the matrix, but the problem that people who study quantum gravitation have is that they partially arrive at the same conclusion. All of these have to be done independently of the metric, although there is no mention of the metric. I have the impression that this is not the standard definition of the general definition of the metric. The question is to know if the introduction of the metric does not change what I have not verified. Maybe it's a restriction of definitions and the problem that limits the definition of basic information to the problem of the answer. I would like to verify how he defines this problem, but he puts it in a case like this. It's possible, it's possible. I'll show you in the pages... He didn't even mention 10 N in the term. That's why I'm having trouble doing it. We'll see if this general definition includes the other and especially if the fact that Saurès will paint the other changes the problem. In any case, it's already a big problem just in the case where if we change the matrix we can understand structures independent of the matrix and that's it. Strictly speaking, it is called quantum mathematics. So, one of the arguments that is cited against one of the problems posed by the definition of the notion of absolute structure is the following.

1:00:00 In fact, it is the symmetric problem of the one of the difference between covariance and symmetry. Here, we can give absolute structures that we would normally give for dynamism. Because we know, and this is a theorem, in differential geometry, that vector fields that originate from nowhere are locally different from each other, so it's a theorem. And so a theory of different variables, including vector fields among these variable fields, cannot be very independent. Problem, and so I refer you to the answer, which is a bit technical, here in page 14 of the article. They give the case of the equations of Einstein and Miller for a perfect fit in these zeros and try to see how we could get away with it anyway to try to see what is absolute and what is not absolute for dynamic structures or absolute structures. So I summarize the abstract framework of the article and what I have exposed to you. Let's say invariant in the covariance group and in the search for the same idea. The argument being that the notion of covariance or the notion of covariance group is a much too flexible notion and it has a mathematical meaning. The constraints of physics as we have given it. And so we have to introduce a contraignant structure which is that of symmetry or symmetry groups. The question is then to try to give a satisfactory definition. This notion of group of symmetries is itself, of course, it joins the notion of covariance in a generalized sense, but this time the covariance having acquired what we were looking for and the question posed by the definition of what we mean by group of symmetries or groups of indistinguishability, that's what we call it, or group of symmetries, it presupposes another construction, which is a construction of the notion of absolute structure.

1:02:30 It means that in order to define the group, to give our absolute structure a sense, we have to define the theory in terms of independence of the elements of the theory, in particular the equations.