Noncommutative Geometry — Jean Petitot

0:00 Percocetics. Thank you. I apologize for my English. The recurring topic in this conference has been how to reconcile mathematically very heterogeneous theories, and in particular, general relativity and quantum mechanics. We have focused on the conflicts between differential geometry and discrete structures, and we have seen many proposals for enlarging the formal framework of Riemannian or Carton geometry, for quantizing some of their components, etc. It seems that these problems are not sophisticated, peripheral, marginal difficulties when meet only at the boundary of the theory. But on the contrary, that they concern very basic, central, foundational difficulties. And this means that we need a change of framework as for general relativity with humanian geometry. And non-commutative geometry is a proposal for introducing from the outset quantum concepts in the very definition of the most fundamental geometrical concept from the beginning. In this talk, I will try to present how non-commutative geometry, how in non-commutative geometry, metric can be reinterpreted in purely spectral terms, using the formalization of before that, how a purely non-commutative generalization yields a natural interpretation of the Higgs phenomenon, how it is possible to reconstruct the standard model and how spectral action couples the standard model with Hilbert-Einstein action. Philosophically, non-commutative geometry is a new framework in that sense, including as a commutative approximation both general relativity and the standard model and giving for the first time a deep theoretical meaning purely non-commutative cohesionality. The main interest of non-commutative geometry is, I repeat, to start from the outset

2:30 with quantum mechanics and to compare directly all classical geometrical concepts of differential geometry, Riemannian geometry, and carton geometry. Now it is rather technical stuff. But for those who are interested in, I put my paper on the desktop of the high map in the library. To understand non-commutative geometry, we are first to come back to Gell-Fan's theory of commutative C-star algebra. I remind that a C-star algebra A is a Banner algebra, an algebra with a norm, it is complete for the norm, an involution, x, x star, with a beautiful relation between the norm and the involution. The norm is then deducible from the algebraic structure. The square of the norm of an element x is the sub of the spectral values of this positive element, x-star. An element is called self-adjoint if it is equal to its adjoint, x equal to x-star. Normal if x-star equals x-star x, it commutes, it's adjoint, and unitary if it's inverse, it's adjoint. In the classical setting, the mathematical interpretation of the fundamental concept of space of states, observable, measure, et cetera, are the following. The space of states is a smooth, for instance, compact manifold. The phase of space is a contingent bundle in Hamiltonian mechanics with a canonical sublective structure. The observables are continuous functions with real values, which can be considered as self-enjoined function with complex values, f equals f bar, conjugate, of f, and the measure of an observable f on a state x, which is a point of m, is only the value of f x.

5:00 So the observables constitute the commutative And Gelfand's theory explains that the topology of the manifold M can be completely retrieved from the algebraic structure of A. Let M be a compact apologetic space. A is an algebra, the C-star algebra of continuous function of M. It is a C-star algebra with point-wise multiplication and complex conjugation. The possible values of F define what is called the spectrum of F. That is the value for which the value of f, that is the value for which f minus c, which is the value of f, is not invertible in the algebra a. And the evaluation at the x, that is the measure, can be interpreted as a duality between m and a. and the point x is associated with, canonically associated with the maximal ideal of A which is the ideal of the function which vanish at x which is the maximal ideal and these maximal ideals are the kernels of the characters of the algebra the character of an algebra being a linear multiplicative morphism between the algebra and the field of complex numbers. N is the kernel of the character. And the evaluation of the character on F is also a duality between the element of the algebra and the character of the algebra. So, the spectrum of the algebra is, by definition, the set, the space of characters. It is a topological space endowed with a simple convergence. Now, if f is an element of the algebra A, we can associate to it canonically a function f tilde on the spectrum of the algebra, which is a space, a space of characters. And we use only the duality. The value of f tilde on the character t is only t evaluated on f.

7:30 reality. And so we associate the function from the space room of A to the complex number. So we have a map which is called the Gell-Fann transform and the main theorem is the Gell-Fann-Leymar theorem, which says that if A is a commutative system algebra, the Gell-Fann transform G is the isomorphism between A and the C star algebra of the continuous function of the spectrum of A. So the conclusion of that is that in the classical case of the commutative C star algebra, there is a complete equivalence, complete equivalence, between the space and the functions of the space. Everything which is geometrical, which concerns the space in a geometrical fashion, can be translated in algebraic properties of the algebra of functions. So you have a perfect equivalence. Okay, now it is extremely easy to look at non-plomutative Thestar Angema. It is a very evident generalization. And so, you can say that you have also some strange spaces which, in the Gelfand isomorphism, are associated to this Thestar Angema. strange spaces, which are completely pathological at the classical level, are the non-commutative spaces. So, in quantum mechanics, the basic structure is that of non-commutative Cessar algebra observables. And the first main idea of Hohen, Hohen, is to wonder if there could exist a geometric correlate for these non-commutative systems. And it is for that, that non-commutative geometry is also called spectral geometry or quantum geometry.

10:00 It takes for basic structure the non-commutative C star algebra and tries to translate in this new framework all the basic structure of classical geometry. as was emphasized by the very important specialist, Daniel Kassler, Anakon's non-commutative geometry is a systematic quantization of mathematics parallel to the quantization of physics effected in the 20s, the theory widens, widens, the scope of mathematics in a manner congenial to physics. And I can reinterpreted differential geometry, Riemannian geometry, Carton geometry in this new framework. Let us take as a first example the reinterpretation of the differential calculus. We begin with a universal, formal, purely formal differential calculus in an algebra. We have an algebra A, and we look first at derivations on A, that is, we look at similar maps from A to the module satisfying the rule of differentiation, the universal rule of differentiation, which is the Leibniz rule. and which says that DED equals DE multiplied by E plus A multiplied by DB. You see that for that, E must be an A-B module. You have to multiply on the left and on the right the elements of E by elements of A, of the algebra, shows that there exists a universal derivation which depends only upon the structure of A, and which is quite simple between this operation of derivation, d from A to the tensor product of A by itself.

12:30 And it is a difference between 1 conservative A minus A tensordial 1. And the universal b-module of one form, omega 1 of A, is only the quotient of, is the tensordial product of A by the quotient of A by this relation. The relation And I g is equal to A conservative B minus AB conservative 1. So, you can construct the universal linear rule of derivation. and it is immediate to generalize to n-form absolutely evident ok so you have a universal algebra now we have to look representation of this universal differential algebra which exists And here, Conn uses the formula, the Venmo formula in control mechanics, which says that to the right derivative, the right design, and observable, you take the commutator with Hamilton. And so, he looked at interpretation where the algebra A is represented via representation in a Hilbert space, H, and define this interpretation of the symbol df as the commutator between the element f of the algebra, but now f is an operator on h, the other representation, and a certain operator which doesn't belong to the representation a. Little f or big f? You have two f's. You have a little f and a big f. Yeah, big f is a commutator between small f, which is an element of a,

15:00 of the algebra, but now it is an operator of the universe space, because you have represented your algebra in a universe space. And big F, car F, is a supplementary operator, which is not in the representation of A, and which defines the differential calculus, which defines. So it gives an interpretation of the universal different structure. So big F is the same thing for all small ones? No, no, no, no, no, no, no, no, no. You have to decide. Actually, yes, you fix the big F. Yeah, in that case, for every interpretation, you fix F. Now, f must be self-enjoyed. To have b squared f equal 0, you must have that the committator of big cap f squared with small s is 0. For instance, you can take x squared equal 1. And h splits according to the eigenvalue plus 1 minus 1 into subspaces. Now the main constraint is that Df, which is now an operator of H in your inner space, must be infinitesimal. And so you have to interpret in this new context what can be infinitesimal. And after a lot of reflections, comes the idea that an operator on H is infinitesimal if it is compact. That is, if the eigenvalues, mu n of t, which are the eigenvalues of the modulus of t, of the absolute value of t, converge to zero. And if this new n, which are called the characteristic values of the operator t, converge to zero as 1 on n power alpha, t is an infinitesimal of order alpha. And Kant proved that this is a consistent definition. And so you have now an interpretation, which is an operatorial interpretation, which has nothing to do with the Weyerstrasse intuition or Leibniz intuition of infinitesimism, and which is capacity.

17:30 Okay. Now, if t is an infinitesimal of order 1, its trace is a series of the type, the sum of 1 over n. So you have the logarithmic divergence. But there is a tool, a technical tool produced by Dixier, which is called the Dixier-Trass, which has all the good properties of a trace, and which extracts the logarithmic divergence. And this tool can be used for integration. So you can integrate infinitesimals using the trace of the sum, which is exactly an integration, integrate integral of dF. So you can integrate in a very consistent way. Okay, so now we are in a situation where the universal symbols for differential forms, E0, DA1, etc., DAN, are interpreted as operator on H as E0 multiplied by the product of the commutators, BFA1, etc., BFAN. OK, I must emphasize the fact that this non-commutative generalization of differential calculus is extremely wide, and it is a sort of wide, not only wide, but wide generalization. For instance, it enables to extend differential calculus to fractals, which classically, by definition, have no differential structure. So the topological monsters, which are fractals, can be differentiated in this context. Okay, so you see how differential

20:00 structure can be interpreted in this new context. Now, I think that one of the main achievements of non-commutative geometry was the possibility of reinterpreting distance matrix in Riemannian geometry and using Dirac operator. I have to explain Dirac Operateur in terms of key-force algebra, because it is a key-point. So, the idea is to reinterpret the classical notion of distance of a reunion manifold, as, by the classical formula, is distance between 2.2 is the n of the length of the path. going from P to Q. And the length is given using the Vs squared. g nu, u, x nu, x nu. Vs squared, we have x nu, x nu, x nu. So we have to find an algebraic equivalent of that if we want to generalize to the non-commutative case. And here we need Dirac Operator and Clifford Algebra. You know that Clifford Algebra relates the differential force and the metric of Riemannian Manifold M. If V is a real vector space with a quadratic form, a metric G, by definition the Clifford Algebra of V, is the tensor algebra of V, quotiented by the relations V tensor V equal minus GV. So the element of order 2, V tensor V, collapses on a stack. Minus G of V. GDG means the radical. Okay. And using the scalar product associated with G,

22:30 one gets immediately the anti-commutation relations, the Poisson bracket, VW is minus two times the scalar product of V and W. So if G of V is different from zero, V is invertible. You can compute the joint representation. Do you assume your metric to be positive definite? It's not obligatory. It is not an obligation. If GV G of G is a different of zero, G is a different. So if G is positive definite, you have an inverse or a very great D is a different of zero. So the fundamental relation between the Clifford algebra and the exterior algebra of V is the fact that for G, G equals 0, if you interpret the product, the Clifford algebra, as the the exterior product, you get the antisymmetry of differential force. And this means that the differential exterior algebra of V is the Clifford algebra of V for the null metric. So the Clifford algebra of your vector space V uses the metric G for twisting the exterior product. It is a mixing between the differential structure and the matrix structure. You have two operations of the the exterior multiplication of forms, and the contraction of forms. You have the equation that is well known. And what is important before V, to look at the map, V, C of V. C of V means the sum of the exterior multiplication by V

25:00 and the contraction by V to get the anticommunication relations of the Clifford algebra. And so you realize your Clifford algebra in the algebraic endomorphism of the exterior form. Okay, and now you can use this for twisting the exterior derivative. The exterior derivative, which is classically dx nu, the relation by the x nu, you use the matrix, And you take the images of the one-form infused. And these are the direct matrices. And so you get an operator, which is the direct operator. Now, if you have Riemannian matrix, you can do this construction at every point, in every tension phase. You have a bundle of legumes, before the algebra. Okay. Now, the point is that if in the classical case, M and the algebra of function, here smooth functions on M, if you compute the communicator between the Dirac operator and x using the matrix you get that it is essentially the norm in l infinite of the gradient but now you can return to the classical definition of distance And if you return, as a classical definition of distance, using what I have presented on the IRAP operator, you arrive, arrive, you show that the distance is the center of absolute value of distance. f at p minus f at q, p and q are two points of n, for all the functions of n, which satisfies

27:30 the fact that the differential df in norm is inferior to 1, is smaller than 1. It is It's a very easy complication. But now, here, you have only the algebra and the commutator of the algebra with the Dirac operator. And this can be generalized in the non-commutative case. You have the C-star non-commutative algebra. You have an operator D, which defines the differentiates which gives an interpretation to the symbols df and you define the distance using this form even if there is no space in the classical sense. So you have the following definition of a non-commutative geometry. A non-commutative geometry, by definition, is what comes called a spectral triple composed by an algebra, a non-commutative star algebra, an Hilbert space, a representation of A in the algebra of operators on H, and the Dirac operator, D. the Dirac operator, D, which is unbounded and whose inverse is infinitesimal. The inverse of D is the Ds in this new setting. So the inverse of D is compact and all the commutators between the Dirac operator and the elements of the algebra and are bounded operators. So this is the definition. Here you have a quotation. First, it is precisely this lack of commutativity. It's quite deep. I think that it is really deep by there, very philosophical. It is precisely this lack of commutativity

30:00 between the line element and the coordinates of the space that will provide the measurement of distance. You see, it is a completely different way of measuring the distance, because it is founded on the non-commutativity between the ds, that is, the Dirac operator, and the vex nu, and the ds, because f are the functions of the coordinates of u. And so the differential now R, T, A, mutual commutator between V and A. I wouldn't do that to you. So I will give now an example of the non-commutative interpretation or formulation of a very well-known physical mechanism, but very strange, which is its mechanism, which uses the cooperation between Gauss bosons and Goldstone bosons to compare the mass in Gauss bosons. So in the classical setting, you have the vacuum state, which is degenerate. You have Goldstone bosons connecting the states of the degenerate vacuum, and X-phenomenon is a trick for capturing the goldstone boson by the gold boson to transform bosons in mass. Here is a model of X-phenomenon. You take as algebra C plus C, so the C-star algebra of the space Y, constituted by two points, E and B. The elements F, the functions on Y, are interpreted in H as matrices FA, FB, where FA alternates on the first part of H, noted HA, and FB on the second part of H, HB.

32:30 The Dirac operator is of the following form, n is an operator, and it is decomposed in two parts, d plus d minus, which are a joint, n equals d plus and n star equals d minus. you have a standard chirality you compute the df it is extremely simple you can compute the df functions commutator between d and f and you find this which is not d because here you have minus m and not and here you have the factor minus you apply the formula for distance you see that the norm of the commutator is the difference the norm of the absolute value of FA minus FB multiplied by the greatest diagonal value of N and so if you apply the formula For distance, you arrive at d of e and d equal to inverse of lambda. It's a toy of a model, but it is extremely interesting, because between two points with absolutely no geometric structure, introducing a Dirac operator, which has eigenvalues, you have a distance between the two points. a non-computative distance between the people, which is a spectral distance, because it is an eigenvalue, an inverse of an eigenvalue. So you see how far you can generalize the concept of distance. It can give a spectral content, a purely spectral content. Okay. On the paper you have the computation, but you have the basis of functions, two basis e, which is 1, 0, and 1 minus e, which is 0, 1.

35:00 There are two elements of the basis, two projectors. You can very, very easily compute one form. So you compute one form. You arrive at the conclusion that every one form is an operator on H of the form, like that, and with two parameters here. And here you have L star and L. to the elementary computation. So you have the general form of one-fold, and you make the Young-Mills theory for that. I repeat that it is a tall model. So the vector potential is a one-fold, which is . So you have this, but here you have the condition for assuring the self-adjoinedness. So you have something like that. You compute the curve of dv plus d squared. It is a very easy computation. And here, you have this factor, 5 plus conjugate of 5 plus 5 conjugate of 5, of 5 squared, the Yang-Nil section is the integral of the square of the curvature, that is a trace, the integral is a trace, you compute the trace of theta squared, it is absolutely an elementary equation. And you get the Yang-Nil expression with this factor, and this factor this phenomenon you have here a degeneracy of the circle where this expression finishes so you have a continuous degeneracy and each phenomenon can be extremely It is very easily interpreted in this context. On the circle, modulus of c plus 1 squared by 1 is a circle.

37:30 You have the good group of the Lugutian elements of the Heidelberg's axe in the very traditional way, which is a well-known formula. And in this context, you explain, as a non-commutative effect, each phenomenon. I have two more calls, five minutes. Ah, ten! So, five minutes of the standard model, and five minutes of quantum policy. Now, I want to explain how Cairns and Lott derived the standard model. Martins said that it was possible because it ties the properties of continuous space-time with the discreteness stemming from the chiral structure of the standard model. You can interpret very easily classical gate theory in the non-commutative setting. You have the algebra of spruce functions of the spin manifold. You have the classical Dirac operator. You have the interspace, which is the space of the square integral sections of the spin-off and the group of diffeomorphism of your manifold is the automorphism of the algebra. But if you are in a non-commutative setting, there is something without equivalence is a commutative setting, which appears. In the group of automorphism of your algebra, of your non-commutative algebra, you have canonically the normal subgroup, which is the subgroup of inner automorphism, conjugation, A gives U, A inverse of U. If you have commutativity, u, u minus 1, is 1, and so the inner automorphism are identity.

40:00 So it is a specifically non-commutative effect. Here is a quotation of Kahn. Amazingly, in these descriptions, the group of Gauch transformation of the matter fields arises spontaneously as a normal subgroup of the generalized deformorphism group automorphism of A. It is a non-commutativity of the algebra A which gives it for free. So I repeat that in the non-commutative setting the vector potentials are of the form of HET, elements of the algebra, and commutator of the operator D, which defines the distance with elements of the algebra. You look at the force, which is a two-form curvature, and you have exactly the classical formula for the gauge transformation. Now, the fundamental point is that in the non-commutative setting, the inner of the Marxism, which are not trivial, acts on the Virach operative. act this way, D, D times tilt, we call D plus A, the potential vector, plus this term, G is a real structure of its two technical effects, but it's D plus something associated with potential vectors. Okay. So, you can, it is really a deep idea, you enter in the non-commutative setting, you define, you are able to define this inner automorphism, perturbations of the Dirac operator, but the Dirac operator defines the distance.

42:30 internal perturbations of the metric. You have internal fluctuations of the spectral geometry, which are induced by the degrees of freedom of gauge transformation. The degrees of freedom of gauge transformation can act on the metric using this new definition. And it is for that that in non-computative geometry, the problem of quantum gravity has another aspect that in classical theory. So you have directly, constitutedly, a coupling between metric and gauge transformations. And it is what is needed for complete gravity with quantum fix here. Okay. Carl's show, show, show, with light, that the black hole, the black hole, the black hole, and that's not our model, can be entirely reconstructed using a problem between a classical model spin, a manifold, etc. M, with the algebra of smooth function M and a discrete geometry as in the case of the toy model of each phenomenon. The noncommutative algebra is C plus quaternum plus the 3-3 matrices, complex matrices. This is a noncommutative algebra. Now, the choice of this non-commutative part is linked to the symmetry of the standard model. You need somewhere the symmetry u1 cross sq cross sq3. Now, I have not time for explaining exactly the model, but you have a discrete part

45:00 which is extremely simple. You take a finite dimensional universe space having as basis the particles, the quarts, the lectins, etc. It's extremely simple. And you let with algebra with the non-computative discrete part of the algebra, the space is extreme, Or, it's finite, finite, hidden space. But using what is known from particles, from two particles, chirality, et cetera, what is known, you put this in matrices. It's very simple. And you have a very, very, I'll give here an example. Well, you have the quarks, up, down, left, right, and here you have an action, which is extremely simple. Here you have the complex number, and here you have the quaternion, and here you have the action. There are linear actions completely elemented. So you put this information coming from physics into the machine. Now you look at the tensorial product of this algebra on the vector of the universe space, which is the normal universe space, and the Dirac operator, which is composed of two parts, the classical Dirac operator and the discrete Dirac operator. as in the case of this phenomenon. Now, the miracle is that, after pictures of computations, it is a North-Bell computation. I have the paper here as well. After a Hogue-Bell computation, you get exactly, exactly the North-Bell Lagrangian. And you know that the formulation of the non-standard Lagrangian is something like three pages. And you get all of them absolutely exactly. So this is an interesting

47:30 mathematical result because the inputs are quite simple. But you use the non-commutative machine, you use the fluctuations, the actions of the gaze transformation of a classical model on the non-commutative discrete part. This is the difficulty of the computation. And again, it's a standard model. One minute. If you want time for questions, as short as possible. One minute. Kant proposed some ideas for quantum gravitation that includes this model to . In our community, it becomes possible to introduce in the model the gravitational Ibert-Einstein action. You have to couple Jandit's Goethe theory with Ibert-Einstein action. In the classical setting, when you use principal bundles on a manifold, it's a scriptural group G. The automorphism has a structure of semi-direct product. The automorphism of the principle is a semi-direct product. It is impossible to obtain that, if you want to geometrize group in a classical way, as a group of different morphisms of something. because the group of different morphisms of a manifold cannot be a semi-direct product. As a result, it doesn't matter. But in a non-permissive sensing setting, it is possible because you have this famous normal subgroup, canonical subgroup, of inner automorphism. So you have the automorphism group of the algebra, the inner automorphism, and the quotient of the automorphism by the inner automorphism, which means the external automorphism. And you can find non-commutative algebra where one group is a dishamortistic group, in the sense of general relativity, and the other group is a Gaetz group, in the sense of down-near theory. The strength of Kahn's conception

50:00 is that Gauss theories are thereby deeply connected to the underlying geometry on the same footing as gravity the distinction between gravitational and Gauss theories boils down to the difference between outer and inner automorphism quite recently Kahn's has shown take an action, which is very easy to define, which is a trace. Here you have b squared, lambda is the cut-off of the order of the inverse long length, phi is the smooth approximation of the characteristic function of the interval. It is an action which counts the number of ideal values of d of the Dirac operator in the interval minus lambda lambda. So it is, in fact, a very complex step function, as a theta function. And the key idea, beautifully explained by Lundy and Robelli, is that you can consider the ideal values of the Jiraq operator as dynamical variables for general relativity. And here too, a very difficult computation gives you, for this trace, for this action, a coupling between the standard model and the Hilbert-Einstein action. And so you have some, I don't know, for physicists it's very good, But mathematically, it's, I think, quite interesting because the gap between geometry and fundamentally theory quite disappeared in the laboratory in Germany. I apologize for the technicity that you had the formal computations in the paper. Thank you. Thank you. We have very few minutes for questions until the bell rings, and we can even wait a second after that happens. Yes, first I would like to comment about the big interest of this remark by Lundi and Ovely, because one of the difficulties of general relativity is that, in general, the true dynamical degrees of freedoms

52:30 are mixed with the degrees of freedom of the diffeomorphism. And this is one big difficulty for considering general relativity and to quantize it. Beside that, I have a question concerning the non-commutative part of the algebra in the Kohn-Lott model. It's defined as a direct sum of three-vector space. And in fact, you have no choice because you want that to be a representation of the gauge group. You need an algebra structure for that, which is not canonical, because the direct sum defines the vector space structure, but how you multiply them, there is some arbitrary layer which should be hidden, and how they decide how this vector space becomes an algebra. That's my question. Oh, it's a, you interpret it as acting on a nilber space, which is a sum, a direct sum of subspaces. And so you ask, this is the vector, like a switcher. So it's the direct sum of L, if you just multiply it, if you multiply it, if you multiply it, This is right here. No, there is no problem. So you have a complex number, and 3-3 complex matrix are for color. Thank you. This is a question about the generality of the NC approach. answer, it would be yes or no if you want to, you don't want to go into the details, but can you get a true form of torsion in this? In other words, what's the relation of this approach to other than the tangent problem? Ah, torsion. I don't know the technical interpretation of torsion. Who knows? But you can't have torsion, because you have exactly

55:00 the classical series. It seems you should have it. Yeah, you have it. But exactly the formula, algebraic formula for torsion, I don't remember. the torsion is usually defined as the non-commutative part of the connection so here it should arise I don't remember the exact but I take it the regular part of the connection is also non-commutative somehow well so in fact you should have only torsion But this non-commutative geometry is deeper than the non-commutative defined in non-Abelian or things like that. It's radically deeper. but it is a perfect example generalization of the reinterpretation of classical notions but in a way that you can generalize, widely generalize One way to do non-commutative changes is you simply start from, you take your coordinates and you turn them into non-commutative objects, then you assume that X mu commutator is different from 0. Can you comment on the difference in this? Yeah, I prefer this presentation, because philosophically, it is a change of what is fundamental and what is secondary, what is primitive, and in this point of view, it is algebra of observable, which is primitive. and space geometry is constructed it is completely different as a generalization which keeps the intuition of space and tries to make the space non-commutative here the non-commutativity of space

57:30 is a consequence of the non-commutativity of the algebra but the other approach via non-commutating variables is also closer to physics. You can see how ultraviolet divergences disappear. You can see the mixing of divergences quite directly. So there are a lot of advantages of the other approach as compared to physics, to how closely we want to get into physics. But I don't know, because the last lecture at the College de France rather than the renormalization. No, you can do this here, but there you go there directly. You know, if you start by the relationship between von Neumann spectral measures and the delta function, I mean, is it like that? If it were, it would be nice. I mean, one is a rigorization of the other. Well, yeah. Do you have a device for eliminating the emergencies, etc.? It's really natural. I mean, I'm not saying that's different. I'm not saying that you're getting different results. I'm saying in the other way, when you start with the model product and the coordinates, you sort of very naturally and very quickly, without the whole machinery, get to physical results. But then, of course, this is a different... Physically, it is exactly the same. at the Lorentzian and the Einsteinian You can always do the same thing in a different way. What I like in this approach is the conceptual structure, exactly as in Einstein, what is primitive. Thank you very much. Thank you.

1:00:00 But he'll put it on the web. On the web, I don't know. Do we have an address? Je viens parler à la table rondeur. Il va y avoir un... Sur le site web du colloque, il va y avoir des détails. With a Mac, it's a pretty... Oh, I do have a date. I could just put it on my jumper. Oh, it's pretty amazing. I have something to add to it, just to fill this and this, or I can fill it for you. Should I do it now? Yeah, I think it's possible, yeah. Before lunch, I should. Yeah. Okay. Thank you. This paper will be complete.