Talk — Part 2

0:00 Geometry is obviously invariant by different morphisms, and we can interpret this invariance or different morphisms also as an invariance by the changes of maps, by the changes of coordinates, since we can show that each different morphism, since we are in the category of categories, in the category of varieties, is identifiable. It seems to me that geometry demands to do that. There is a category, we agree, we do not make any difference. Well, that's to specify the status of this covariance that is often discussed. We will talk about the different varieties. All these sensorial fields. When you have a diffeomorphism, it trains you. So there is an equivalence between the different... There are a lot of people who reduce covariance. It's not quite the same thing anyway, because a diplomorphism is really a map with a bunch of coordinates. Sorry? It's the map. I agree, but yes, that's the first difference, except that you can also ... But the other difference is that for me, all this is perfectly defined without ever talking about coordinates. For the first time, I go as far as I can in physics by avoiding using coordinates.

2:30 At some point, you have to do such a calculation, you will be forced to use coordinates. There are two kinds of abstract notions, one is to talk about topological transformations and look at topological objects, or the other is to look at objects with transformations, that is to say, there are two differences. We just abstract the transformations, we take everything abstract, or else we take everything with the transformations and we look at how it works with the transformations. There is another place where we can be a little more specific. You say you don't know the coordinates, but I don't know the coordinates, and we can do better. That is to say, we can replace the vector by the data in the tangent space, the pose, and in each corner there is a pose. Which would be the null cone? The cone of the other one? The cone of the pose. But that's not enough, it only defines the conformity part of the metric. If you give the cone of light at each point, Yes, but it only gives the sign. Since if you have two different metrics on the same variety G and G', the tangent cone at each point is the same. Its interior is the same too. Ah yes, that's enough to define the whole causality and the chronology. But you still don't have the metric. I also agree with you, it seems to me that physics only needs the conforming part of the metric. What does it mean that we need, let's suppose that we need, in addition to the conforming part, the total metric?

5:00 It means that we need each time an echelon of time length, but we don't need it. When we do a measurement, we never compare a certain length to an echelon length. If I change the metric, if I multiply the length by 2, It means that the length of the heels is going to be multiplied by two, my length too, but it seems to me that we should reformulate physics in a completely independent way from profound transformations. And it remains at this point that the causal structure that you can very well represent by, if you want, a cone field. Now I see that there are still two ways, or you say, well, I say that it is not that it is necessary, or I have to take into account all the transformations to ... I have a question to which you might be able to answer. Indeed, in 1900, I don't remember if it was in the 10s or 11s, he proposes a theory. He proposes, indeed, to add as an invariance of space-time to normal isometry, a transformation of scales. And that's where he introduces the idea. What we answer to him, the idea is very real, and it worked well because the connection associated with this representation of electromagnetism, The theory of the gauge by replacing the transformation of the scale. If I take a fundamental length of an echelon, which is the one that gives the value. That is to say, such as any thermal transport. That is to say, the connection associated with this thing is trivial.

7:30 If we build a theory with a connection, it means that this problem of the transport of long-distance echelons is no longer a problem. In fact, if we have all the connections, maybe it will not be of great interest because we know that the problem... I would still be interested in... To show that we can formulate physics without metrics, just with the conforming structure. Just to finish with that, and I'm done, why was I making this parenthesis? It's because if we have two varieties, by different morphisms, under the reserve that they are topological, we can bring that back to... We want to compare two spacetimes, it comes back to compare, and we want to compare their causal structure. We're going to define a relation between causal relations. Causal mapping. Each one has a relation, and we're going to define the relation between the two.

10:00 V1. Yes, that's it. Basically, that's it. I'm looking for the... That's exactly it. I mean... No, but I'm looking in which direction, because... Yes, you're looking in both directions. Well, in short, you'll find it in the paper. If the fact that a vector of V2 is causal... It implies that the transformed, because it is necessary to find a difference between the two, implies that the transformed here is causal and of the same nature. And so, what is interesting is that we can have V1 without that, it is not a norm. So there are a few examples, the gap where this occurs, I have the impression that, I'm not sure it has a physical relevance, that's why I was a little demotivated to read this article. Technically, it exists, but to get to the last time, I still think there is a problem with language, because normally the idea is to build a relationship.

12:30 It's not the same thing to say that morphism is a relationship, it's really... It's simple, there's a lot of things in there, but it's really a matter of time before you know it. I just wanted to answer, indeed, but here, the morphism that we would like... Yes, it's not completely the same. No, because here, indeed, you have a first variety, that's it, that's it. You have a first variety, that's it, with, here, you have Cone, which is sent here, in the tangent space, to another Cone. But this variety has a structure that is specific to it and we ask that this cone finally contains the other. It would be necessary to walk in the same direction. I don't know if the notion of sauce in each point is strictly different. If there is a matrix. That is to say that ... So if a cone is contained in ... Well, it poses a problem to me, this paper. I have not worked hard, I do not see the relevance. And the examples that show are so distorted that I do not have the courage to go further. Here, one thing I'm really sure of is that the metric defines all the cones and that if two varieties have in each point, they are, for this relationship to be verified, it means that in fact there is a phi-diffeomorphism that goes there such that there exists one and obviously this one will not be the inverse of the other because if we ask that

15:00 I have a condition, there is an isomorphism, a difluomorphism such that morphisms which are not the opposite of each other, there is a means of causation, but in the end I am more interested in what is necessary and so what is interesting, what we know well since Confucius, is basically

17:30 And from this metric, we can define the causal structure, even interpret it as a partial order relationship between the events. What is interesting is the reverse approach. That is, if we give ourselves a set with a partial order, what do we manage to reconstruct? Do we manage to reconstruct the conforming structure, etc. etc. Obviously, a priori, if we give ourselves only a set with a partial order, But even if we give a partial order, do we already have a space-time, can we define a space-time, can we interpret the elements of this set as events in the space-time, will we be able to know something about its structure? These are the problems that are currently being examined, so simply a set with a partial order, some actions, under what conditions this set can start from? What would be interesting, in my opinion, is that we talked about events. Events constituted as points. What will happen as soon as we will want to do quantum physics, but especially when we will want to try to quantify it, is that the notion of points will disappear. To do everything we have done with causal conservation, so it reminds a little of what Crenn did in his something, which are the relations of inclusion.

20:00 I think there is a way to do that in a more intuitive way than to force the product that corresponds to it. There will be a way to axiomatize that in a more intuitive way, firstly, and secondly, it would be to do less explicitly the geometry expressed by abstracting space-time. And that will allow us at the end to find the space-time. I think that one thing to do is, in one way or another, to extend all this to a kind of notion of causality, not in its own way. Then we will want to expand it in such a way that it can describe this case as a particular case or a limit case, but that it is defined. Instead of making a theory of causal ensembles, as people do now, causal, it is a set, asking oneself if then, in a certain limit,

22:30 I'm going to do this exploration to see if it has already been published. We fix that in another session. I had a question. Who is my daughter? She is about to go to bed. Can you tell me in the chat? No, it's very simple. She is there. I'm waiting for her. She is there. I'm waiting for her. She is there. I'm waiting for her. I'm waiting for her. So, the Polar Code is the set of things that rely on the vectors of the world, isn't it? Yes, it is. Okay. It's the same thing. It's the same thing. It's the same thing. It's the same thing. It's the same thing. It's the same thing. It's the same thing. It's the same thing. It's the same thing. It's the same thing. It's the same thing. It's the same thing. It's the same thing. It's the same thing. It's the same thing. It's the same thing. It's the same thing. It's the same thing.

25:00 It's the same thing. It's the same thing. It's the same thing. Thank you for your attention. Thank you for your attention. Thank you for your attention.

27:30 Thank you for your attention. Thank you for your attention.