Weyl, Cartan & Méthode du Repère Mobile

0:00 Let's look a little bit more closely at Cartan's geometry of generalized space. Well, geometry is based on the notion of a group. This is following a climb. So you attach a homogenous, or it's often called a climb, space to each point of the manifold, the base manifold, as it will come out in the fiber formulation. the space in which G acts freely and transitively is defined by a frame in P geometry is viewed as the studies of properties of M expressible in terms of these attaching maps independent of the choice of the employed attaching maps this is another part of our story this was only clarified by Charles Erzmann in 1950 Second principle of foundation of Carton's Geometry of Generalized Spaces is the method of Rittgen-Württemberg view, a method of studying sub-manifolds within a given manifold M. And here we see that the method of moving frames and the exterior differential calculus fit together like hand and glove. Finally, the third component is the generalized theory of connections, which is just to connect the different homogenous spaces introduced under one and two. So here's just a brief pictorial look, visual aid, to present the notion of a moving frame, which is highly intuitive. So to each point of the manifold M, You attach, in this case of dimension n, attach n linearly independent basis vectors. And then you've got something called a field of frames with the one-forms, the dual of the field of frames, so that at any point p, you've got the Koenig or delta between the one-form acting on the frame. Then in a moving frame, right, in an open subset y, of the manifold M of each point P is N plus long twofold,

2:30 just for notation. Here's the fundamental theorem of the method of moving frames, according to a later work of Carton, 1937. There exists a bi-unique correspondence between two given f and f prime of the same number of parameters, the necessary and sufficient condition of which is a fixed displacement, bringing into coincidence each frame f with the corresponding frame f prime, such that the correspondence in question establishes the inequality one by one of the relative components of these one-forms of the two families of frames. In consequence, by attaching to each point of a manifold m with a given number of dimensions, intrinsically determined frame, one can determine whether two manifolds, M and M prime, are equivalent. Or in other words, by knowing those small quantities in each of them that determine a small displacement, one knows their differential geometric properties. What's the call for the question? Is displacement just in a perfect match? Yes. A connection. well the problem lies with Cartan's treatment of connections because Cartan assumes a correspondence between the transformation of frames pertaining to infinitesimally close points P and P' in the manifold M and the infinitesimally small displacements under the action of the group G in the attached spaces these fine spaces at each point P of M exactly is the nature of this correspondence. Well, according to the great differential geometer S.S. Chern, who was a Cartan student, it was Elie Cartan who recognized that this notion of connection admits an important generalization, that the spaces for which the infinitesimal motion is the tangent spaces of a Riemannian manifold and that the group that operates in the space plays a dominant role. Well, that's just stating the ambiguity once again.

5:00 Associate is the generalization of the notion of an affine connection, and then we'll see exactly what the ambiguity is. Associate with each point P of M, an affine space, to which P also belongs. And let an orthonormal plane be located at D. Then M has an affine connection in Cartan's sense if there's a law relating the affine spaces at two infinitesimally close points, P and D. What Cartan has to say about this in his famous paper that it called the first part in 1923, is quite arbitrary. It only has to enable us to say that such and such a point in the affine space associated with P corresponds to such and such a point in the affine space associated with P prime. And that such and such a vector in the first space is parallel or equal to such and such a vector in the second. The assumption of law of continuity that coordinates of P prime in the affine space originate in P are infinitesimally small will enable us to say attached to P is, in a certain sense, tangential to him. There's the ambiguity about what is tangent, and what a tangent space is in Cartan's treatment of connections. So, in developing his theory of generalized spaces by the method of moving frames, Cartan called tangent space what is now known as fiber, while the base space has a tangent space from its differential destruction. So you can take two points of view in the classical theory of athorn connections. With Loewe, Chivita, and Bile, you can consider each tangent space as a vector space. And then an affine connection is a law of parallel displacement of vectors along curves. That's also how the bio-institute length connection is in the tangent space. In Cartan, each tangent space of M is an affine space on which the affine group acts transmitting.

7:30 An affine connection is a law of development of tangent spaces, referred to moving frames along curves. So if you're looking at the bundle formalism of a four-dimensional space-time manifold, you're looking at the linear frame bundle, about which Mark told us it's today in some detail. It's the space of all linear frames over M. Its projection maps a linear frame, or an ordered basis, into each point P of M. Its typical fiber and structured group are both the general linear group. the frame bundle is the principal bundle associated with the tangent bundle onto the manifold. So a connection in the sense of levacevita is a connection in the frame bundle. Now, following cartan, consider each tangent space as an affine space at P. Then you enlarge the frame bundle whose structure group is the affine transformation group acting on the fibers of the tangent line and containing the general literary group as a subgroup and the affine group is taken as the structure group of the tangent so a connection in the sense of cartin is a connection of the bundle of affine frames over hem and what's the relationship between these two connections there's a natural homomorphism from the frame bundle onto the affine bundle that induces a mapping of the connection on the affine bundle into an R4 value one form on the frame bundle which decomposes into two parts Vita connection in the second part of the one form, theta, that provides a unique way of projecting a vector in the tangent space to a point x in the frame bundle, down onto a vector in the tangent space to P and M. This theta is called the solder form on the frame bundle, such as solders the frame bundle to the manifold. That was Erisman's contribution

10:00 So the presence of a solar form means that the principal bundle of gravitation, which is the frame bundle, has more structure than any other general principal bundles, which play a role in Yang-Mills type gauge theory. And the covariate exterior differential of theta is the affine torsion of the connection, soon to be vanishing in Einstein theory and non-vanishing in so-called. Einstein-Kartan theory. This is the paper, the famous paper of Charles Peresman, probably the most famous paper of Charles Peresman, a great differential geographer from Alsace. And what he does in this paper is to clarify the relation of parallel transport of a vector from t to t prime and m to group actions in the separate spaces of kartan connections, g over h, just a little bit more of technical stuff and you'll see a nice picture which will make it all perfectly clear let pi be the projection of the principal bundle onto mg the structure group so So P is the total space of the bundle, whose fibers, pi inverse, are obtained by the action of all the members of G. So you have the tangent space to a point U in P at any point, such that that tangent space at point U has two parts, a horizontal subspace and a vertical subspace. then a connection on a principal bundle is an assignment to each point u, member of the bundle, fiber, of a horizontal subspace of the tangent space such that the vectors in the horizontal space are pushed forward by the right action of the group that's just what that was saying and here's a nice picture So you start with a principal bundle, P over M, and you start with a, then you add a parameterized

12:30 curve in M with an initial point. And you pick an initial point, U over P, in P, lying over the initial point of the parameterized curve. And then you just lift the curve into in that horizontal subspace. So a connection on the principal governs parallel transport of things along smooth curves lying in the base manifold. Did we start at 12? All right, so this is really just this is not essential, but you might wonder what all these one-forms are doing. Why do we have to work with one-forms when we're doing differential geometry in the cartons? Well, the reason becomes clear in the fiber formalism of Riemannian geometry. So for each orthonormal frame of the metric on M, there's uniquely associated a set of differential forms open subsets Y, such that these relations hold. These are called the structure equations of the monumon geometry. Those don't mean that much to you until you look and you see that the exact similar structure appears in a bundled space B of this dimension by a projection B into M, and a set of these forms on B. and these are linearly independent at each point B so to each part of the normal frame of the metric on M there's a cross section map from the open subset up into B whose pullback map sigma star here gives exactly those relations that natural isomorphism between the two bundles or at least the base manifold would be the associated bundle. But this pullback map is not defined for vector fields. The use of these forms then enables this bundle isomorphism theorem

15:00 to reach diffeomorphism in M or from M to M prime, carrying one Riemannian method into another. there's an associated dipheomorphism in the bundle, associated bundle, carried a set of forms defined in B into those defined in B-time. All right, what's all this got to do with Weill? Well, in 1925, Weill complained about Carton's treatment of differential geometry. he said the climb space he called it sigma so p we've been calling it g over h is not a pure product of the manifold m as is the tangent space it requires grounds of determination still lying outside look how vague this language is he's not talking about affine bundles or even here All that Weill is recognizing here is that this fine space is not determined in the manifold. So long as that is the case, the Cartan schema as a single manifold, theory of a single manifold, is not a possibility, and also not as a world geometric foundation for physics. This comes from a work that Weill wrote in the early 20s, finished in 1925, It was supposed to appear in Russian, and some university of Kazan stands to sponsor a thing for Robicheski. But it never appeared and was published only in 1988. Well, in 1929, Blatt comes back to his criticism in a paper, one of the first he writes in English. he's at Princeton now visiting of course he would go back to Germany and replace Hilbert and then come back to Princeton but in 1929 he's visiting in Princeton and the purpose of this paper is to compare the Princeton school of differential geometry the geometry of paths of Beblin and Eisenhardt and Tracy Thomas Cartan and also to geometry. But in 1949, in this paper, he notes that the tangent plane in Cartan's sense

17:30 is not uniquely determined by M. And he says, so long as this is not accomplished, we cannot say that Cartan's theory deals only with the manifold M. Conversely, the tangent plane in P, in the ordinary sense, is a linear manifold of line elements in P, a centered affine space. its group gene is not a matter of convention. This has always appeared to me to be an efficient view of the theory. His last criticism of Cartan is reviewing Cartan's famous books on theory of groups and who apply to geometry and the method of moving frames. And Bob reports here that he did not quite understand Cartan's general procedure, though the examples are clear, can you conclude, I must admit, I found the book, like most of Cartan's paper, hard reading. Well, some of you who know some of the history of gauge theory, you might say, well, didn't Viall adopt Cartan's method of moving frames when he formulates a direct equation in the set of general relativity in this famous where we, this is the proper origin of Gage Theories, in 1929. That's what they're playing with Lachlan-Orappity in The Donning of the Gage Theory, where there's an English translation of this paper. well no you define the spinner spaces of the Dirac equation at each point B of M by what Weill calls an orthonormal axoncoids, tetrad in a rapid translation but these spaces are point by point deeply embedded in the tangent space of P and this is Weill's language zero order, first order, et cetera, et cetera, et cetera. These are not moving frames. The Lorentz group, in which the spinner representations of the Dirac equation, are defined by these axon-kreuz, that they live embedded in the tangent space.

20:00 So, via contra cartin, summing up Weil's original opposition to Cartan's treatment of differential geometry stems from his perception that Cartan's generalization of Levechevita parallelism whereby the spaces on which infinitesimal motions are defined need not be the tangent spaces of a Riemannian manifold and in which the group operating in the space plays the dominant role no longer had a base in And that's that problem of constitution with these primitive, sorry, pure infinitesimal geometry, where you have this visualizable evidence of primitive elements and linear operations. Hence, Cartan's generalized spaces could not be constitutive of obvious appeal. But in short, they're not suitable as world geometries. You can't build them up from evidence in the space of intuition, basically, the tangent space of the matter. After 1930, of course, that paper, Geometry and Physics, that I quoted from earlier, Bayer had some pretty serious criticisms to make of the entire geometrization program scientist and Carol was a birch at the time. So he's given off by 1930 on generalization programs. And his objections center more now on obscurities in the Tartan program that we've seen later clarified by Charles Kersenhoff and Schoenhoff. say that by 1949 Bayer had made a full penning. He says, this is a paper that he wrote for Einstein's 70th birthday celebration. This is really quite remarkable. It is not advisable to bind the frame of reference signals of P to the coordinates covering the neighborhood of P and M. In this respect, the old treatment of the affine mechanical manifolds is in C. Of course, that's just

22:30 Vile's position between 1925 to 1930. Nobody can predict what sort of geometric structures may be thought of, and hence it would be foolish to claim that our pattern of associating Klein spaces and the displacement is universal. Whatever the structure, it must in some arithmetic way, relative to a frame of reference F, whether that frame consists of a coordinate system of the manifold M, or a coordinate system of M plus admissible frames of reference for each associated inclined space, or is something even more complicated. So Weill has learned his lesson to see the fruitfulness of the Cartan approach, and in particular, he was impressed by a proof of churn analysts of mathematics in 1944, where Charing explicitly uses the Bible-Bungel pharmacy to give a proof of the Gauss-Bernet theorem for an even-dimensional two-line manifold. It's a classic proof in modern differential geometry. Charing refers to it over and over again in his writings, and many people do it as well. A much more prestigious proof. by way of conclusion. Recall that this world geometric objection about Watt and Bile, which in Cartan's treatment is that the Klein spaces had no clear relation to the underlying manifold. But if you think about it, is the moving frame approach of Cartan, which is essentially, as we saw in his writing on the fundamental theorem of the method of moving frames, Bringing a frame at f at p into coincidence with a corresponding frame f prime at p prime, infinitesimally close by, any less rooted in constructive geometric intuition than Weyer's paradigm notion of parallel displacement of a vector? It seems not. It's just happening in a different space, this affine space, not the tangent space of the quantum mechanics. Pusserl, I think, allows this. Moreover, the Cartan approach follows up upon Poincaré's conception of geometry, where

25:00 the notion of room is an a priori concept of room. and famously in this paper much of which appears later in Science and Hypothesis in some of the texts Potentially we have in our mind a certain number of models of groups and experience alone assists us in discovering which of these best pertains to the reality Well, hypothesis Weyl's world geometric objection to Cartana derives from a dying conception that is generally co-variant and gauging-variant world metric for this, containing also electromagnetism, that this world metric is the fundamental constitutive element of physical reality. This is a famous quote from the first paper, actually it's the second paper of the program, in 1980. He says everything real or actual that transpires in the world is a manifestation of the world metric, which now of course includes this scale freedom which gives electromagnetism. Physical concepts are none other than those of geometry. Whereas, as we've seen, Karpin held that the fundamental hypothesis of einstein's theory is not as many people have believed that it is possible to formulate the laws of physics in any arbitrary coordinates that's needed in simple technology but that in any sufficiently small region of space-time the laws of classical physics that's expressed in his specialty are true in the first approximation. And you can see why Cartan would say that, knowing something about each geometry of generalized spaces. In sum, Weyot has essentially adopted a background-independent approach to field physics, whereas Carton's approach more closely approximates that of quantum field theory and super-state theory so the GR is recovered in the weak field approximation as a perturbation leave aside Einstein Carton theory but this is why

27:30 to answer my question that I posed at the beginning this is why contemporary geometrization programs, in particular stem not from bile, but from kaitanya. But it leaves us with questions. Does gravitation really exist as a gauge theory, that is, a connection on a principal bundle? And can such a theory be background independent? It does as a connection on a principal bundle. Thank you. Thank you very much. For the meantime. OK, so, well, almost simultaneous, so yeah. OK, Matt. In your last transparency, so you use background, background, independent. do you mean in the same sense that co-variant or in the sense that it will be independent with respect to a choice of a class of co-variant space-time I mean to say that in the sense that the world metric for Weill is the only dynamical object it's the only object from which to constitute the rest of physics Yes, yes, of course, of course. So it seems to me that in this meaning, all the theory of captain is co-variant, in a sense, because all objects are different. Yes, that's the point of using the differential forms. Sure. Yeah. Okay. So my question isn't really philosophical, just a little more technical. I'm wondering if you could give us an example that would help develop our intuition about what an affluent frame is and what it is you gain, maybe this sort of loveliness of geometry What is your gain by going from just having a tangent from space to having space? Well, you gain, first of all, you can employ immediately the exterior differential calculus

30:00 with the moving frames so that everything that you do is, as Mark said, is going to be fully covarian. And it gives you also, I mean, this was the point, it gives you also the, these are the Klein spaces with respect to these frames attached to each point of a manifold so that you study the manifold M by what's going on in moving both frame-to-frame and point-to-point in each of these, these open subsets. I think there is a question about it. Yeah, please. So my question is in the aspect of your thought when you were considering Byron's original objection, and the extent to which that ties into this sort of commitment to the tan of space being the space of intuition and the output. And it seems to me that that could be pronounced as much as possible. So if you think that the tan of space is somehow very directly linked to the space of intuition, It's a space situation for a subject that can be located at that place in the manifold. Why should it follow that your position also involve these more generalised spaces? I mean, you're always going to have a plan of space. That just comes with a manifold. so Miles' objection seems to be surely he's right in this stuff all of this machine in the cartons is in some sense being added on and that seems to be like a completely different intuition to subjective stuff and it seems like Miles' objection this commitment And that seems independent to the connection to, I think, the right kind of space with the state of the nation. Well, that may well be independent.

32:30 You may be right about that. I was trying, though, to there are unclarities, clearly, in Cartan's treatment that violence focuses on. But what I think is an additional to the unclarities Friedman is this line here about the real geometric foundation of physics. And it's quite clear that for Bayle, such a foundation can only be doped up in the tangent space. Because that is the space where the ish centrum recognizes the superposition of two vectors or the congruence of two figures. Thank you very much. I just wanted to add a remark to your question, what you gave me back to those terms. If you want to describe spinners on a manifold, direct equation, you have to use the Carton-forminism, and you use it all the day, all the time, for, you know, to describe supergravity, add fermions, and describe the interaction of fermions, which is a gravitational switch, and then it's just unavoidable, there's no operating time on it. So in general, when you say it, it allows you to add better. Yes, because you also not only have the time and space, but you can add other kinds of models, like screen models. Yeah, just to finish the question, what's the relation I have a student who's writing this thesis on, right now, on carton geometry and different formulations of general relativity and so forth. That's a long answer to the theory of carton geometry, but I think you have to do it. You'll have to ask somebody more knowledgeable than I am about the calculating formulation. It's very related, yeah. So you have both, so both brain field and the Lorentz connection fit together in this add-on connection that's being described here. So we don't understand why those two entities are so important and how they fit together. The carton is just very, very nice.

35:00 Michael I might have missed something because I stepped out of the room for a moment but could you go back to the slide where where Bile capitulates there's penitence as you put it and then I'm still worried there's still something I don't quite understand if you start out thinking that there has to be this notion of intuition at the basis of any mathematical construction that we do. So fine, we start in the tangent space, we say that's intuition. Then the question is, what more can we do? Now here he says, well, it must be described in some arithmetical way. I mean, it sounds like it might be a purely algebraic, non-geometric, and here perhaps he's more going along with the Hilbertian less intuitive or more symbolic, as he puts it, notion. It clearly is referring to some more formal treatment of geometry here in the second part of his quotation. And I think, as I said in my remarks, he cites this paper of Cherenz in this paper. He was tremendously impressed by the power of the fiber-lumbo formalism, which had no connection to intuition as far as anybody knew. and he saw the power of the method and that's what this is about he's capitulated here but he's giving up this basic commitment to intuitive construction so my question is then you brought in Poincaré to help out and say Poincaré also makes group primary but that doesn't quite do it because of course for Weyl the group is also primary he's following in the Helmholtz-Lee theorem in his own pure infinitesimal thing but one could argue that that tradition retains a link to intuition and here that's precisely what we've lost or it in other words if we start from where vial is and we want to bring cartan in we have to don't we need to somehow think about why that's an intuitively admissible step and not just the kind of algebraic and maybe cooked up yeah and I don't I mean it seems to me there should be a way to do that, but I haven't yet seen what it is.

37:30 No, I agree, but the problem here is, what are Weil's views on differential geometry after this time? I mean, he's won over to the new five-run formalism, which is shiny and new and very powerful. He's not talking about intuition and geometry. OK, I got that. Do you think, since you, at the beginning, said, you're still, I understood you to be saying, you're still holding to that idea of an intuitive, although I noticed you said intellection, intellection, and a construction, but you wanted to retain some of that intuitive. So do you think there is an intuitive way? Sure, right here. Oops, sorry. It's right here. Why is this moving frame approach any different than, you know, less rooted? of constructive geometric intuition. Well, that's a negative. I mean, can you say something positive about why it is rooted? Well, the same kind of evidence that Weill would insist upon in talking about parallel transport vectors is involved here. You see the superposition of one figure or frame over another, how they coincide, if they do coincide, what the angle is between them, if they differ, how they're rotated with respect to one another, it's directly inspectable because these comparisons take place at a point what about the association between the whatever it's called the affine bundle so now you're talking about things that are happening in the tangent space what is the nature of that association between this more abstract thing and the tangent spaces is that a purely algebraic analytical association a geometrical association. I think that it's only from the tradition of Riemann that Weill hung on to the tangent space as the locus of construction. I think it's a purely historically contingent fact. And I don't see why the same kinds of constructions cannot take place in other bundles besides the tangent bundle. That's what a geometry of connections is really.

40:00 So if you take the curvature of this connection, which is made of two pieces, you obtain the torsion and the Riemann curvature, which are linked to the translational part and the rotational part. And usually, general relativity is expressed with the Lorentz connection. But you can also express relativity with a connection which has torsion and no curvature. You meant the lavagevita connection. But you can use a connection which has pure torsion and no curvature. That's the distant parallelism. And they are equivalent. Do you have a comment on that? because it's quite surprising that you can use either torsion or curvature to obtain the same theory. Yeah, I know you can do that. I don't know what to say about that, actually. Uh, nothing. No particular comment. It's a wonderful thing. You should have an explanation somewhere. Well, the explanation is in mathematics, but the conceptual explanation, this is what it is. Well, you can show it mathematically, but there is no deep mathematical reason, nor physical reason, why this is so . Yeah, maybe. He's not really thinking about that. Okay, I think Oliver. Well, I had a question very closely related to Microsoft, I think. I mean, the connection with intuition seems to have a positive and a negative part. There's this positive part that the tangent space is the right sort of space. We have a very close connection to our intuition. It's the right sort of dimension. Our experience is sort of momentarily and temporally extended, and that's on the three hands from the spatial world. And then it has this negative part, right? Because we can only make these comparisons at a point, we shouldn't have these absolute comparisons or relations between quantities at spatially separated points. Now, in your defense of the Cartan generalization being still no less rooted in constructive geometric intuition,

42:30 I mean, that's all to do with the negative aspects, right? That, you know, we can't, it's insisting that we can't, there aren't these absolute comparisons to be made at this point. Surely, it's the positive aspect that's being put into question. You know, if the space is, you know, I mean, there's no constraints in the dimensions or the nature of these client spaces that you're putting up in the point. So they're homogenous spaces. Well, at least in Cartoon's geometry, the client's spaces have to be the same dimension. Right, yeah. So you're replacing, for example, a tangent plane by a tangent sphere or a tangent, like, involving space. I would say that you're just changing your mind about the best approximation to the local space-time. So you're just using any space of constant curvature? You could use spaces that are more challenging than spaces that are true. I mean homogeneous space. But actually one's that seems to be most important. My students' thesis articles are constant credit. So that's sort of been that knife that my student has that same square diagram that you should I just showed at the beginning of that, but what's the name of this belief? Derek Wise. Well, suppose we limit ourselves to custom curvature, could we then have some kind of generalized almost lead theorem? I don't know exactly what I'm asking, but... I don't know what that means to lead to this question. Well, it says something like, if you have a group act in theory, transitivity on the manifold, and there's a unique Riemannian metric, such that those are the rigid motions of it? My guess is that we can in recline space. That's right. So that might, then, be the connection with Hamel's Poincare. So you don't just, if you do have constant curvature spaces, you have a connection with intuition that is the

45:00 group itself has some kind of connection with something like the group of displacements, the bodily emotions. So it's not just some kind of algebraic thing that I've cooked up. I mean that would be a different generalization of Hemmol's Lee than when Weyl does. Oh yeah, you can't put Weyl's treatment in the fiber by the thermos. All it's got is the Euclidean tangent space. And to have a, in my general. So maybe that'll work. to understand what it is. Can you all have a question? Yes, thank you. So, again, I was completely wrong about this thing. I'm still unhappy about the negative thing. I mean, I see no big argument that just because it's not directly connected to the sort of comparisons we make the point, that any critical theory that does involve these absolute comparisons I mean, so long as you, so long as there are clues in it, you can make the point. You know, what good argument is there against it? I mean, so Einstein's objection to the bar, one of the things that's so good for this world, you know, we can only compare angles to the point. Why are you making that as well? I think it's, well, I can give you a weasel answer to tell you that it's a matter of aesthetics. But it's more this, this idea, intelligibility. Distance comparisons are primarily not intelligible. one wants to try to understand them in a point-by-point, path-dependent manner. It doesn't mean they're not perfectly good physics if it works, but they're just non-inteligible.

47:30 Thank you very much. Thank you.