 It's working. First of all, thank you, the organizers for the opportunity to speak here. Well, I'm quite more or less marginally in long-range stuff. But although I did a few, few works, but I'm hoping here to talk about some concepts that might be useful, or maybe some people they might get interested and think about it, especially people that they do quantum chains and stuff. But there are some surprising connections to also some more classical oriented people that they work on the Casimir effect. These are works that I've been, this is the concept that I've been working on that in last year, since the year and a half. And I decided to put a lot of stuff here. It's quite dangerous things to talk about all of these things. But somehow, since there are some connections among all these concepts, I think it's good to see also the connections. But my main effort will be just to say that there are some definitions and concepts that they can be really useful. And well, there will be also a few formulas that to just say that things really work. So I will start with formation probabilities. The definition of formation probabilities in quantum critical chains. Then I will discuss about formation probabilities in conformal theory, how we can really calculate this stuff for critical systems. And then there is an obvious connection between Casimir energy and formation probabilities that when I found out that there is this connection, I was quite surprised why it was missed by people that they used to work in these kind of things. And then a few numerical results to show that the things works. And then at the very end, I might just flash out another concept, which I call it post-measurement entanglement entropy. Then conclusions. Well, the formation probabilities, I define it like this. Take the ground state of a spin chain and write it in a particular basis. Well, of course, the ground state of the spin chain will be a summation of all the possible configurations. So you have an exponential number of configurations in that particular basis that you are writing your ground state. And then every configuration will come out with some probability. Very basic things in quantum mechanics. So you can imagine that you have a spin chain. And then you just say that I'm going to see that what's the probability of having string of spins up. Or you might be interested in another pattern, another configurations. But OK, these guys are going to come with some probabilities. And this probability is what I call these formation probabilities. And the special case of these formation probabilities has been studied in the integrable community for a long time, actually, for almost three decades. There are a lot of people that they contributed to these studies. But I started to get more interested when I realized that one of the results of these guys was that if you calculate the formation probability, for example, for an easing model, then this is actually a logarithmic formation process, the logarithm of the formation probability. This guy will behave linearly with the size of the subsystem. So here, the quantity I'm interested is that you have an infinite spin chain. And then you take part of the system. And then you say that when I'm increasing the size of subsystem, how this probability is going to change. And of course, in most of the cases, it should decay exponentially because my Hilbert space is exponential. So this guy will be linear. But there will be another term when you are at the critical point, which looks like I'm really sorry, this L and S are actually the same thing. So there will be also a term which depends logarithmically to the size of the subsystem. When you have a logarithm, you expect always that this coefficient of the logarithm be a universal quantity. So the universality comes from this point. Well, there was a study by Jean-Marie Stéphane a couple of years ago. He studied this quantity in a generic spin chain. And he was able to show that actually when you study this guy in a quantum system, in a Euclidean version, you can think about actually having a slate here and try to just find this probability like that the probability will be actually the partition function of this system, two-dimensional classical system with this slate over the partition function of the total system. Well, this kind of partition functions are some kind of quantities that we can actually control in a field theory, and especially in conformity field theory. So it turns out that the coefficients are going to be a central charge. Well, for those that are not familiar with the central charge, when we have a central charge somewhere, that means that we have now an idea that what's going to be the universality class of the quantum critical system that you have. So we get excited to see the central charge. And well, I think those that are familiar with entanglement entropy, they might realize that there also we have usually a central charge when we study the quantum critical system. Well, this one. OK. OK, now it works. Well, good question. So the point is when you fix, first of all, you say that I'm interested to calculate the formation probability for a particular observable. So you first choose your observable. For example, in this pin chain, you might say I'm interested in sigma z. In the imaginary experiment, you want to see that the probability is that you want to see that the probability of these guys are going to be up in the sigma z basis. And then you also ask about what configuration you are interested. So these two things, choosing the observable and the configuration that you start, they are going to fix a particular boundary condition here. But you need to figure out which configurations and which observable are going to be conformally invariant. You induce the conformally invariant boundary condition here. This is a very difficult question. We have control on particular systems. But I think it's something that, a priori, it's very difficult to know about it. You know, it's something that, well, it's an exponential number of configurations. You need to deal with the exponential number of configurations. That's not an easy question. Well, then, apparently, not necessarily related studies, people have been studying cash energy and cash forces on two-dimensional critical systems for many years. And there are plenty of things are known here. But my interest started by the work of these two guys, these two papers. And what they were claiming was that if you really induce some sort of, if you put two objects in a critical medium, these two objects are going to impact. There will be a cash force among them. And in principle, if they are at the critical point, and if the objects that you are putting here, which means that the boundary conditions that you are forcing here are conformally invariant, you can calculate this stuff using conformal theory techniques. So the cash energy is going to have two parts. One part is the geometrical part that you can derive using this equation. These are some technical quantities that I don't want to spend much time on that. But for those that are interested, this is a Schwarz derivative. It's quite famous in conformal maths techniques. And then these are F-annulus, which means that these are actually the free energy of the conformal theory on the annulus. So what you do is that you have this medium, this medium you map it to the annulus. So this guy goes here, and this guy goes here. So in principle, the partition function of this guy is going to be the partition function of this annulus. Plus another term which comes from the conformal mapping. So this is a technical thing. This is something that is already known. But somehow, probably now it's obvious that this guy should be related to formation probabilities. Because as I said, in the formation probabilities, you also have some kind of sleet-like things. If you think about, for example, the formation probability of two disjoint intervals here and here, what you need to do is actually finding the partition function of this guy without these two sleets. And this is kind of a Casimir energy. So this is the main part of the talk. So what the quantity that is going to be exactly like a Casimir energy is this quantity, which this P is the formation probability of these two disjoint intervals. This PCA is the formation probability of this guy and this without having this guy. And this PCB is the formation probability of this guy without having this guy. And these things can be written with respect to some partition functions. So this ZAB is actually the partition function of this guy. And then ZA is the partition function of all of these guys without having this sleet. And ZB is the partition function of all guys without having this sleet. And Z is the partition function of the full system without having any sleet. So these are, as I said, technical things, but it's kind of obvious that there is a relation between formation probabilities in the critical system and the Casimir energy of two floating objects. But here, the floating objects are going to be two needles. These are quantities that people have been studying in the Casimir energy. So for the first exercise, what I did was like, OK, let's calculate the Casimir energy of these two guys and try to see what's the formation probability. So to calculate the formation probabilities, I just started to use the techniques now in the Casimir studies. So somehow, this talk will be more or less like a glorification of the Casimir energy. Well, to just tell you that some long-range things also appear, if you put these two guys a bit far from each other, the Casimir energy will decay like this, which means that your, and this delta one is the smallest scaling dimension present in the system. It's a technical thing, but if you like, you can think that this is going to be the scaling exponent in the easing model. It's going to be the scaling exponent of the energy, or there's a scaling exponent of the magnetization. So if you have a critical system with different kind of exponents, you can actually play and find different kind of long-range Casimir forces. And this is also something that already known for a long time in the Casimir energy studies. Well, I hope that these horrible equations, they don't scare you, just emphasizing on the relations. Well, you can try to check these things in a simple critical spin chain, which is the xy chain. You pick the sigma z, you say that, OK, I want to calculate the Casimir energy of this, sorry, the formation probability of two intervals, these two intervals, and see that these equations are really working or not. Here, I'm showing the result for the easing chain when the gamma is actually 1, and then you pick the sigma z. And you look to the configurations, all up, or all down, or up, down, up, down, up, down. So they are all just sitting on top of each other more or less. And this is the analytical result of the dashed red line. It's pretty surprising, because the equation that I had was pretty complicated. In this case, I have the xx chain. In the xx chain, I'm taking the configuration up, down, up, down, up, down, up, down in the sigma z basis. And I'm looking at the probability of occurrence of this configuration. And this is the analytical results, and the points are actually the numerical results. One good thing is that the benefit that these things can have also for the Casimir studies is that, at least in this model that I'm studying, you can really study these things for almost indefinite sizes. If you tell me that I calculated for 1,000 or 2,000 sizes, you can really do these things. But these kind of calculations in really classical systems is almost quite nightmare numerically. So this is something that actually helps also to see that these Casimir things are really working numerically or not. Well, another benefit of this is that, well, I'm going to show another example here. It's an easing model, which you put the guy inside. One of the benefits of this kind of studies is that, as I was telling you, this guy is the smallest scaling dimension that is appearing in the spectrum of this critical system. So when you are interested to study the quantum phase and the quantum critical point, you want to know what's the universality class. The first thing that we usually want to know is the central charge. And as I showed you, it appears already here in one interval. So you can really fix part of the things here to know that what's the universality class of the system. But if you want to know further to know that what's the spectrum, for example, in the easing model, we know that we have two critical exponents. We have the energy exponents and the magnetization exponents. So the central charge is not enough. You usually need to know also the spectrum. And this spectrum also can be seen here because these guys, they have the information about the spectrum of the system. So by calculating one interval formation probabilities and two different interval formation probabilities, you can actually fix your universality class. So this is something that, in parallel to the studies of the entanglement entropy, because a lot of people, they are interested in calculating entanglement entropy because you can actually know about the structure of your universality class. But this is a different quantity. It's not as complicated to define at least as the entanglement entropy, but it gives the same information. It has also, there is also another thing that was motivating me to study this kind of things. And that's Shannon information. There is a quantity that you can actually calculate numerically mostly. And it's, well, it's the Shannon information. And in the Shannon information, remember, I had this ground state with these probabilities. So you take all these probabilities and put in the Shannon information. And I mean, the definition of entropiedos. And then surprisingly, you realize that you have a volume law, but with the extra logarithmic behavior, which looks like, again, the coefficient of the logarithm is dependent on the central charge. So this is another quantity that shows the central charge. You can calculate the central charge out of it. Well, we have been trying. We checked this stuff in many models. And we also tried to persuade some experimentalists to calculate these things, but we haven't succeeded yet. But anyway, the final things that I want to say is that another quantity that I've been studying recently, and that's post-measurement entanglement entropy. So this is something that I'm interested because it's more like a tripartite system. Imagine that you have a periodic quantum system. It doesn't need to be periodic quantum system. What you do is that you pick a part of this guy, like A, and then you make a projective measurement here. So when you do a projective measurement here, there is no dynamic. Forget about the dynamic. So you have a ground state. You make a projective measurement in part of the system. And then the rest of the system collapses to another wave function. And when it collapses to another wave function, the question that I'm interested is, what's the entanglement of this guy with respect to the rest? Because this guy is already out. So these kind of things are also interesting when you are studying the tripartite systems. It appears also in a concept. Some things, it's part of another concept, which is called localizable entanglement. And that's one of the reasons I'm studying. So these kind of questions are things that also related to what I was telling before, because you need to have here configurations with respect to the conformal symmetry of the system to be able to calculate these kind of things. And so I think I'm going to run out of that. OK, there are some interesting setups here that you can think. For example, you can actually do the projective measurement here, and also here, and then ask about entanglement in the way of this guy with the other guys. This is interesting because somehow you make the projective measurement, and you end up two completely decoupled guys that they are not especially near each other, but they are quite entangled. So you can really make some calculations regarding this thing. So what I was trying to say was that using formation properties, one can fix the universality class. Formation properties are intimately related to the Casimir energy, and Shannon information can also fix the universality class. And this post-measurement entanglement, I didn't show how one can calculate also the post-measurement entanglement entropy in field theories using the Casimir energy. But just these kind of calculations are also doable by just using that formula of Casimir energy that I told you before. So everything boils down to calculating the Casimir energy. Thank you.