 Good morning, but today I'm going it's not working. I don't know. Yes. Yes. Okay today I'm going to discuss the relation between the entanglement entropy and the renormalization group flow. The idea is to show you that it's possible to prove C theorems in terms of entanglement entropy, okay, in two, three, and four dimensions. Okay, let's... No, this is not. Let's start defining C theorems. The basic idea is that the renormalization group gives you a systematic way to study changes in the physics when you change the energy scale or the distance scale. Perhaps the simplest way to think of this is if you put your system in a lattice and you want to see the changes in your theory when you change the lattice space, for example, okay. And, okay, this information at the end is encoded in the changes in the coupling constants of your theory. These are known as the beta functions. And then if you want to have a picture of the space of theories, perhaps the first step is to identify fixed points, okay, in this space of theories. In this fixed point, the theory looks the same at all scales. We say that these fixed points are scaling variance, okay, and then the renormalization group flow interpolates between fixed points, okay. But what is interesting is that, in fact, in the flow, not all critical points can be joined, okay, with each other. There are constraints provided, okay, by C theorems. So take this as a definition of what it is C theorem. This state that certain C charge decreases from the UV to the infrared fixed point, showing the irreversible character of the RG flow, okay. So here I have listed three conditions. This C function has to satisfy in order to have a C theorem, okay. The first one says that you need a quantity which is independent of the regularization you use, okay. This we usually say a universal quantity. The second requirement is that C has to be dimensionless and well-defined at the fixed points, finite at the fixed points. And the last condition is that C decreases along the renormalization group trajectories, okay. And okay, now let's see why we can say that any universal dimensionless decreasing function C will do the job will satisfy the three conditions there. If you write the first condition as an equation, this is what you get. This is telling you that C is independent of the regularization scheme you are using, okay. The total derivative of C with respect to tau, it has to be zero. The second condition C dimensionless can be written in this way. And now if you combine these two equations, what you get is this. That it says that changes in C when you change this here R, here is a characteristic length of your system, okay. For example, in the case of entanglement entropy for regions could be, I don't know, the length of your interval, okay, of the region. And what it says, this region is that changes in C, the way C change when you change the length of the system is proportional to the change in C when you change the coupling constants, okay. So if you find a quantity that is decreasing with the length of your system, you are done. You are sure you have these three conditions fulfilled, okay. So let's see how, yes, say it again. The function here is the chain rule. I mean, you have the total derivative. You can write it in terms of these partial derivatives and this gives you a function. Yes, but the point is that here, what you have here is you prove that from condition one and two, the changes in C due to changes in the length, characteristic length of your system are equivalent to the changes in C, changes in the coupling constants. This is the only thing you need. I am showing this because what we are going to show and also happens in the zomological proof. What he proves is that he finds a function C that is decreasing with a length, a characteristic length of the system. And then you can ask yourself, okay, why something that is decreasing with the characteristic length of my system is decreasing along the normalization group trajectory. And this is why, because C is dimensionless and is independent of the regularization scheme you are using. Okay, this is the motivation for showing you this relation. Yes, it should say tau here. It's, I mean, tau, it's the scale, the energy scale or the distance scale at your viewing your system is the characteristic length of your system. So in general, of course, you are knowing, you cannot, you can, this is showing you that both things are related. Okay. Perhaps it's redundant, it's a redundant way of writing it. But this is just to show you that many times people ask why if you have something which is decreasing with the length of your system, you have something that is decreasing along the normalization group trajectories. That that's why I'm explicitly showing this, this relation. Okay. So the first C theorem that appeared in the literature was due to samological in 1986 for two dimensions. And what he proved is that the C function at the fixed point for in two dimensions corresponds to the Virasoro central charge. And he also found the C function in terms of these two point correlators. This is the trace of the stress tensor. And using reflection positivity, he showed this function is, the derivative of this function is negative. Okay. The idea is that of course this, the samological of C function at the fixed points, okay, takes, is finite and is given a Virasoro central charge corresponding to the conformal field theory at the fixed point. Okay. This in a way, try again. Let's skip this. Again. And now let me, let me go to the entanglement entropy side and let's see if we can give a new prove or an alternative prove of the same theorem, but in terms of the entanglement entropy. For that, I will use these two properties of the entanglement entropy. The first one is, tells you that the entanglement entropy given two sets, okay, with the same causal domain of dependence, you have the same entanglement entropy. Okay. This is easy to understand because what you are doing is tracing over the degrees of freedom that lives outside the region. And of course, the complementary region of A and A prime is the same as the global state is the same. Of course, you get exactly the same result. Okay. This is a way due to causality. Okay. And then we are going to use a strong subadditivity that I have already mentioned yesterday or the day before. So these are the two basic ingredients I'm going to use. Okay. Let me go back or forward. Let's see how the proof can be done in terms of entanglement entropy. So the basic construction, you have to take two intervals which are both to each other. Okay. And you put the boundary on the light cone. And the idea is to use the strong subadditivity to get some relevant constraints or information, okay, about this quantity here. So if you see the, why I'm saying I'm using the independence of the Cauchy surface because I can rewrite these are the two regions I'm considering. The entanglement entropy of this region here is equivalent to the entanglement entropy of X, Y. The entanglement entropy of B is equivalent to the entanglement entropy of Y and Z. Here we have the union and here we have the intersection. Okay. And now if you take the limit, this, this is big R and this is a small R in the limit. Okay. Big R going to small R. What you get is this equation here that has second derivatives of the entanglement entropy and first derivatives. From here you can read this function C here, which has first derivative, negative first derivative. And you say, okay, so this C function, remember the requirements I have listed before. We need three things. Okay. First, we need a dimensionless quantity. You need a regularization independent scheme quantity. And the third one, you need something which is decreasing along the normalization group trajectories. And in fact, this C function defined like this satisfies the three requirements I mentioned before. Okay. And yes. Yes. Well, here, what we need is Lorentz invariance and we are using Lorentz invariance and we are using, you are assuming unitarity, but well, you need causality, I mean, to have this independence of the Cauchy surfaces. And you need Lorentz invariance to express the entropy as a function of the length of the interval. Okay. Otherwise, it's not true. But these are the two things I'm using. I'm using Lorentz invariance and causality. Well, I'm not using, if you want, in the use of independence on the Cauchy surfaces, you are implicitly saying that your theory is unitary, but causality. But it's not, there's no step here where you need unitarity. Okay. Yes. And okay. Then if you want to generalize, okay. Let's compare the, so now we have two C functions in two dimensions. We have the the entropic C function and the samological C function. Both gives you a, are equivalent in the sense that, yes. What? S of r. F? S of r. Okay. S of r. Yes. The, the, I mean, you know the entanglement entropy depends on the length of your r. It's just the length here and bigger here, the union. And small r is this length here. So the size of a is just the square root of the product of both. Okay. So here you have the entropy twice the entropy. The same, you are taking a and b have the same length. So you have twice the entropy here. And this is the, the union. Well, this is the union and this is the intersection. Okay. And the infinitesimal limit, you get, you get this expression for, for big r close to small r. Okay. And from here, you can read this function satisfies as negative derivative. So, so far we have, we have two. Well, something which is interesting is that this entropic C function at the fixed point gives you, it's again proportional to the bidazotto central charge. Since you know how to write the entropy at the conformal point. So this is the expression of the entropy at the conformal point. If you evaluate your C function at the conformal point, what you get is something which is proportional to the bidazotto central charge as in the samologic of proof. Okay. And here it's the way it looks, the, the C function for a Dirac, a real scalar and a Mascherana field. And then you can ask once you have two, two different C functions, you can ask, which is the difference if, is there a best one, are they related to each other? If you can deduce one from the other one. And the answer is that the, the entropic C functions in fact are not related to the samologic of C function. In, in what sense? The idea is, I mean, when we were trying to, to answer these questions, we didn't know that Capelli, Friedman, Friedan and La Torre in 1991, they, they already answered the, the, these questions. They didn't know about the, these entropic C functions, but what they, they, they, they show in this, in this paper is that in fact, once you have one C function, you have an infinite number of C functions that can be built in terms of this spectral density rho here. Okay. And the, the way you can construct any C function is using, okay, a numerical function that satisfies these conditions. So any function, any C function has, that has this form satisfies the three requirements I mentioned in the beginning. And, okay. So, so you have a huge family of C functions in two dimensions. And you can say, okay, this entropic C function belongs to this family or not. And the answer is no. It's completely different because, and one way to, to deduce this is you know the relation in an example, I mean, you can show this is not true because you know the relation between these spectral densities for a scalar and a Dirac field. And this relation implies that any C function which is, which belongs to this family satisfies the relation here. So, as this is negative, what you have is that the C function for a scalar has to be always above the C function for a Dirac field. And for the samological C function, this is true. Let me see. The first two curves corresponds to the samological C function and these two to the entropic C function. And in one case, okay, you have for a real scalar and for a Dirac field, which is the example I am giving you here. And for the case of the samological C function, what you have is that this one here is for a real scalar and this one is for a Dirac field. And in the case of the entropic C function, you have exactly the opposite. It appears first for the Dirac field and then for a scalar. So, you are sure that the entropic C function doesn't belong to the same family. So, it's a different constraint even if all both at the fixed point looks the same, okay, gives you the same or are proportionate at least. So, nothing. This is just a comment because we were, for a long time, we were trying to relate both C functions and at the end we realized that it's not going to be possible, okay, because are completely different constraints. So, now let's go to three dimensions and what we have learned in two dimensions is that it's very useful to put the boundary of your regions on the light on the null cone. And we also have a hint in three dimensions, we knew we have to use circles. And why? Because from holographic C theorems, from the f, the conjecture of f theorems and also there was a proposal by Liu and Mesay for a renormalized entanglement entropy always considering circles, okay. The constant term of the entanglement entropy of a circle was the candidate proposed in three previous papers. In fact, in this paper here, it's not the constant term of the entanglement entropy for a circle, but the free energy. But at the end, you can show this is exactly equivalent proposal of the, I mean, you can show that the free energy, to calculate the free energy, it's equivalent to calculate the entanglement entropy for circles, okay. So, we were sure the circles were the good candidates and we want also to use strong sub-additivity because it works into dimensions. We know the boundaries had to be put on the light cone, but the problem is that now two sets are not enough and the reason I'm saying two sets is not enough because intersections of circles and unions of circles are not circles. So, and why you need the same kind of regions on both sides of the inequality because you need to cancel diversion terms. Otherwise, the inequality is trivial inequality gives you no information, okay. So, the solution we thought is that instead of taking two, we can take an infinite number of circles. This is the planar construction, okay. You take circles, rotate it, an angle two pi k over n around a point, of course, different from the center and in the infinite and limit, the sets you get looks like circles, okay, centered at the same point. So, you say, okay, I can use, here I have written the strong sub-additivity inequality for three sets and, okay, it works in the sense that you have the same number of regions on both sides of the inequality, but the problem is that in this planar construction, you see this wiggle region has casps, okay, and we know that in three dimensions for circles, we don't have logarithmic contributions. So, in this part, the left hand side of the inequality, you don't have log contributions in the entropy, but you have log contributions to the entropy in the right hand side of the inequality, okay. So, it's not going to work because you cannot cancel the infinite on both sides of the inequality. So, the solution, go to the light cone, because as you approach the light cone, the angles, okay, go to pi and also the perimeters, even if at first sight you say you believe that in planar construction, this wiggle region has the same perimeter as a circle, it's not true. And, okay, you can prove that if you put your construction on the light cone, the limit, the infinite and limit is enough to, in this limit, the entropy of this wiggle region corresponds to the entropy of circles, okay, it has the good limit. And, okay, this is the expression you get for once you take the infinite and limit, and again, you take the infinitesimal limit, big R, go into small R, and you get this inequality now. You get a restriction for the second derivative of the entropy. This tells you the entropy is a concave function, okay. And again, from this inequality, you can read a C function, an interpolating function with, which is decreasing, okay, with first derivative negative. And, as you know, the expression of the entropy at fixed points, if you evaluate your interpolating function at the fixed point, what you get is exactly the constant term in the entropy. But now in, okay, remember we have three requirements, so we have already a decreasing dimensionless and decreasing function, but still you say, okay, this quantity at the fixed point is or not well defined. And the problem is that we know that the area term in the entanglement entropy is not universal, I mean, depends on the regularization you use. It's easy to see that if you change your definition of R in the lattice, for example, you can change the value of the constant term. And again, you can find a solution, you can find a solution introducing the mutual information, which is a well defined quantity in the quantum limit. And the idea is that you can define a kind of, I don't know, you can use this mutual information as a geometry regulator for the entanglement entropy. The idea is that you take these two regions and you take that in the limit when epsilon goes to zero, the mutual information is just twice the entanglement entropy of the circle, okay? Because, okay, the union is almost everything, so the entropy is zero and the entropy of A becomes equal to the entropy of B, okay? Here we are using that the entropy of the complementary region is equal to the entropy of the region if you start with a pure global state. So you have a way to define the entanglement entropy in terms of mutual information and then you are safe saying that you have a good definition of the quantity at the fixed points, okay? And, okay, now you say, okay, you can go to higher dimensions, you can go to four dimensions, just taking now spheres, okay? Boosted spheres and putting these spheres on the light cone. But before, which are the proposals done by other people, for higher dimensions, you have these the early result by Cardin in 1988. He proposed the coefficient of the Euler density term in the trace anomaly at the fixed point as a candidate for a C theorem, but just for even dimensions, in odd dimensions, you don't have such anomaly. Okay, for odd dimensions, you have these holographic C theorems that tells you these are very general because they have also not only a proposal for odd dimensions, but also for even dimensions. This is the same as Cardin's proposal. But for odd dimensions, what they say is the constant term of the sphere entanglement entropy, the good candidate for C theorems. And also we have the F theorem propose final term in the free energy, okay, of a three sphere decreases between fixed points on the renormalization group flow. And in fact, they show that in many cases that it works, but they didn't have the proof. So let's skip this part. And okay, let's go to four dimensions. Okay, if you try the natural generalization, as I said, is to take symmetric configurations of boosted spheres in the limit of a large number of spheres. And so you do exactly the same thing. You apply strong subadditivity. And what you get for any dimensions is this expression here. This is the general expression you get using this construction, okay? And okay, in any dimension, the only problem is that this is true only if you can replace the entropy of these wiggly spheres by spheres, okay? That in three dimensions was true, but you have to prove that in any dimensions this is true also, okay? So this if, in a way, means that you have to answer these questions here, okay? Does the inequality contain cutoff independent information? This means diversion terms cancel between the two sides of the inequality. The second question is whether the wiggly spheres entropy can be related to sphere entropies. And the last is, okay, suppose everything goes okay and you can replace entropies of wiggly spheres by entropies of spheres. Still, you have to make sure that the inequality teaches you something about central charges at the fixed point, in the sense that if the inequality is strong enough to reach the log coefficient in the entanglement entropy. Since we know that the strong subadditivity at most gives you second derivatives of the entanglement entropy. So you naively in four dimensions use this inequality and you evaluate for a fixed point, you know the expression for the entanglement entropy. What you get is that the inequality is incorrect, okay? You get the wrong sign. So, evidently, this replacement of entanglement entropy for wiggle spheres by entropies of a sphere was incorrect. We spend a lot of time trying to see if the angles in these in four dimensions gives you log contributions, okay? And geometrically, you can answer this problem and the answer is that there's no log contributions coming from the angles in four dimensions. It's hard to prove it, but still you can do it. So, the problem was not coming from the angles, but the answer is that the problem is that these wiggle spheres have gives you a finite contribution that is not canceled in the inequality. So, the solution for this is to use something called the Markov property. We are going to use the Markov property of the vacuum state and the statement is the following. The strong subadditivity for the vacuum state saturates when the boundary of your region lives on the null plane, okay? This is true for any quantum field theory and in particular, if you have conformal symmetries, this is true also in the light cone. So, what you can do at no cost is to insert in your inequality these zeros, okay? This relation, I mean, for the entanglement entropy, the strong subadditivity when it's applied in the UV is Markovian, okay? And then it's, you can introduce in your inequality these zeros and in order to define these new quantities. So, you are going to introduce this quantity that is just the entropy minus the entropy of the UV, okay? And you can do it, you can replace without changing anything else this quantity here in your inequality since the vacuum on the light cone is Markovian, okay? So, you are inserting zeros in your inequality and the good news is that now these new quantities have the good limit in the infinite end limit. So, these new quantities goes to, from wiggly regions to spherical regions, okay? Have the good limit. Then you can replace in your inequality here, it should say S for wiggly regions, but you can write S for spheres, okay? And, okay, this is the solution of the problem and what you get is this inequality here that tells you that the coefficient, the logarithmic coefficient in the infrared has to be smaller than the coefficient in the UV. So, this is what we know as the A theorem. In general, perhaps this is better to write this general expression here in terms of the entropy of the area and this tells you simply that the second derivative is negative. So, it tells you that this new quantity that has the entropy that has subtracted the entropy of the UV is a concave function and you can rewrite this expression in terms of the area and just gives you that the second derivative of this quantity is negative, okay? So, in this way, you put, if you want on the same footing the three proof, okay? For two dimensions, you get the C function in three dimensions, you get the F theorem in four dimensions, you get the A theorem, okay? And the three of them can be treated in the same way, just putting your sets on the light cone and applying strong sub-additivity. These are the ingredients we are using for proving the three theorems, okay? Then you can ask if we can go to six or five dimensions and the answer is no. And the problem is that, okay, everything works okay, but the strong sub-additivity is not strong enough. Second derivatives of the entanglement entropy are not enough to reach the log coefficient. So, it gives you an area theorem if you want. You think that the second derivative of the entropy in terms of the area is concave, you have two restrictions, okay? I mean, the slopes of the curve in the infrared has to be smaller than the slope of the curve in the UB. And you have also a restriction for the height of the tangent line at the infrared. It has to be positive. This means your curve is concave. And these two restrictions are not enough. I mean, it tells you nothing about the logarithmic coefficient in more dimensions. So, this is, we need more inequalities or or something is missing. Go farther.