Quantum Quandaries (last part) / Category Theory & Physics

0:00 Theoretic stuff can be translated into set theoretic stuff or not. Yes, it can. So the way I think about it is that set theory is sort of like, what do we call it, assembler language, very basic operating system that's just enough to get your computer to run, but then you don't want to always be programming in that low level language. which you want to build up more abstract levels to reason on. And category theory is sort of like version 1.0 of your operating system. And the sort of the area, the thought was this. One of the reasons we kind of like set theory and feel happy with it is because it seems closer to sort of classical oppositional logic and we think of that to be, in some sense, representative to a rough degree of how we think about things and think about the world. and hands and always recomposing in nice boolean ways and in set theory you have a structure which sits happily with that and then when we move to some other you know if it's true to some other structure like that a category theoretic structure is more natural representing quantum mechanics and that might be so mathematically but there's a sense in which it's not helping us with our given that our thoughts are sort of like in the old style boolean structure we've still got the here's this stuff how would you have to think about it when you think Booleanly well yeah I certainly have things to say about that so the quick answer is stop thinking Booleanly but the longer answer is is that there's a beautiful relationship between category theory and logic where different categories have their own internal logic so the logic of set theory is Boolean logic and there's a way to make that you're able to generalize that concept to other categories. So you can go over to some other category, like the category of Hilbert spaces, and ask, what's the appropriate logic within that? I mean, you can still use your old Boolean logic outside that category to reason about it. I'm not telling you that you have to rewrite your brain, but the point is that the appropriate logic, in a certain sense, for dealing with Hilbert spaces is not going to be Boolean logic anymore. because it's not a Cartesian category. The concept of times and the concept of am changed their significance. So there's been a lot of work done on this

2:30 in the world of so-called topos theory. So a topos is a category in which you can do a lot of logic like set theory, but with one huge difference that the principle of excluded middle may break down. And that's been intensively investigated. What hasn't been so intensively investigated is categories like the category of Wilbert's spaces, which is not a topos, in which the appropriate logic seems to be more radically different than just what you may consider the excluded middle to be sufficiently radically different. But actually, in some ways, that's less of a drastic departure than the failure for and to distribute over or, which is what happens in the final logic. I have a question but I just want to first make a remark backing you up with what you just said but one thing you've shown with the paper is that geometry is a very natural way to think about these strange quantum properties so not only do our minds like and and or our minds like pipes so I can say that you have to find the right logic for thinking about quantum theory you have to learn about how it works for different categories are geometrically natural things like these they're not counterintuitive it's just a very different intuition that you use for these that you would use if you're thinking of this thing as a function my question was this business about the the tensor product and the failure of commutativity and the the no duplicating and cloning kind of thing well could you say something about what that has to do with quantum entanglement yes yes so so one feature of a yeah i meant one feature of a cartesian quote a cartesian category is that there's a certain precise sense in which you can know everything there is to know about a product of two objects by sort of knowing what's going on separately in each half that's a fairly vague statement but i I don't think I want to, but you can define a concept of a point in an object, and a point in a Cartesian product of objects turns out to be a point in the first one together with a point in the second one. But in non-Cartesian political categories, that doesn't work, and that's exactly attributable to the existence of these states that can't be .

5:00 in the object, the object being the open space. Yeah, that's right. The whole mechanism of points works quite differently for non-Cartesian categories, and it hasn't been developed as well. So I'd be a little reluctant to exactly say that, but certainly the failure of ability to understand the points of a product from points of each It's sort of a hallmark of non-Cartesian categories. I have a few other questions. This will reveal my ignorance, but if I'm a mad dog four-dimensionalist, which I'm not, and I take the whole ball of wax, the frozen formulas and everything, are you taking a particular view of GR when you're taking the object as spaces and the dwarfism as space-time? I'm certainly taking a particular view like that. Havory theory does tend to do that. It certainly does not at all say that any particular slicing of space-time into slices is being preferred here. But it is focusing your attention on a certain way of thinking about space-time, where you can think of space-time as going from some slice to another slice. So if you don't ever want to talk about that, then this whole formula is not really the right thing. It's purely topological. It's nothing to do with a constant metric or anything else. No, no, I could soup up my cobordisms to include a metric on it. It wouldn't need to be purely topological. It doesn't need to, but the notion of space is just boundary, lower dimensional boundary. Right. It could or could not have a metric, But I'm definitely sort of committing myself to studying space-time by thinking of if it's going from some past to some future. I don't care which one's the past. I don't care about you, Eric. So I want to challenge you a bit. It seems to me you can't possibly capture everything that's surprising about quantum mechanics in this way. Because if you admit that, for instance, the fact that you can do quantum computation, It's just supposing there is a speed up over classical computation. You say that's something that needs to be explained. Or you admit that there's something surprising about Bell's theorem. And I think this can't capture everything because there are subsets of quantum theory,

7:30 such as, for instance, if you take just the Gaussian states and the Gaussian operations. And I think they'll have all the characteristics you're looking for. They have a no-cloning theorem, they have teleportation, they have a no-deeding theorem. so that sort of theory will fall into your framework but that sort of theory doesn't allow for speed ups and complication doesn't have Bell's theorem I guess what I'd need to do before I can really answer that question is I'd have to learn a bit about what you're talking about and I would want to cook up the category where the only morphisms in it were the Gaussian processes and I'd want to understand what that category was like That would be very interesting to see. But yeah, so I'm not trying to solve this, right? Are there any general results on the logic of non-Cartesian monodal categories in the way say that truth objects are not 01? There's so far, there's a lot of stuff about category theory which is out there, which you can use to think about logic in the general monoidal category but it doesn't it hasn't really been assembled in a treatise somewhere partially because people haven't really been thinking about it so much from the point of view of logic in a category so I can hand you a big fat book on topos theory where you can learn all about how to do logic in a general topos and how to get your existential and universal quantifiers to do the right things in the topos and at that level yet here. I think there could be, but it will have to be quite different. So in fact, right now I'm reading a thesis on an attempt to come up with something called quantum topos theory, which is an attempt to do that. But I think it's going to take quite a while for people to really get a good theory. I don't think we're going to do it. Just a small follow-up to Tom's question. So I've never said, I don't know if it's true or not, you can always swap your object to be your arrows and pick what you had for them. An arrow, an object, and what you had for them. And if that's true, whether that would affect how you answered. I don't think that's true. No, I don't think that's true. What they may have been saying was that you can, something that's sort of

10:00 like that is true, is that you can always turn around the direction of your arrows. So you can just redefine who's the source and who's the target. You get something called the opposite category. No, I don't think you can do that. Maybe you're thinking of there are certain categories where I think the set of the distance between two objects is itself a bunch of interesting events. Yeah, those are called closed categories. That's a very interesting one. Yeah, that's a very interesting one. Closed exponentials? Yeah, yeah. So there's a set of maps between two sets. There's a vector space of linear maps between two vector spaces. so there, instead of just having a set of arrows you have an object of arrows between two objects, and then it's called closed because now it should keep living in this world in that category you had a question it was essentially quantum topos in Cobodism, your morphism are all inversible, no? no, they're not for example, that one's not so they're reversible in a certain way, there's an upside down version of this, which goes the other way. And that's very important. But it's not an inverse, in the sense that if I compose this one with an upside down one, it's not equal to a cylinder, which would communicate as equal to two cylinders. So that wouldn't be the idea. So there's another feature that I didn't talk about, which is both these categories have a notion of duals of warpisms, different than the duals of objects I mentioned. Where here, it's the adjoint of a linear operator going back between Hilbert's faces, which is not And here it's turning this thing upside down. So that's another interesting feature that they share. The category set just doesn't have. If you have a function, there's no way to turn it around in general, unless it's in vertical. Yeah, so that's another clue that there's something for it now. I see no more questions. So let's thank... Thank you. We need to give it five seconds. And a great time to the organizer for inviting me in this nice environment. What I will give you is very basic, is more or less the view of space-time by a physicist, and the present view of space-time using different mathematics tools

12:30 which may indicate some directions to generalize the present physics. And I think there is no need to convince you that we need more general physics, new physics, and the question is, there is no need to convince you also that we need more mathematics and which mathematics, which new geometry or algebra for these new physics. And of course, this should result from an analysis of what are the present mathematical structures which are linked to space-time. So why a new physics? There are some conceptual motivations that you know. We have presently two frameworks, which are relativity and quantum physics, which are, each of us, in some sense, incomplete, and we have the indication this morning. They use different mathematics, but maybe this is not such a big difference, because we will see that there are some common mathematics which can be used to describe both. Of course, the standard model is not well explained, and the big difference is that space-time is not quantized like matter. So, presently, we have classical matter inside quantum space-time, and gravitation is not quantized like other interactions. This is a resume on this table, but you know very well that, so I think I will skip that. But there are also factual motivations. I mean in observations, in experiments, you have some unexplained facts, and probably this should be a hint for research. And if you try to find new physics, you should combine, of course, new conceptual principles, new mathematics, and try to think about these factual motivations. Some come from astrophysics, dark matter and dark energy, you know very well that. If not, I can comment at the end. The problem of the Pioneer Probe, which seems to suffer a constant acceleration, which is not explained. The question of black holes, this is not very factual, maybe they raise very interesting questions linked to the nature of space, time, quantum physics, information, and so on.

15:00 and you have the problem of antimatter and many other problems which arise in primordial cosmology or if you prefer, if you try to know what are the initial conditions of the universe and this is also the questions of the time and time around. So we want more or less to understand these things and this is why we need new physics And I think there is a fundamental question, which has already been debated here and will certainly be, that we have a message in general relativity that gravitational interaction is, in some sense, the geometry of spacetime. So the question is, is this message fundamental or not? So if yes, this means that we should keep it from a new theory. Why are the other fields, electromagnetic or strong interactions, not geometrical in the same sense? And if we want to quantize gravity, this implies that we must quantize geometry, which is a difficult task, as you know, especially concerning the problem of time. answer no to this question, and consider that gravitation is a field like the other one. It's only a coincidence that we have a theory which describes it geometrically. And in this case, you would think that nothing is really geometrical, but then why gravitation seems to be? And something which calculates maybe a third answer, but maybe also a synthesis between both, view, which is also background independence, and I will comment on that later, so I will go ahead. I will also comment on relationism. Of course, we have met this question of quantization of gravity, and the first question is, is it necessary? It seems that yes, and there are many motivations for that, you know that perfectly. The question is, if you want to To quantize gravity, in some sense you have to quantize geometry, or maybe to quantize algebra, which is linked to geometry. What do you have really to quantize? And you know that there are many answer to these questions, and I will try to present some issues to define what should be quantified.

17:30 In any case, if you want to search for a new theory, you have to un-learn the mathematics. many ways to do that. You can, for instance, add new dimensions, and you know that there are many propositions like that. The idea is that in the present physics, you have some kind of internal dimensions, for instance, the phase in electromagnetism, and you want these internal dimensions become of the same nature as the geometrical one. But as you know, so this is not a success in the present theories of this type. You can also consider new symmetries, and you know that symmetries in mathematics are represented by groups, Lie groups, and Lie groups are also Lie algebra, so you will be led to group theory and algebra, And the symmetry are, we have already some symmetries, like the diffeomorphism group, the Lorentz symmetry, the spin symmetry, the conformal symmetry, and I will make some comment about that, and of course the supersymmetry. You can also keep the same framework and consider additional structure, like for instance, torsion. For the moment, we describe space-time by Riemannian manifold, which has a connection with curvature, but no torsion. And there have been many theories proposing that we should also include torsion, and torsion is linked to spin. We can also add new objects and for the moment there are many propositions just to add new objects, not geometrical but some kind of material objects just to explain some problems like some color fields to explain inflation or to explain the acceleration of the universe. or vector fields like the proposal by Beckenstein to explain the rotation curves of the galaxies because most often this is a very adult proposition but maybe it's an indication that we have here some kind of effective theory of a more deep theory you can also change the structure of space-time just replacing the notion of manifold by something different like in the non-commodative geometry. You can change kinematics, and of course this is linked to the previous, but instead of changing space-time, which is a manifold, you will change the structure in some sense, you will work on the tangent bundle of the manifold.

20:00 And for instance, you have what is called double spatial relativity, and this theory has not a very well, very firm basements, but the idea is to change the structure of velocity or moment impulsion transformation. You can also use complex physics, you can quantize gravity, you can quantize geometry itself, and I will go into some of the details of that, not all the things. So, what is our present view of space-time, which in any case is the departure of any enlargement? So space-time, for the moment, according to relativity, is a Riemannian manifold, modulo-deofomorphic, and with some constraints like orientability. What is a pseudo-Riemannian manifold? This is a differential manifold plus a metric tensor. And as we will see, to this metric tensor is associated a connection, so we will have these tools at our disposal. is a topological manifold plus differential structure and a topological manifold is simply a set of points with some structure which defines the neighbor relations at the bottom so first I will speak about differential manifolds and very quickly I just want to show that if you have a differential manifold it carries with it a lot of structure which can be expressed by natural bundles and then to a differential manifold you can add some structure like metric and connections for instance that's what you do in general relativity to express the gravitation but you can use different definitions of these objects or relate them in a different way or also introduce But the formalism of fiber bundles, which is already present here, because there are many fiber bundles associated to differential manifolds, is also very relevant for more general geometry.

22:30 So a manifold is a set of points with topology, and I will speak about differential manifolds. So in addition to this topology, you have a differential structure. topology allows to define the continuity of function and then the differential structure allows to define the to differentiate the functions and what is the derivation of a function? this is a vector field this is defined as an operator acting on the algebra of functions with some rules And what is interesting is that as soon as you have a differential structure, you have a natural structure which is that of vector fields. And if you take a vector field, which is a derivation, and you make it act at one point, one point of the manifold, you simply have a vector at this point, a vector at the point M, and at a given point, all the vectors define the tangent space at this point. So this is very basic algebra, but here you have a vector space, and since you have a vector space which is simple algebra, you can use all the tools of algebra, which are duality, tensor product, and so on, and this is exactly what we will do. First, what is interesting is that the radius of all the tangent spaces, when you vary the point of the manifold, form a nice structure, which is a fiber-bundle structure, it is called a tangent bundle. So this is a fiber bundle, which has a basis, this is a manifold, which has a basis, which here is space-time, and there is a projection of this manifold to the basis. This is the definition of fiber bundles and this differs from the usual Cartesian product because here you have the fiber and you have the basis. If this were a Cartesian product, you would have two projections of the manifold, one over the basis and one over the fiber. But here you have only one projection over the basis, which is the space-time in this case, and this is what defines the fiber structure, the fiber-bundle structure, which plays a very important role here.

25:00 When you have a fiber bundle, you define a section. A section is just the choice of one element over each point of the basis of the manifold. For instance, a function is a section of some basis because at each point of the manifold, the function takes a value, and the value is just a choice in the fiber, which in this case is just a set of real numbers. So, in general, we will consider vector bundles. Vector bundles are fiber bundles whose sections are vector fields. And, for instance, if you have the tangent bundle, a section is the choice of a vector at each point, so a section is exactly a vector field. And very generally, in physics, a field will be a section of some fiber bundle with space-time as basis. And then you can define the space of all sections of your fiber bundles. In this case, this is a space of vector fields, and this has also vector space structure and model structure. This is a space of section of the tangent bundle. This is a canonical way to transform a vector space into a vector bundle just by insisting that you have a copy of this vector space over each point of the metaphor. Then, sorry, I don't want to have any details, but just are there additional conditions on how many vectors at each point? Yes, of course, of course. What is interesting in a fiber bundle is that in each local part, you can have what is called a trivialization of the bundle, and locally this looks as a Cartesian project. So locally you can make projection on the basis and on all the fiber, but it is not possible in general to extend this projection to the wall-made form. In the case it is possible to have a trivial bundle which is nothing but a partition project.

27:30 So this is a richer structure. Since at each point you have a vector space, which is a tangent space at this point, you can choose a vector basis of this tangent space, which is called a frame at this point, and this is in fact four basis vectors, so you have a new vector space, a new space which is the space of all the frames at this point, and if you choose one frame at each point of the manifold, you have some field that you can call a frame field, but it's called a moving frame, or repaire mobile in French, because this is a word by Carton, who insisted first on that. And this is a choice which, to each point of the manifold, associates a vector basis of the tangent space here. And, of course, you will define a frame bundle, which is just a region of all the sets of the frames and each point for all points of the meaningful. And this is a very important object, too, because if you choose one point and you decide to change the frame, How do you change the frame? You change linearly the four vectors, so you have a group action, and this is simply the linear group in four dimensions in this case. So a change of frame at one point is an element of the linear group, and this linear group is fundamental for manifolds because it acts here as you see on vectors, and it will act also on a tensor and everything. And then you have the frame bundle, which is defined like that, and project onto the basis, which is the space-time. And considering this action, this means that this is a principal bundle. The principal bundle is a bundle where each fiber is more or less identical to the group which acts on the fiber itself. It's the group itself minus the definition of an element being the unity. But it's more or less a fiber bundle whose each fiber is exactly the group. And so we have this group which acts in a very important way.

30:00 So we have three vision of vector fields. They are defined as operators acting on functions. They are also defined as sections of tangent models, and also what is very important, especially because we will be interested in covariance and differomorphism, according to the theory of Lie groups, we can consider that each vector field is the generator of a differomorphism. I will define later what is a differomorphism. This is a map of the manifold itself, which presents a differential structure. So this allows us to consider the algebra of all vector fields as the lead algebra of the group of deformer fields and this is very important for our concept And in fact, we have defined functions over differential metaphors we have defined also the tangent bundle whose sections are vector fields and at each point the vector field becomes a vector since we have usual algebra, we can use duality and we know that the dual of a vector space is a dual vector space so we have linear forms acting on vectors so from the tangent space we can define the quotagen space which is a dual of one forms acting on vectors and, of course, we have the cotangent bundle which is, in some sense, dual to the tangent bundle and then a section of the cotangent bundle is a one form and a one form acts linearly on the vector field to give a function. So this arrow is duality. Of course, from the tangent space on the tangent bundle and the vectors we have defined the frame bundle like I mentioned before. We can also define a coframe bundle, which is just a bundle of passes from one forms. And, starting from these two spaces, we can make various tensor products. And this is the way we define tensors of the many forms. And then, since you have defined tensor by usual tensor product, this is just a generalization to the fiber bundles of the tensor product of vector spaces.

32:30 operators which act on tensors and you'll define them later their whole connections and what is interesting you can also make not exactly tensor projects but anti-symmetrized tensor projects and this gives you special tensors which are anti-symmetric tensors and they are called multi-vectors or multi-forms and what is interesting is that if you consider the anti-symmetrized tensor product called the wedge product, these form algebras which are called the exterior algebra and the project in this algebra is the wedge project. Ok, this is what is important, that all the structures, and of course there are more, are canonically linked to any differential manifold. So, differential manifold is at present the basis of our definitions of space or space-time, but each time you speak about differential manifolds, you have all these tools at your disposal. And what we will see is that if we want to add additional structure, or if we want to modify the structure, we are free to choose to do that in any of these natural bundles associated to the manifold. Precisely, what are additional structure? There are mainly two interesting structures for physics, which are matrix and connections. Metric is just adding a tool which allows to evaluate the lengths of the durations in space-time and the angles which in space-time are velocities. A connection is just a tool which allows you to derivate the vectors and also the tensors. And in some sense, as I will try to show you, it's more fundamental than the metric. And this is why some versions of quantum gravity try to quantize the connection rather than the metric. If you have a differential manifolds, metric and connections are completely independent structure. You can have a connection without metric, you can have a metric without a connection, but they can be related because if you define a metric this metric will select

35:00 a peculiar connection in the set of all possible connections it is called the Levy-Civita connection and in fact when you have a metric with the Levy-Civita connection this is what defines really the pseudo-Riemannian structure and the connection is very important because this is the connection itself, which define, for instance, what are the geodesics and how the various fields can propagate. So, general relativity is expressed as a pseudo-Riemannian manifold. Pseudo-Riemannian means that the metric has a particular signature, which is 1, minus 1, minus 1, minus 1, which considers the difference between time, light, and space-side dimensions. So the gravitation is expressed by the metric or equivalently by the Levitch-Givita connection, which is associated to it. And in this case, this connection is also called the Riemannian connection. but you can build some other theories by considering a metric and a connection which is not the Levy-Chivita connections by considering the metric and the connection as independent structures so you know what is a metric, it's just a tensor which acts on vector fields to give a function which is called scalar product And according to general relativity, this metric tensor is the gravitational potential. And this is a general rule, as I told, that a physical field is a section of some bundle. Here, this is the bundle of a symmetrized symmetric tensor of type 2-0. Means, they act linearly, pre-linearly on vectors. And, of course, there are also equivalent ways to define the metric, for instance, you can define the metric as being a function on the tangent model or on the tangent model, but this is not very important here. So, I just want to make small digressions about covariance and differomorphism because I think this could be a major topic of discussion in here. So, I want to give my version of that, and this can be used as a basis for discussions.

37:30 So, what is a differomorphism? A differomorphism is a map of a manifold to another manifold, differential manifold to another one. Here we will consider a differomorphism of a manifold, seeing the space-time, to itself. So this is a differential map which is also invertible and the inverse is also differential. So it preserves the differential structure and to a manifold is associated the group of all differomorphism. If you transform a manifold by a differomorphism, you obtain a manifold which, in some sense, is the same differential manifold. They are completely isomorphic. So, if you want to distinguish them, you have to be very subtle. In fact, the categories allow to distinguish between identity and isomorphism. category, theory, is a good tool here to make the distinction, and this distinction is completely pertinent if you want to understand covariance. So, it's very easy to show that if you have a different office, it induces a change of coordinates also. In other words, the transformed manifold can be seen as the original manifold written in a different system of coordinates. And this is what is called the passive view of the differomorphism, of the same differomorphism. And you can consider the change of coordinates as gauge transformations. So with this point of view, a differomorphism can be seen as a gauge transformation. And also, as I told, differomorphism are generated by vector fields. If you want to consider an infinitesimal differomorphism, this is exactly what is called a vector field. is the Lie algebra of this group, of the group of diffeomorphisms. But you have many objects like vectors, tensors, forms, which are associated to the differential structure and any diffeomorphism maps this object in a very natural way. This is called the pullback or the pushover, the push forwards, sorry, push forwards.

40:00 And for instance, if you have a function, the transform function evaluated at some point, which is a transform of the point M, has just the value of this value. So this is a very simple definition, and there are canonical ways to extend the pullback of push power to any type of vectors, and so on, and so on. And so if you have a manifold, differential manifold, with some tensor fields on that, one can be the metric, another can be an electromagnetic field, or what you want. And if you transform the metric by the differomorphism, and also if you transform all the tensor fields canonically, the two are completely indistinguishable. There is no way to make a distinction between the original objects and the transformed objects. So, in particular, if you have a metric, so if your manifold is, in addition, a Riemannian manifold, then you can also transform by a given diffeomorphism, and then you will have a new manifold, but also, in all possible concerns, they are the same Riemannian manifolds. So, this is called covariance. Covariance is diffeuomorphism invariance, but in fact this is more or less the definition of geometry. What is usually meant by geometry has to be diffeuomorphism invariant. And if you want to call some physical theory, geometrical theory, it has to be ifeomorphism. This is true, of course, of general relativity. But if you consider spatial relativity, what is spatial relativity? It is based on a manifold, which is the R4, with a specific metric, which is the Nikoski metric. but the spacetime of general relativity is not R4 with the Minkowski metric but this is the equivalence class of all base manifolds under the diphtheromophage which means that you can use any map you want to map a Minkowski spacetime but this is also true in geography For instance, the surface of the Earth is a manifold, but you can represent it by any map.

42:30 And you can resume deformities in variance by this sentence. The map is not the territory. The real geometry of the surface of the Earth, for instance, does not depend on the map you use to analyze it. So, if you have a manifold with some fields including matrix and its transform, they describe the same geometry. So, this implies that the only things which have some reality are the relation between these things. So, in this sense, the general relativity is relational, and geography also. If I want the geography to know what is my position, I may use coordinates, but I can also define my position with respect to other things. And other things in general activity would be other fields. So, even my position is something which is defined in relation with other fields, and basically the metric field. The gravitation, the metric field, has a field linked to spacetime, but you can also forget that, and you say, this is a field, and if I want to express my position of velocity, I will use this field as reference. And, in fact, all processes in general relativity can be expressed as relation between these fields. And this is, for instance, expressed by the old argument of Einstein. But you can resume all of that by saying that in relativity, real objects are covariant objects. So, what is a spacetime? manifold with metric field and many other fields which can be electromagnetic or matter. So space-time is the equivalence class of this Riemannian manifold with fields under the thermomorphism. The solution of general relativity is such a space-time that we can call a background, but the theory is background-independent

45:00 because when you write the fundamental equation of ancient equations in general relativity, there is absolutely no preferred background. There is no manifold or class of manifold which is preferred. But any solution, of course, defines the background. If you consider spatial relativity, this is different. Of course, it is covariate, because any geolatical theory must be covariate. But the spacetime is completely defined. Minkowski space-time and since it is covariant this is the equivalent class of Minkowski space-time but this is a background so spatial relativity is not background independent and if you want to have a perturbative view of general relativity this means that you will consider fluctuations around the privileged states that you will call the fundamental state or vacuum if you want usually this is taken at Mikoski's space-time, but you can make other choices like De Sitter's space-time but since you have to make the choice of a fundamental state, space-time, perturbative general relativity is not a background in the planet. This has a very interesting consequence, I will go very quickly, but this is very interesting. general activity in some sense is a dynamical theory for the metric so the interesting objects are the possible variation of the metric that I write in this way but a variation of the metric is real, is physical if this is not a simple gauge transformation so if it is not the result of a diffeomorphism but we know what is the form of a variation which is the result of a diffeomorphism. Because a diffeomorphism is expressed like a vector field. So a variation of the metric which is due to a diffeomorphism is of this form. This is the lead derivative. So any variation of the metric which is of this form is not physical. It's a pure geogage. so it must have absolutely no effect and if you want to have an action for general relativity this constrains the action

47:30 and this action is constrained to be it is covariant of course to be an integral of a space-time of the volume form multiplied by some tensor here