loop quantum gravity overview

Thank you. Thank you very much. Not being myself is a strengthier. I'm very honored being invited here, and I thank you very much for being here and listening. I will talk about loop quantum gravity, and I will assume that most of the audience here know very little about this search program. So what I'm going to do is to give you an overview. With the main ideas, the main results, and just a few words about the main current lines of investigation. What I'm going to do is to start by discussing exactly what is a problem that is addressed. And introduction to a multivator explained what are the loops using here and why, which I will use also to introduce some technical notions like the spin network basis that just happens later. And then the main part of the talk I'll describe the theory. First, the kinematics and then the dynamics think as if I was in the dozing QED by first giving the fox base of the electronic photons, the relational operators, and then the QED vertex for the dynamics. And then I will list some of the main applications. And in particular, I hope I will have some time at the end to talk about the recent calculations of end point functions, which I used to test the low energy limit of the theory. So what is the superiority is a research program that has been developing now for about more than 20 years. There are roughly 200 people and many books allow me to advertise my own book. What is a problem which is addressed here? The problem is how to describe the elementary degrees of freedom of a quantum field theory when there is no background space time. The way this problem is addressed is a few hypothesis. So let me put this up front first that a radical conceptual change in the way we treat space and time needed. But on the other hand, that this problem can be addressed within a current, within a context of current physical theories, a generativity plus a standard model of some small extension of the standard model. And the reason is that because this problem already is there, when you try to define a high energy, a quantum field theory of any theory which includes generativity, as I will discuss. Now of course, this goes together with another hypothesis, which is an old idea that the bad divergence of the theory is not like the ones of the old Fermi theory of weak interaction, a sign of sickness of the theory by itself. But rather, indication that the perturbative theory is not correct in this context, the perturbation around a smooth solution. It's wrong because it introduces, it's wrong back, it introduces in the theory, some makes that gives a freedom which are not there at high energy. And this, I will show a posterior is confirmed within the loop approach, where one sees that in some sense transplants and degrees of freedom are not there in the normative theory. Now the guiding principle, like in so much of model physics, symmetry, and it's the symmetry and the dephomorphism group, the action of the dephomorphism group, the active action of the dephomorphism group on the fields of the theory, which I'll discuss in the next transparency. Consistency with quantum mechanics and generativity and having a fully different motion invariant theory, explicitly, if a motion invariant theory, gives very strong constraints on the theory. The difficulty here is fine one theory that has this property, not fine which theory that has this property. So for the moment the problem is finding one. The main result, which I hope to illustrate is the definition of a theory or more precisely a framework for defining theories, both in the canonical and in the covariant form, a quantum theory that has the property of being variant and the dephomorphism. Before the beginning, let me make a comment about generativity, because this is a source of misunderstanding of the communication between different communities. In a sense, generativity is two different theories. First it is a specific theory for one field of this action, which is of course well supported in purely collet law energy, but we don't know if I, I energy is corrected and how. So there's nothing wholly about this particular action of generativity from this point of view. However generativity is also something else. It's a modification of the way we understand space and time. Of course the idea of the discovery of Einstein is that space time is the same thing as a gravitation of field. I like to say there is no space time. There is only a gravitation of field. So in a sense we have moved with generativity from a picture of the world in which the fields live on a given fixed meaning. On a given fixed matrix, it's a matrix, which the fields live on top of one another. And the problem of doing quantum gravity is I think the problem of understanding how to do that in the context of quantum theory is solution to how to do that in the problem of classical theory is in fact the one provided by Einstein in 1950. Now in the theory this modification is expressed by the invariant and the theory under the actual, if the surface group, the field of the final manifold, but then the theories invariant and arbitrary is most displacement of this field. So in a sense the manifold, the physical interpretation and manifold is washed away by the invariant and surface theory. Now let me remind you I think that most of you are more or less familiar with old ideas about non-parturbitivity quantum gravity in the canonical framework of the wheeler, the wheat or the covariant framework sort of most associated to the name of Oking. Here you have a function of a geometry invariant and if you are all feasible, will you have an integration here you have an integral of O4 geometries with a without an eye. Now there is a beautiful idea that has inspired much work but they have never worked by themselves for defining a theory one can compute things. If you try to compute there you don't get anything or if you expand around some background you get back all the divided divergences. So in a sense you can view loop quantum gravity as a way of doing all that to a level precision where you can start computing and getting numbers out of these expressions. So why loops? Loops are a old idea in physics of course and there is a long list of major people, some of whom are here who have sort of argued for the idea of expressing age theory in terms of loops. In a sense it can be traced back to faraday, right? Faraday introduced field theory as a convalence of lines in space and then the electric field is just a tangent of the line of the faraday line in one point. Now if there are no charges the lines may close, these are the loops and the gravitational generativity also can be expressed in terms of a gauge like connection and electric field. So the gravitational closed faraday lines and these are the loops we are talking about. Now question can we describe a quantum field theory in terms of these loops? Well there is also yes to some extent and let me again take you back to physics of the 70. Take a young mill theory on the lattice in the canonical picture. So three dimensional lattice this is as far as you keep the lattice space finite it is perfectly well defined equal to field theory. So if you see the states and the states operators, dynamics and so on. Now in this context you can define a state by taking the product of the loop variables which are the main variables of the of the theory along a loop loop in the lattice. And this state here is well defined finite norm state in the the heapspace of the theory has a proper of being an eigenstate of the electric field. As is simple, where the eigenvalue has zero everywhere it has support on the loop itself. So it is literally a quantum excitation a single quantum excitation of a faraday line. Now these loops this loop states by themselves do not form a basis in this field space but a simple generalization of the same object does. And this is a so called spin networks you take a spin network is a graph on the on the lattice here. Colored with some representations on the graph links and representation of the gauge group and some intertwiners. Some invariant tensor in the associators and nodes. Now for each one of these object you can construct a state basically by contracting the variance tensor representation elements. So it is again have the same properties so the eigenstate of the electric field where the eigenvalues concentrated on the graph and they are basis of not a normal basis in the in the Hilbert space. So as far as you are in the lattice you can transform everything to this basis and represent entirely young mil theory in terms of this faraday lines and operator acting on them. Now can you do the same on the continuum. The answer is no essentially because these are two singular if you move if you use a minitase in one of these loops in the continuum you get. You assume this is a not a normal basis you get a you know a diagonal state and if you take this is the definition of a real space you get you define something much larger than this Hilbert space that you want.