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

0:00 In my opinion, this is a very, very broad range of learning examples, and once this is done, then we will have the definition of what we mean by impossibility. And once this is done, we will have this definition, or a satisfying definition of what we mean by covariance, or we can reject it, but that's not the case because that's enough. It is difficult for the community to work with covariance notions, but we have given this covariance notion a more satisfying meaning. So, basically, now the discussion is open. For example, if I resume, these are formulations that are a bit... I agree with... Some of them are a bit alambic, some of them are a bit atypical, but for example, the only question is whether or not there is a definition of covariance that is entirely satisfactory, as is the case with all of these rather vague discussions. For example, here is another work done by Friedman in older articles, articles from 1973, where he says the two requisites to which a theory of physics must respond. He has a requisite 1. All inertial ratios are physically equivalent or indistinguishable, but this is not enough. We need to understand relativity, and therefore R2. If two referentials are indistinguishable in this theory, they can theoretically be identical, according to this theory. And so we can now look at the symmetry group at Anderson. The principle of special relativity is the conjugation of R1 and R2. And the symmetry group is the largest group of transformations in reality, which preserves objects beyond relativity.

2:30 Equivalent referentials induce a second group, the indistinguishable group. The group of all transformations linking two equivalent referentials and leading to indistinguishable models of t. And again, they develop in terms of indistinguishability. So if a theory satisfies the two requisites of the data, the group of indistinguishability of t is the group of symmetry of t. So basically, it revolves around this... Construction, as we hear it, by groups of symmetries, preserved structures, and so on. Just as I was saying, you were talking about a conference of Pierre Cartier. Pierre said that it relates a little bit to the story of the little gorillas. It's exactly that, let's say... All this analysis of problems posed in the framework of Riemannian geometry analysis by the notion of groups, I think at the beginning of today, it's even a little strange, because of course we can talk about groups of coherence, but the way, at least for me, it's not important, but in other ways, let's say. One way or another to talk about it is of course to say that there is a pool of coherence, but I would still like to ask this question because it is at least one way to make the difference between coherence and invariance. Maybe, well, we must discuss, we must also discuss what is the physical aspect, maybe there is a specific physical reason. It is important to keep this approach of groups, symmetries, etc. But it is also Elyse Moline who poses this problem. Is symmetry the right thing or is it something else? Or is it something to discuss exactly in this context?

5:00 Because of course, from the moment... I already presented it the other time. In what sense? Do you understand? In the context of Riemannian variety and general relativity, it is not that all transformations are composable, is it not? If we take two Riemannian elements, we can always make a path, but if we do not talk about something topologically bizarre, we can always make a path, but still not composed of... There is a whole structure of transformation, which is partially compatible, one by one, one by one, but not in a group. Of course, from this structure, we can always come back to a group, if we just disregard all this. The simplest example is what we call the fundamental groups, the topological spaces. It requires a lot of elaboration. It requires a lot of preliminaries. In my opinion, it clarifies a lot of things here. There are co-reliance and co-reliance groups, but in a simpler way to say that there is something like a groupoid in the case of co-reliance, which can always be presented as a group in a trivial way, that is to say, when it is called a formation, if we do it, of course, we can always, apart from a groupoid, we can say, we identify all the objects and say it is the group, but a little weird, it is not a co-reliance, etc. There are two possible orientations. The theoretical mathematical orientation, but there is also a much more elementary way of transforming the DiffM, which is not a real group, but we have all sorts of means, we are not obliged to make categories, it is the quotient in the classic group of... No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no.

7:30 The problem we usually encounter is that we never talk about the observers and we always group around them. The confusion between local observers and map changers is that many teachers continue to teach in PEA as a group of map changers that has never existed and that we call, at the same level as Michel Morphine, but we will maintain the confusion. In fact, they do not have to talk about the coordinates of these things. Yes, in fact, it is perfectly justified. If we talk about it in a global way, the form of a category that is adapted to topology, it is the category that we are looking for. I took the coding class of my teacher. I learned it in this class at the University of Montreal. That's exactly what he says. He says it's a teacher's intro. He says that there is this problem of... I think it's important to know how to define the word in question and I don't think it's... By the way, in fact, you must have it somewhere. It's not a coordinate change, it's not that. No, no, but when I speak, when we speak in terms of coordinate changes, when we speak in terms of coordinates changes, we look at a group of map changes, we look at a group of map changes, etc. There are times when you have to go to places where you don't have to go there. It's just what we want to tell you. Now, the question of knowing if we are going to access the categories, etc. is another problem. I believe that categories must intervene if they are imposed. I mean, if they are imposed naturally. At the moment we are doing topology, we have a category. But we impose ourselves in the sense that if we can... It could help to describe all these structures, let's say, by means of... Algebraic, co-physical, with groups, etc. The only use of groups is not enough. On the other hand, with categories, there are propositions, there is geometric symmetry.

10:00 It's a way of reconstructing the calculations of geometric geometry in an algebraic way, but that's for sure. It's not the same as the twistor theory on the ground. You run, there's a way to do it. Yes, but for example, in the case of this theory, it's quite plausible. But at a certain point, we don't need to know if there are emotions that are naturally corporeal. At that point, we build. What Pérotin says is that we have to build a fiber pointer, and there we have a real big theory, full of elaborated theories, Galois theories, in the polemic sense, but in my opinion, for the moment, I have the impression that it would be something that is... I'm trying to... the question today, the question is, is it broken or not? Do we try? Is the problem of distinction, is the notion of covariance a very clear notion, and that all this discussion is a... There is a long discussion on this subject. Can we have a satisfactory distinction between what they call symmetry groups and covariance groups? I think it's perfectly distinct, both in relativity and in string theory, but the two things are totally different. Yes, but if you look at relativity from the point of view of gauge theories, the covariance of relativity, in the language of gauge theories, you will call it gauge symmetry. That is to say, a diffeomorphism of a variety from the point of view of string theory. Geoge is a Geoge transformation, that is to say, what it transforms, we can call it a change of father, and so in fact, a Geoge transformation is something that does not change the world in the same way as the 3D world, a change of coordinates does not change absolutely nothing, does not change at all any geometry entity.

12:30 And therefore no physical quantity if we take theory and geometry. Yes, but the argument that has historically made sense is the 1917-1918 Krashman argument, which was taken up and discussed by Einstein. Krashman says that the group of covariance has no physical signification. Why? Because we can give any theoretical theory covariance. But in what? For example, you can make Newton's theory covariant in space, but not in space-time. You can make the theory of Kehl-Ouzak-Klein covariant in space-time at 4 dimensions, but not at 5. So, if you want, covariance equals geometry. If you manipulate real geometric objects, i.e. vectors, connections, tensors, scalars, it's necessarily covariant. You don't even need to ask the question. If you manipulate... Yes, but if you don't manipulate, you manipulate. It depends on what you mean by geometry. There is geometry in the sense of... Mathematics, you say. Yes, but in the sense of... When you say, I don't know what, when you say it's the geometry of space-time, the geometry is still loaded. It's not any geometry. Geometry is a structure defined on a variety with thinkers. Points... Points are a variety. The points of gravity are the points that constitute these varieties, which are geometrically related to each other. Now they are events. And these events will have an existence, they will have an existence that is only virtual when they are for this variety. They will not have an existence in the world of reality as they are in the world of the individual.