Feynman Rules for Quanta of Space

0:00 Thank you for being here, such a full audience, a distinguished audience. I'm very happy of having the privilege to talk in particular because there are a number of new results which I'm going to present. So I'm happy to have this opportunity. Let me put my hands ahead. Unfortunately, I don't have the observational results like Roger is sort of hinting as he has. I wish I had. There's no. You wouldn't know if there was an observational result. So the results are a little better about the theory. This is the plan I am talking. I apologize for having being a little bit small, but from the back, you can see. The main part, anyway, everything is a slide I'm going to say, so even if you don't see exactly, I don't know what it does. The main part of my talk is a presentation of a theory, a model. model. The model is pretty explicitly and entirely given, so I will write explicitly the transition attributes and then I will talk about the state of the model, which define a quantum geometry. And the conjecture is that this model is a quantization of generativity, has generativity and its classical limit. So we present some of the elements, the various elements, that indicate that the classical part of the talk. Before, I will just say in which sense I am going to discuss the relation of the physics and geometry. And just to make a list of main ideas that has converged to the construction of this theory here. And after that, I will discuss some properties of the theory, whether there are divergences or not, what kind of divergences you have to do with them, what are the scales in the theory, and the relation to making smallness which is one way of understanding this theory, this theory is a strict relative to the matrix models which can be seen as a quantization of two-dimensional quantograms. So this is a quantization of four-dimensional generativity in the same sense. Hi, welcome. Let's talk about matrix models. And I will just at the end flash a couple of slides no more about calculations that So the main part of the talk is just to tell you what the theory is and write the main

2:30 information of the theory. So let me start from number one. What sort of problem? Discussion here is on relation to geometry and physics, of course, there are many relations geometry and physics and each one of the other very distinguished speakers the way they address this relation of the physical geometry is is diverse roger has just told us sort of this magic speculation about the possibility that from the castle structure just focus on the castle structure of gr will learn something about very large-scale cosmology and robert will tell us about, I suppose, his work on the geometry that you can read out of strings, and if you take string theory, it's a description of everything that's of geometry on the private side. And Alan Cohen has been developing his non-commuted geometry as a way of reading out geometry from the standard model, the same way in which Einstein was able to read out Minkowski geometry from Maxwell equation. That's one way of reading what he's doing. And then, of course, we will learn from a much more mathematical point of view, from Consenicci, what can we learn from mathematics or physics. I'm not doing any of that. So the approach I'm taking here, problem addressing, is a total to that. What I'm discussing is what we can say, what we can guess, what we can deduce, what we and in theory, about the micro geometry of space, our physical space, three-dimensional space, on the basis of the theories which are today well-supported empirically, and in particular, generativity. Generativity has changed in a sort of dramatic way, the way, of course, we understand the geometry of the physical geometry of space. And if we take this seriously, if we take this sort of not because of Einstein construction but because of the century of empirical support growing and growing, and today even more growing empirical support for generativity, we take it seriously and we try to say what we can do on the basis of that, on the basis of generativity, what we can say of the microstructure of space-time, what you also do. include quantum effects, what can you do? And this is the spirit in which I'm talking.

5:00 Of course, we can do a lot, right? You can take generativity and use it as a quantum series and effective field theory, it's perfectly consistent theory, which works up to a certain scale. And if you scatter two particles, say, imagine you have different regimes depending on So there is a space of possible regimes, and if you can use a generativity as an effective field theory for studying what happens at a relatively low energy, and then when you go higher in energy, you have more complicated final diagrams. At some point, the theory as an effective field theory doesn't work anymore. Because you have too high energy, and I think this is in terms of normalizable, essentially. energy, we still know what happens. Because if you fix impact parameters, you go very high in energy and you throw two things one against the other one, they fall black hole. So you can use general classical generativity again to compute within some approximation what happens. When we really don't know what happens, it's just a narrow wedge here in And these units were E equal B, where we have no idea what it is, okay? So, what we want is sort of compute generativity and fx is quantum theory up to this region here, which means finding a quantum theory whose classical intergenerativity. And the way I'm looking at it is a theory which is a theory of GR, namely it has the the same degree of freedom of GR, and it implements the same physical principles on which GR is built. Well, because, unlike Einstein, but because it's essentially that sort of nature keeps confirming us that this is a good piece for a good direction for understanding space and time. Now, there's an objection, which you can hear immediately in all sort of particle physics world about the problem itself, which is that this is impossible. It's a number of arguments that says that if you take a local quantum field theory and if you try to do a quantization of generativity as local quantum field theory, it doesn't work because essentially you mention g of the constant and everything that goes around.

7:30 So you need matter to cancel, matter loops to cancel the gradient loops, or else, or put it in something different. Now, these arguments are correct, of course, but they are in the context of local quantum theory, namely the assumption that the geometry of physical space is essentially that of a manifold. And that is an assumption I'm going to argue, which is contradicted by generativity itself once in the city of quantum theory. So the aim here is to find a quantum field theory whose limit is classical generativity, but perhaps can be said more clearly to find a general covariant formulation of quantum field theory which does not have this sort of particular form of locality, which is say they had axioms or the whiteboard axioms, the way they define a quantum theory. And we know general comparative quantum theory, so the same was here, the definition of topological quantum theory. This is a general scheme in which we can think about this object here. And the theory we present is the theory that enters in this general scheme here. Now, is this a theory whose classical linearity I don't know, I'll present this as a conjecture, so I'll present a model and give you some indirect arguments saying that in fact it might address the specific problem here. So the main ideas that go into this sort of listed here, not mentioning any of them, almost everybody working quantum gravity has this formula in mind, right? the quantization generativity is a sort of Feynman sum over four geometries, perhaps bounded by a certain three geometries. This first material paper by Niesner in 57 was developed by Hawking in great detail, but if I drop in the eye here, Niesner attributed it to Nira. So that's something which is in mind, of course this is extremely defined, so one way of understanding what the meaning of quantum gravity is trying to give sense to this expression here. Now, a main source of idea, and in fact, if you play a role, what I'm saying is Regen work. Regen in the early 60s realized that there is a natural truncation of general activity, which is Regen happiness, which you can think in this way. You can think to take a manifold which is flat, everywhere, except that on two-dimensional

10:00 surfaces the curvature is concentrated, and so constant times, where curvature is concentrated, and in fact is represented by a single number, which when you go around it, the monomy of the connectivity connection around it is determined by a certain number. And what Reg has shown is that there is a simple description of truncated generativity in terms of manifolds with this sort of defect where curvature is. Now, a key ingredient in what I'm doing is panel spin geometry theory, which I will discuss. And as I'll say at the end of my talk, this is strictly related to the matrix models, which were in large part developed here in Paris, I should probably have many other names I should add here. In fact, in 92, Oguli made an effort, matrix models can be seen as a quantization of generative in two dimensions, right? It's a sort of sum over two-dimensional surfaces, which are generated, the sum can be generated by an auxiliary quantum mechanical theory, the theory of matrixes. UGUI-92 tried to generalize that to four dimensions, and wrote a certain model, and what I'm presenting can be viewed as a modification of the UGUI model, that's one way of viewing about it. The main object I'm talking about are forms, and the idea of forms was introduced in 94. The actual model I'm describing was introduced in 98, it's in fact a small modification of a previous one, a previous one, a previous one. So, the final form of the theory is in presenting it is really recent and has been clarified in the last two years. So let me go into technicalities. Let me begin. Before actually giving you the equation of the theory, let me tell you how to think about this. And the best way to think about this, not necessary, but it's easiest to train what you're talking about, is to think in terms of a TIA topological quantum theory, because in a sense it's the only example of general covariant quantum theory that we have so far. So what is a TIA idea? A TIA idea is to think as a quantum theory in a fully general covariant context in the

12:30 following way. As a rule that associates a Hilder space to any three manifolds and a map between Hilder spaces between each four manifolds that has the three manifolds as boundaries. So you have three manifolds, possible different topologies, different topologies have different hipospaces attached. And then you consider cobordism, you consider four manifolds that have given three manifolds as boundaries, and to each one of these cobordism you have a map from here to here. So intuitively you have the space of states living here, and this is the transitional amplitude for the states going from there to there. In fact, these things form a category, a tensile category, and so on. There's also this behavior space category. You can view a definition of a topological quantum theory in the sense that he has a function from this category to this category, if you like. These should satisfy a set of actions and properties, of course, consistency things, the main one being that when you glue two C, C1 and C2 to a common part of the boundary here, the amplitude associated from going from here to there is given by the one going from here to here, the one going from here to there, but the gluing here is given, of course, in the hubris, with the hubris problem. So that's, keep this in mind, and this is what we're doing, but we're giving it a twist. I give it a twist to that. Because essentially all the examples that we have of these are very simple, very trivial, and have no local degrees of freedom in a specific sense. They're all sort of flat theories. Now, I don't want flat theories, so I want somewhere to put the curvature, and I learned from Legend that one way of putting the curvature is putting it on defects, d minus 2 defects on the manifold. So let me play the same game, but instead of manifolds with T minus two defects. So three one-dimensional defects here and two-dimensional defects here. Now, it turns out that the simplest, a convenient characterization and a way to give precise meaning to this manifold with defects is to describe the three-dimensional manifold with one-dimensional defect in terms of graphs and a two-dimensional manifold with two-dimensional The next slide I will illustrate a little bit why metaphors with defects can be described

15:00 naturally captured by graphs and two-complexes. Their topology is naturally captured by that. But stay with me for a moment. And so what I'm doing is replacing three-manifolds with a graph and replacing four-manifolds with two complexes. So two-manifolds. Put a graph here and two-complexes here. In a sense, I'm still doing it here, topological quantum theory, but now with local degrees of freedom. So in some sense, it's not topological anymore, depending what you mean by topological, I'll keep calling this topological quantum theory. Now, I promised I said why a manifold with defects is a graph or a two-complex. Let me try to illustrate it with a picture. I hope you follow me. If you don't, it doesn't matter for the rest, but it explains why graph and two complexes do the job that they want. Let me start from three dimensions. Take a three-dimensional manifold and put one-dimensional defects. So imagine for concreteness that you triangulate the manifold with the trailer, and the one bones here, the one-dimensional lines of the triangulation, the segments here, to take them away from the manifold. This is where, in three-dimensional edge, a curvature leaves, all these things here. Curvature is when you go around, it's captured by the long way when you go around these things. Now, take the dual triangulation, so put a point in each tetraegulum and a line in each that crosses the triangle here. Now, the two objects, the two relevant objects here is the manifold with defects, which is the original manifolds minus the black lines here, and the graph, which is the one skeleton of the dual triangulation, the blue lines here. And these two objects are communitorially almost the same thing. In fact, the pi one is the same, the fundamental group of the two is the same. So you can capture how things are glued together here by this graph here in the triangulation, and once you know that, you know where this depends. The same thing except that now, in addition to the graph, it was also to put the faces. This faces here, the two-dimensional things here, which are dual in the triangulation, are dual to the

17:30 triangles of the triangulation, which is where regi-cremature is, right? So again here, the combinatorial topology of the two contexts capture the topology of the manifolds So, this was just motivation. What we have is two complexes defined combinatorially, an abstract set of points, lines, edges, edges and faces, and the boundary relation in glass. And I want to define a general covariant topology property here as a factor from the category of these, where these are the options, these are the arrows. The boundary of the two complexes is a graph, pretty naturally. You can define things naturally. And TQFT is a rule that associates a hidden space to each one of these, a map between hidden spaces to each one of these. And I one of these things and then show later what it has to do with geometry, what it has to do with generativity, what it has to do with geometry. So first, we have just a bit of notation, I call nodes and links, the one, the zero and one dimension of the graph, and the vertices, edges and faces, those of the two columns. So this was all as preliminaries by concern. Let me now go to the actual theory. This is the definition of the theory. So this is it. This is an explicit expression for the traditional of the theory. And of course, I'm now going to illustrate this. And first of all, open I will show you why it defines a generalized topology in the sense that I just said. discuss why it defines a quantum geometry. And finally, I will show you that in spite

20:00 of the total appearance of nothing here seems to be related to generativity, in fact, there are, as I said, indications that this is a quantization of generativity. So let me start from the beginning by just giving this a little bit more open, explicitly how this works. First, what do you solve this thing? C is a two-complex, right? So C is this object here. All vertices, faces, edges attached to one another with a boundary. And the boundary, you have a graph and you have the links, the links are called L here. So for each L, to each L, I associate an SU2 group element, H. Each are SU2 group elements. Z is a complex number, C is the two-complex. So for each two-complex here, I have a function of a certain number of group elements, one And how is the thing constructed? Well, first of all, to each edge here, I associated a couple of SN2C integrals, so one here and one here, to each couple edge phase, I associated one SN2C integral, and the character of the SN2C elements in representation J, this will come in a moment. And to each phase, I also see the sum, the sum here, over spins, over, if you want, the equivalent facets of the use of the representation of SU2. This is dimension. And this is sort of GH, G minus 1, GH, G minus 1, the homony of all these group elements going around the phase, right, they go around the phase. This is the edge in the boundary of the face, so this is an SL2C group element, and I take the character of this group element here, this is an SL2C group element, right, H-I-V, that's a, S-U-I-V, which is a fix, a sudden group of SL2C. The character of SL2C in a representation, a unitary representation, the unitary representation of SL2C are labeled by a continuous parameter and a discrete parameter, and I choose the So the discrete parameter is just a J I'm summing, so I fix it to be the spin I'm summing over. And the continuous parameter, I'm fixing it to be gamma times J, where gamma is a fixed

22:30 number. It's called a nifty parameter. It's always dependent on gamma. I should put the gamma sum. So the theory is one of the parameters in the series is gamma. At the end of the talk, you'll see exactly what is gamma. You'll see it in the larger angle when it's starting from, so it is a simple interpretation. Then there is some technicalities. There's a combinatorial factor like Feynman diagrams. you where this h are here when you go around the face and you hit a boundary edge here instead of putting ghg you put just the h an h hl which then is the only thing not integrated over so what you remains just a function of each. So this is the very explicit expression for this one that is here. Now, first, why are these, why do we define a topological problem between the sense of Atilla? Well, it's pretty immediate. These things for each, given the two complex, X, this is a function of a bunch of SL2, of SU2 . So it's a vector, a generalized vector, and a L2 of SU2. In fact, it's a little bit more tricky, because it's easy by inspection to see that they are gauge invariant. If you think in terms of lattice QCD, you have group elements to each link. In the node, you can make a gauge transformation. They are gauge invariant from just looking at the formula. So they are invariant under that. So in fact, they're not a function of L2 of SU2 to the but that modulo the action of the gauge group of the nodes, so this is in fact the space. So here it is, this is the first element of the functor, if you want, to each graph, which I'm doing here as a boundary of C, I associate a river space, and then if the boundary is made by two disconnected pieces, so if the boundary is gamma union, gamma prime, then you have a function on the tensor product of H, and this is a space, so immediately you have a function of a map from one, a little map from one to the other. And this is the second element of the material construction. In fact, you can just sit down and check all the actions, the way they glue together and everything works fine, so this defines a factor in the appropriate sense of all the consistency conditions we go through. Let me say upfront that for the physical interpretation, we want to consider also, not only, but also

25:00 a sort of limit, I just sketched the idea that this is like lattice QCD. You want to consider a sort of continuum limit in which the two complex between an initial graph and a final graph is arbitrarily complex, is arbitrarily refined. So, you want to consider a sort of limit for infinite refinement, and this limit is well-defined, if it exists, if it converts, it's well-defined, because the two complexes, the set of two complexes is partially ordered. You can say when two two complexes are larger by embedding them, one complex is larger than another one if you can embed the second into the first. And they have an upper bound, and they will be given two, you can always choose a certain one such that both two are less than the other one. And this is sufficient to define a limit in the sense of net. So this limit is well defined. And it's also clear, this continual limit, how to take it. In the community, there have been a long discussion on this sort of construction. if the continuous limit of the theory is obtained by refining the two complex or by summing over all two complexes. And it turns out, there's a little theorem that is the same thing. Namely, you can show that this limit here is the same as a sum over all two complexes of an amplitude which is slightly different, and it's simply obtained by taking the previous formula and summing over the spins, not from zero to infinity, but from one half to infinity. So, if you drop the trivial representation in the sum, then you just can rewrite this sum and if this converges, this converges, these two things converge, they are the same. The same thing. Okay. So, this is what I've said so far is that I've given you a definition of TQF, a concrete construction of a TQFT in this generalized sense. So far, it's sort of up in the air, it starts to come down and come to geometry. And the first thing I want to argue is that this talks about geometry, and three-dimensional geometry. And the reason it does is thanks to, essentially, a theorem that Roger proved in the early 70s and which turned out to be crucial in all that.

27:30 In fact, the development of the subject hit upon this theorem without even realizing it. So let me show you how it works. To each graph, fix a graph in one of these graphs. Remember, the graph is sort of what remains of the three-manicle here. And on the graph here, we have this Hilbert space. It's not the Hilbert space. I'm just repeating it here. Now, this Hilbert space has some natural structure being in L2 of SU2. There are the derivative operators, which are just the left invariant derivative of SU2. The derivative operator And there's one, in fact, this is the algebra, so this is an index, a vector index, and there's one of these vector things for each link. In fact, this operator here is not well defined in the single space because of the gauge invariants here, but you can easily construct a gauge invariant object by contracting two of these if the two links have the same source, okay? These links are oriented. So these are the natural momentum operators, the derivative operators of this super space here, this thing here. Now, suppose I take this super space with all these operators here, and I just interpret this as a quantum mechanical system. What is this classical, I take the H bar, I construct clearance states that minimize the spread of these things. What sort of phase space it describes? Well, that's exactly what Peros' theorem tells me, because the formula is two. In fact, let me give you not the original version, but a sort of sharpened version of the same thing. It's based on a theory by Minkowski, the same Minkowski special activity, but there's nothing special activity here. It's three-dimensional theory. So, by bringing together Minkowski result and Perro's theorem, the following is true. Consider this GLL prime, right, this operator, fix a node here of the graph and consider all the possible couples LL prime on the non-diagonal element of this matrix here with LL prime, and also the diagonal ones. These are the ones that Roger didn't want to look at in the 70s, and the reason being I was interested in conformal geometry, which still needs 40 or 40 feet. Consider also the diagonal part of this, and then what is true is that on this silver space, these operators here, the set of these operators here, can be interpreted, and I'll say a moment

30:00 what it means can be interpreted, as the angles for the non-diagoral ones between normals coming out of a polyedrum normal to the faces of the polyedra these normals have the length given by the area of the face of the polyedra so this phase space that one obtains by taking the limit is exactly the phase space of the shapes of the polyedrum so what do I mean? I mean that in this space there are coherent states that minimize the strength and that if you look at the expectation values you get exactly those relations, which are the relations between the angles and the areas for a given polyelium. So this is the phase space of the shape, geometrical shape of polyelium here, which means that this phase space with this operators here defines the metric, the metric in that point. If I have metric, I have most of the volume of this polyelium here, just a function of this L here, from the metric at most of the volume. For instance, For a valiant vertex, the volume is given by this expression here. Now, the genome commute among themselves, right? Obviously, they are just derivatives in a left invariant . But suppose I want to look... I want to find a basis that I analyze this thing as much as I can. I search for a maximum commuting set of operators, and areas and volumes are a maximum commuting set of operators, and this is the space, so I can diagonalize them. And then you call the basis diagonalize them in this way, where gamma is the graph that defines the space in which I'm living, J is the quantum number of A, and V is the quantum number of B. The whole thing is I was doing that when I was a student. So let me get back to the thing here. In fact, this is one of the most important slides. Maybe this is the most important slide of the talk, at least the kinematic part of the talk. We're talking about quantum geometry. This is a definition of quantum geometry. There's a huge space. There's a set of operators, and they have an actual range of angles and areas and volumes.

32:30 There is a basis in the hidden space, which is this basis here. J are the eigenvalue of the area, B are the eigenvalue of the volume. The eigenvalues of the area are easy. I go back one slide there. It's just a diagonal part of this one. So when L is equal to 3, it's just a casimir. So the eigenvalues, of course, are J, J plus 1. Sorry, yes, the eigenvalues of J, J plus 1. This is the quantum number of A. This is the quantum number of B. So this basis has a simple interpretation, right? it represents this is a graph and i look at the basic states labeled by a certain j on the links and it represents a a a a quantum geometry in which i have a quantile space with that volume and between two points of space there is an area a surface which has a a an area with that with that even by the quantum number there now careful the g don't commute among themselves so the actual geometry is fuzzy you cannot think of these as actual polyhedra in this small because the angle between these things you cannot reconstruct the geometry of the polyhedra just from here and volume, you have the other quantities, the angles and so on in the semi-classical limit which now is clear what it is, it's large quantum number When these things are large, you have quantum states that give exactly a macroscopic, just a bunch of deteriorator attached to one another, but in the small, it's quantum. And these are the quantum states of the theory, okay? So this is what I promised in the being. The quantum states are not photons, particles, strings, stuff, work, living in some space-time at all. these are quantum space of space-time itself, right? And geometry is quantized, so this is what I said there, geometry is quantized in various sets. First of all, because the eigenvalues of these things are discrete. In particular, there is a zero eigenvalue, then there's a first eigenvalue, there's a gap between the two, for the area is that, for the volume, this is a plot of the eigenvalue, so the volume for different spins, for different a minimum one, so there is a minimum amount of area of volume that the silicon talks about. Which is why there would be no atomic divergence down the road. There is no degrees of freedom

35:00 smaller than something. This talks about the gravitational degree of freedom, metric degree of freedom, so we are talking about metric, but in such a way that they are naturally quantized in the same sense in which the energy of the harmonic oscillators is quantized. There's no energy less than one half omega, h bar, h bar omega. So, again, that's not discrete. The operators don't commute, so you cannot just measure angles at the same time. And most important, a generic state in the space is a linear superposition of this object here, not just one of these objects here. So, I've given you the final rules. I've told you why it defines a general bar in front of this theory. that this thing here lives in a hidden space, which has a natural interpretation as a quantum three geometry. What about generativity, okay? And in fact, if you look at that, generativity doesn't seem to appear in any sort of way. In fact, if you think this is a completely natural thing you can write if you have two complexes, SU2 and SL2C. What is there is just a way of putting SU2 inside SL2C and choosing some representations of SL2C to write. Now, this is in the basis of group elements, through the space you can analyze group elements. You can go to this basis of screens that analyze area and volume, and we write the same expression in the pattern. It's just a change of basis, and it looks like that. So now you are expressing this in the interface in the basis of the JLB, the values of area volume. And inside, you can use the same variables, and there is a combinatorial factor, or a retrieval factor here, and all that we need is here. It's a product of vertices, the vertices of the two complex, of a vertex amplitude. So this vertex amplitude is the dynamics of the field, gives the dynamics of the field. The same sense in which the vertex amplitude of QD gives the dynamics of QD zone here. So we can stay here to see what sort of a dynamic it is, and look what it is. You have a vertex here around, you have the faces and the edges, so around, if you cut the thing, you have a graph, the boundary around the vertex, and this graph, you have J and V, so you have a three geometry. You have a three geometry, imagine a collision here, right, a set of collision, one here, one here, one here, one here, one here, that bound a four-dimensional region of space-time.

37:30 Now, think of regi. In regi theory, that's exactly the way you view generativity. It's a truncation of generativity where you have this four-dimensional synthesis, a geometry around, and you write the action in regi as the exponential of a sum, the same as the product, of what? Of a regi action which depends on the geometry of the boundary. So in regi, you can write, there is a regi, look at the regi paper and you find the action of a boundary geometry that is just a boundary attached to one another. So it turns out that if you just compute this thing here and study what happened to the large JVs, it took long, and the real theorem, which was proved essentially by people in the UK, but also some French people, like everybody here, that says that this thing here in the large JV is exponential of the radiation. So, in other words, this is a truncation of the product of things like that is the equivalent of a sum of things like that. So this in here is a sum over truncated four-dimensional geometries with a weight which is something that approximates the unit of action in the refinement link. So this is the first strong indication that we are talking about generality. the main one. This concludes the definition, the construction, the theory. So I can use some transition amplitude. I've shown you how to think about them in terms of topological quantum theory. I've argued that of the human space of the boundary described a quantum geometry, three-dimensional quantum geometry. And the attributes are, in fact, just a truncation of this thing here. So they were talking about quantum geometry. Now, three comments about the theory. Two short ones and one little bit long if I have time, but I've lost about time. But this is what I've done so far. Three comments. Divergencies. Now, the key point, okay, you can say, well, what have you done? Essentially, you've done lattice QCD for generality. No more than that in a sort of nice, pretty natural form that Lattice-QCD, Wilson-Hatcher also can be written in nice, pretty forms, and stuff like that. What have you done more? Well, what have you done more is that when I'm summing over the

40:00 geometries, okay, these intervals of the group, in fact, correspond to sum of the geometries which are naturally cut at a minimum scale. I showed you. There is a gap. So the degrees of freedom summed over don't include transplank and degrees of freedom. So therefore, there in theory. In fact, there are no underlying divergence in theory, right? There's this interval, there's the sum of a j, which can go to infinity, there is nothing that responds to the underlying divergence. If you compare it to an anonymous calculation of the probability theory, you see that the momentum interval sort of cuts at the Planck scale automatically. So that's the main result of the new structure. Here's a theory, which maybe is generativity, which talks about quantum geometry, and it has no underlying divergence. Does it mean there are no divergences at all? because there are infrared divergences. Infrared divergences come from the sum of J, the sum of the spins that go through . And in fact, they're there, and if you're computing, we've been computing for a while now, there are divergent things. And these divergent things are not attached to loops, associated to loops, like in standard final diagrams, they're associated to bubbles, because the spin, you're summing over one spin for each phase and in order to automatically get a consistency condition of edges, so the only way you can grow is if there's a bubble. And they have a nice interpretation if you think in terms of regimen calculus, they're spikes. See, imagine two-dimensional regimen calculus, you have a two-dimensional surface with triangulation here, you take one point and you move it very far away, the area of these phases here becomes arbitrarily large and that's the infrared How to deal with that? There are two ways. One, which is completely natural, and in fact, it has been sort of only initially worked out, but it's not yet known, is to do what Torai-Viego did for Ponsano-Renge, or Crain-Yetar did for Ongoi. Namely, we write the same thing, instead of SU2, SU2-Q-deformed, and going to use inappropriate deformation of SN2C. And this would cut these infrared divergences off. And what would it mean physically? Well, we know what happens in three dimensions, we sort of formally know what happened in four dimensions. There are both indirect arguments for that. Adding this Q corresponds to adding a cosmological constant.

42:30 If Q is the root of unity, the sum of the representation is cut. There's a maximum representation, okay? So which means here there's a maximum size, which you can't go too far there. And this maximum size correspond in fact physically to the, you know, the Einstein equation with modular constants in the solution of a certain radius, sort of a maximum dimension, the logic constant sort of a maximum dimension here. So that's one possibility which I like. Another possibility which is studied here in Paris in the Orsay group especially is to just try to re-normalize them away. This is the state of the matter here. Now, where are the scales of this theorem? What are the scales here? As I've defined it, and this formula that I gave you, there are no scales. Everything is just integral from AC2, SN2C, and some representation and characters. So everything is without scale. But in the moment in which I interpret this operator here as matrix, I need a scale. So in other words, there is a minimum area for instance, a minimum area here with a square root of three-half. I interpret this as an area. This is an area in some units, of course. So the theory comes with its own unit, which is the units in which this is that. Let me go and look for a moment at the scale of the theory, so that in centenars, I have to rewrite this or that. And how much is it? Well, So I can easily estimate it, because this theory can be derived from a number of different ways. I've just given it. It's forming from the sky. One way is to take generativity action, go through a canonical quantization, a main constraint, a more and more infrastructure. If I do that, there's a momentum per acre, so I want, I know the pass on bracket with a loop element, so I know that has to go to h bar, et cetera, et cetera. So I know exactly the constant here, and it turns out to be eight pi gamma G. Well, gamma is gamma at the dimension of the beginning, H bar. So GH bar is a prime constant. So it's essentially the prime constant times a number here. Why don't write directly a loop in this form? Because this, because I don't know how things scale. Even if there are no divergences from that fundamental scale of the theory to large distances, there are negative corrections. So I don't know if the G that comes at the Planck scale is the same G that I see at a

45:00 large distance. There will be radiative corrections between the two. So this naive relation between L-loop and L-plank might be affected by radiative corrections. This is just a parenthesis. But a closer parenthesis, and I forget about the relation with microscopic G, there is physical scale of the theory which is the size of the quantum space, the minimum quantum space if you want. If I use the Q-the-form version there is a cosmological constant which means that there is another dimensionless parameter. So if you want there's another dimension full parameter beside a loop there is also these two, this gives sort of the relation, the relation between the two. So there's a minimum scale and maximum scale if there is a, if I use the Q-the-form in the case. In all cases, there is also this dimension on this parameter gamma, which I mentioned before. Nothing else. Nothing else. So, dimension full, there's just this Planck scale, okay, which is intrinsic. Now, the comparison of this QCD becomes a little bit more clear. In QCD, on the lattice, you can find this here on the lattice, there's a clean definition of this here. What do you do? You take a lattice, you assume that the lattice spacing has a certain value in centimeters. So the lattice spacing enters indirectly, right, in the action because you have to scale the beta in the action and take A to zero. And then you have to take the limit. The continuum limit is defined by taking this A to zero and the lattice large. Okay, you have to go large and large lattices, of course, because the creation function becomes larger and larger. Here, there's no A. There's no lattice spacing because there's no background metric on the space. That's the entire game. The two-complex is like the lattice of lattice QCD, but without any notion of what is large, what is small. The notion comes when you put the quantum field over it, the j, the dynamical variable over it. This is naturally cut to some scale. So there's nothing to take corresponding to the lattice QCD A going to zero limit. The only limit is large termination. Okay, last comment before talking about calculations and applications, I don't, we lost some time to think, I don't probably outfly over it, the relation with matrix models, so this would take two transparencies, I'll be a little bit fast, and I apologize for those who get lost here, but I think I want to stay here, it's fast, but nevertheless, stay. So, stay with me at a couple of transparencies, this is generally, right, in the original

47:30 Now Palatini tells us you can rewrite exactly an equivalent thing by considering as an independent variable of the metric and the spin function. Kaplan and Bohlen use the tetrad formalism, and you can rewrite this in terms of tetrads and an SN2C spin connection. And this is how this thing appears in these variables here. This is a QCD action, and when you go to the quantum theory, you add to the QCD action terms, sort of highly violating term, which has no effect on the classical equation of motion, but it plays a role in the quantum theory, multiplied by a constant, theta QCD, which doesn't have no classical interpretation, but it affects the quantum theory. right? That's the famous one. Imagine you do the same here. So with this F here, there's an object to do here, you add a parity violating term with a constant in France. It's just the same without the star, like here. And that's the action we are talking about. And this gamma is a gamma, the words parameter, this gamma that was in the original formula at the beginning. These three parameters that float around in the theory, the real numbers. Now, this can be equivalently rewritten in this way by sort of dropping the star plus one of the gamma on this side here And look, this is like a BF theory where B has this peculiar component Now, BF theory is a theory that Google quantized when we tried to go from two-dimensional matrix model to four-dimensional matrix models So, generativity is a BF theory where, however, B has this particular form there. Keep this in mind and take Oguli. Oh, okay. B has this particular form here. Now, how do I write, what is the constraint that B has to satisfy in order to have this particular form here? It's this one. In some gauge, there's a little internal gauge fixing to be done, is the following. B is what? B is a two-form with value in the SA to CE algebra, the Lorentz algebra. So it has the same index structure like as usual. So it's a Lorentz generator. So I can split in a rotation and boost. Okay?

50:00 I choose a frame. Rotation and boost. And this is the condition that can be. If this condition is satisfied, then I cannot be in this form. So generativity is like BF theory will be satisfied. It's a fact. Now let's go back to Ogui 92 work, which I summarized in one slide, in one line here. What Ogui says is that take the F theory without this constraint, and formally quantize this theory by writing a functional integral, integrating in dB, which is a delta of the curvature, give meaning to this formal expression by instead of the connection, by choosing a two-complex, a triangulation, or more general two-complex, and instead of the connection putting group element, S into C group elements on the edge of the two-complex, and interpret this delta function as a delta of the allotomy around each phase. So this is a well-defined expression for the partition function of BF theory, written by And I can take this delta, expand in representation, and I get this expression here. Let me bring it to the next slide. obooly quantization of BF theory. Generativity is BF theory plus this relation here. Now, this is a character of SL2C in the representation PK, it's a continuous parameter, K is a discrete parameter in the representation. So it's a trace in this hidden space here, the carrier of SL2C representation. If I reduce this with respect to the reducible of the SL2 subgroup, it breaks up into this irreducible here. Now it turns out, it's a little lemma, that this relation is satisfied in this group of loyal states, in the weekly, in the Hilbert states, in a subset of these things here. Which ones? The ones where P is gamma K, so this P and K are related to one another with this gamma. Gamma is the limit parameter. and J is equal here just one component of this reduction here which shows actually why this trace converged here in this sense because it's an infinite measure of trace and you're just squeezing it down to find measure of trace. So just take this formula, reduce P to be gamma k and reduce this trace to be only on the subspace which can be done

52:30 by inserting this S2 interval it's just a trickery for implementing this reduction of the trace That's one of the main ways of getting to that expression. Just take a Bowie theory, which is quantization of BF, and restrict these traces to the subspace where the relation that reduced BF to GR volts. So this is an immediate family of the Bowie generalization of matrix modes. Why is this interesting? I mean, why is this just blah, blah, blah, blah, playing around with things? No, why is this interesting? Because what Oguli shows is that exactly in the light of the matrix models, this sams here can be obtained as Feynman intervals of an auxiliary field theory. Right? From the perturbative that you write in the auxiliary field theory, the matrix, you do the final expansion, you do the perturbative expansion, And the actual terms that you get are this traditional amplitude here. Now the same can be done here. In a wooing work, the auxiliary field theory is not a native theory. It's a field theory on a group. The narrow idea has worked a lot on these group field theories. And here you can do the same. You can write these things here which define the thing as final amplitudes in a perturbative of a genuine standard field theory, not on space-time, but on a group, which is an auxiliary field theory that generates that. Why is it interesting? Because now you have all the machinery of standard field theory to work on this. Now you can do all your, you know, you have learned about quantum field theory, you can apply to analyze this thing here. In particular, you can analyze whether you can analyze these things or simply how they scale. And you can study scaling, you can study everything on this. And this is what the Orsay group is doing with Rilasov and Deresky and other people, some are here, Merlach is here, others. We look at this as genuine Feynman diagrams in an auxiliary field which I have not written here. Okay, this was a little bit abstract. Maybe let me come down to Earth. And this is the last topic, just two slides. How to compute with this, what sort of computations have been done, and what sort of applications have been done. Just

55:00 two slides, one on calculations with the theory. Calculations which are so far only meant to reinforce the conjecture that the classical limit of generativity, no more than that. So one thing you can do is, well, in the perturbative theory, you can compute the denominator of the gravity here. Can you do the same? Yes, you just have to think a lot harder than we have to do it. Essentially, you take a boundary region of space-time and you put your x and y in the boundary here. And what you want is to, the point is this, This, the two-point functions, depends on the background, obviously. Okay? This is the two-point function of gravity over the encoste background. The theory here does not have a background. It's defined as background-dependent. So you have to tell the background to the theory. So you have to choose a state, a boundary state, in the hidden space that I gave you. You choose a translation of the field. You choose a hidden space that represents flatness, appropriately. And then you compare the amplitude of this state here and the amplitude of the state with insertion of two of these things so that we have to make it. So this is depending on the rate of the state here. So you do the calculation. It's very complicated. And what you can do is that where you can show this is a large J limit where these things are sufficiently far apart. You get exactly the graphical operator of the graphical. So essentially you get the one over the p squared scaling and you get the tensile structure for indices A, B, C, D here, correct. Now, I should be honest, this calculation here has been done in the computing system of this theory. In the Lorentzen system, people are working on that, but they have not seen . This technique, in principle, allows you to compute n-form functions to arbitrarily order. This is doing a perturbation function, And in principle, compute the corrections to the ones that affect the field series, right? So you should see the plants that are appearing under the caps. Another calculation that has been done in the technique is seeing that the spirit is completely different is you think about cosmology. You want a homogeneous cosmology, say close, is a proper homogeneous cosmology evolving to another isotropic in which is cosmology. So you can start in the theory, do a calculation, first order in just one vertex here, and compute the amplitude going from here to there, and

57:30 yes, in the minute you find essentially the freedom and appreciation. This is all motivated by the fact that we believe that this theory has generativity inside. In a sense, it has the gravitational wave terminology and has the basic cosmology inside. What can you learn Well, I mean, a lot of, there's a lot of literature about doing what for entropy using the kinematics of this theory, not the dynamics of this theory, the calculation that gives the equation formula and the entropy comes out finite. The interesting part is that, the idea here is that this microstate that generates entropy of just a quantum fluctuation of the gravitational field, which is sort of At a thermodynamical interpretation Just because there is a horizon To me the result is not Totally fully clear I don't think we have a clear explanation Why this path away could be Much more interesting is the Cosmology applications Now if you take basically The key physical result Here which is the discreteness of Air and volume And you plug it into standard quantum Cosmology, what you obtain literature of that, I just try to summarize the results in one slide, is that the Friedman equation is corrected by this factor here, where rho is energy density and the rho critical is just a constant which depends on the H bar G, the Planck scale, which means that there is a balance. Okay? Now, this is the context of homogeneous cosmology. So, Roger Penning is not going to be happy because he says, unless you go outside of homogeneous cosmology, you're not going to see what is interesting, which is the complicated geometric here, but in principle the theory is able to go around it. So far the strong result is this correction to the three-dimensional equation, which is the only hope so far to see physics, the observable physics out of here. There's also a correction that the acceleration equation which derives from that, which shows that A double dot can be positive in fact. So you can have a relation. Okay. So I've got to the end. Let me just summarize what I've said. I have given you a given theory by writing explicit transition and physics. I've argued that you can think of this as a general covariant one to presume in the sense

1:00:00 of Attia, and where instead of manifold cobordism, you work on graph two-complex cobordism, which you can think as manifold with defects. I'd argue that the human space that is defined on each boundary by this theory has a natural interpretation of quantum geometry, so there's a clean way of thinking about the geometries quantized into the smoke, whose basic ingredient is not a speed geometry theorem. This theory has no intervallic divergences, so it's natural in the form of the absence of intervallic divergences. Is this generative? I wish I had a theorem, I don't, of course, but there are a number of implications. One is that you can get to there from the generative A stronger one is that you can explicitly write the single vertex amplitude n is a discretization of the integral for two in the large j limit and you can sort of compute out of it the So in a sense it seems that the standard knowledge of generativity is there, so there's a number of indirect indications. About physics, the only physical result in court, because there's a number of approximations in it, that it's coming out. So far is this correction of the Friedman equation, which is of course far from being observed. So as a physical theory, this obviously has not yet any support. Thank you.