 So I'm going to continue talking about fractals. And so these are points which we covered yesterday, just a recap. And also I have corrected my slides. I hope things which we did not see can see today. This business about scaling only gives one possible solution in the self-similar, just y equal to x. And in the self-affined, just a power of x. Which is not interesting. So the answer is that the function has to be non-differentiable. Only for if you have a regular and a smooth function, self-similarity will give you nothing interesting. But if you have a non-smooth function, and the mother of all non-smooth functions is this wire stress function, which you can actually construct using the Fourier series, which is why it is so interesting. It is continuous everywhere, differentiable nowhere. And as a result, it looks just like itself. So if you take a tip of it, magnify it, it actually looks like the whole function. And this has a scaling property. This is how the wire stress function is defined. And you see that A and B have to be bigger than something to make this series convergent. But apart from that, if you multiply this by an A here, you get A to the power n plus 1. And then I can multiply y from the left by B. And that becomes B to the power n plus 1. And because this is an infinite sum, I just redefine n. And it will look exactly the same. So I have this property that B by Ax is just y. So it has a scaling property. You can choose A and B, provided they satisfy this condition. And it's actually not easy. You can see this paper is 1998, whereas wire stress invented this function, I think, in the 19th century, if I'm not wrong. And the fractal dimension eventually was calculated in this late time. And this is the fractal dimension of this curve. And it is smaller than 2, because A and B are organized such that A is a little smaller than 1. And B is bigger than 1. So this will give you a fractal dimension smaller than 1. And you see that you have to deal with very rough functions to accommodate a scaling variance. Look, what I do is that I multiply x by an A. So it becomes A to the power n plus 1. Right? And then I multiply y by a B. So this becomes B to the power n plus 1. Then we'll redefine n to n minus 1. Some will look the same. So the right-hand side will be y. Left-hand side will be B, y, A, x. Yeah? OK. Yes, but OK. But yes, you are right. But those are very small in the sense of the whole function. It's just constant. So the zero changes you are saying. Yes. The zero changes, so the zero value. The zero value will be just the cos x, which is a completely regular function. It has no effect on an irregular function. So it's a completely differentiable regular function. Pardon? It comes from requirement for convergence. I cannot give you a proof now, but I'm sure that this condition, well, it's not just one, which you might have on the first instance. OK. The next point is that you have some other types of fractals, such as this guy. This is a diffusion-limited aggregate. This is a DLA. And the way this sort of shape is formed is that you take a seed, and then you let another particle come in from infinity by a random walk. So this was there in this picture. My pictures are not very good. And then this guy moves up until it hits the existing cell. Then it sticks there and doesn't move. And then another particle comes from infinity, and as soon as it finds the existing cluster, it sticks to it. And it just grows like that by aggregation of many particles. This system is called diffusion-limited aggregate. I want to put some applications on this to show. The problem is that this doesn't have a battery, it must always be connected to electricity. And this is clearly a fractal, and it's kind of fractal you see in nature a lot. And its fractal dimension is 1.539, something. The fractal dimension of this shape depends very much on the sticking probability. So this particle comes in from infinity. It's the first point. Then it may or may not stick. If you can have a sticking probability equal to 1, it means it has to stick. Or you can have a sticking probability between 0 and 1 so that it may or may not stick. This shape is s equal to 1, and it has the lowest fractal dimension. This is 1.5. If you reduce s, this will have a higher fractal dimension. Anyhow, this is a very short introduction to DLA. The question is, how can I calculate the fractal dimension of such a shape? The reason I go into this in detail is because this method is interesting. So what I can do is that I have a circle around the origin and I look at how many small circles I need to cover the whole shape. The number of small circles I need to cover the shape obviously grows with R. R is the radius of this circle. As I make R bigger, I need more circles to cover it. And the number of circles which I need to cover it when R is very large is clearly related to the fractal dimension because that's exactly what the definition of fractal dimension is. On the other hand, if I make the test circles smaller, that means the A is the radius of the test circle. The number of test circles you need to cover the entirety of this shape grows keeping big R fixed. Of course, you cannot make the smaller circle smaller than the particle size, the particle which is coming from the... but we are always much bigger than the particle size. So you can see that this... the number of circles you need is somehow connected to R divided by A to the power F. Next is that... suppose this... I do that for the center, but then the center of this shape is obvious. I want a method which is independent of the center. I may have a shape with an unclear center. So what I do instead, I say that I choose some particle randomly, such as this particle. Then I look at the density of particles around it. So the probability that I have a particle at X is rho of X density of particles. So the number of circles now I need to cover this shape is related to the joint probability of a particle here and a particle distance R, which is just that. However, now this center is chosen randomly. I must integrate over the entire space and then I get this integral. And if I do that, then I have this power of R to the minus 2 coming in because over the whole space, the density just has to fall with 1 over R squared as you go further and further away. Hence, I can calculate... If you can calculate the correlation function as a function of R, then you observe that it must fall with R in this way. Of course, the problem is if I can numerically calculate the correlation of a shape, then it may not be a scaling behavior as we have here. This only comes out if already you have some idea that the shape is a scaling variant. Otherwise, you would get the standard e to the power minus R over Xe. So this is the expected form of a correlation with a correlation length. That comes out only if you have a scaling variant shape. It's ready. Thank you very much. Just one minute. So now the point is that I can shut my eyes to what I'm calculating the fractal dimension. I can use this application to calculate any fractal dimension of any shape. For example, a photo. So if you apply it to this photo, you get fractal dimension of 1.334. However, you can further use this method to do something interesting, which has been done in this, and we did it for our paper here. Any image, any two-dimensional image is really a photograph of a three-dimensional entity. So you have had an object somewhere. You took a photo, and the photo is two-dimensional. The question is, can you project from the photo, can you project back and calculate the fractal dimension of the three-dimensional entity? This you can do if you make assumptions about the shadowing. So you have to make some assumptions about the shadowing. That is, in this picture, for example, all the grays are the gray colors of the original lady. Some of the grays are because some parts are closer, some points are closer to the lens than the others. So you can reconstruct a three-dimensional object and work out the three-dimensional fractal dimension. It's a terrible sentence, I'm sorry. If you have a fractal which lives in the space, something like this, it's actually a fractal with a dimension within two and three. But when you take a photograph of it, you will see just the two-dimensional object. So you must somehow be able to project out from this to a three-dimensional entity and calculate the fractal dimension, which is done by the same method, except that you have to add this gray scaling to the previous calculation as well. And for her, this comes out to be 2.867. Yes. I tell you something which usually is to think about it. I don't want to give a proof because I cannot. When you look at an image, so humans are able to do this. We don't know how. You immediately assign a scale to the image that you see. So the real sizes of the image are not relevant. You assign a scale to it. So if you scan this image around you, many things are happening and it's a completely jumbled-up picture. Many things. It's just a picture of Einstein on the back wall and then students sitting on the front row. So some play with a scale is happening in the images we see. There have been many papers claiming that a conformal transformation is happening in our brain. Neurons already do a conformal transformation on the image. So that's all that I can say. Otherwise, it's just a game. If you don't want to buy that story, it's just a game. There is no self-similarity. You just apply a method where I shouldn't have. So the next step is that I have some fractal curves which I want to calculate the dimension of, which I have all these traditional methods which I discussed, which are fine, but I would like to use a completely different tool and that is conformal field theory. So I'm going to go slowly through conformal field theory to the extent that we need for our purposes in the next part of my talk. However, before I do that, I have asked Nahid to lend me her laptop in order to show you all these things which I failed to show on these simulations. So could you help, please? Okay. So some things which we should have seen earlier. Here is a Brownian motion generated on a lattice. Lattice is fine enough so that you just see a very smooth path. You don't see the lattice spacing. Brownian motion is a motion which chooses between the four possible ways on the lattice, left, right, up and down with equal probability. Each time the random seed is different, so it behaves very differently, but still you can see that it's a self-similar shape. You cannot tell what's the difference between one version, one member of the sample and the other. Okay. This is close at hand. Here is a DLA. Difficult perhaps to see, but particles are actually coming in from infinity and they stick where they can to the existing structure. So the structure grows until you get a very large push. Why there are loops? I don't think there are loops. I think that the problem is the image. It's not large enough for you to see. It gets very close here, but I don't think it connects. No loops. It's not allowed to form a loop. Theoretically it can, because we have a finite particle size and it's possible that one will come and block the way into an area. But theoretically that is not correct because it has to do with the finite size of the particle. It's small enough to never do that. Okay. Now some fractals. So it's forming a fractal by adding triangles to the existing shape. So as time goes on, a fractal is formed. We start. This is the Koch fractal. The same iteration, but now on a square shape. This is the Serpinski triangle. Okay. I have two more programs. One is related to the talk by professor Dar. And this is a sand pile. You drop a pebble of sand on a square. Each of these colors shows the number of sands which are present on that square. And if it goes beyond three, so three, two, one, zero, then an avalanche happens. Here is an avalanche. So you see that some points go black, means that they are over the height. So an avalanche happens. Truly it looks like a living entity. It's very strange. And the other one is growing surfaces, which I'm not sure if I get a chance to get to it. For growth processes, there are various algorithms as to how you grow a surface. Particles come from infinity, land on a surface, and the surface grows. And surfaces like these, which are used very much in modern technology, are formed by that. The internal thing is something much cheaper than the external surface. The surface is grown in the lab or in the factory for this purpose by actually letting single atoms come and sit down on the surface. Now, how it works depends on the model. So this is now Eden model. I will tell you what the differences between these models are. Eden model is that a particle comes from infinity and just sticks to the nearest boundary. So it looks a little bit like growth of a bacterial colony. Now, this is random deposition, which means that nothing is... no rule is fixed on it. Just random particles come in and hence just little needles seem to be growing. Now, ballistic deposition means that particles come in, but they can also stick to the side. So a particle can come in, a stick here or a stick there, both with the same probability. As a result of this side sticking, you see that it has cavities inside it. And here is the LA in one dimension. So this is the LA process again, except that it's not circular like the one which I showed you with the central seat. This entire line is acting as the seat. Yes, that's correct. So I have two different types of rules. One is that I choose a point and I let a particle come down and it comes down in a straight line. And then after it comes down in a straight line, then I have a rule as to how it sticks to the surface. The rule of sticking to the surface may be just stick on the top of the first column that you arrive at. It may be that you can stick to the side as well or that you arrive at the highest peak and then go into the next edge. You are allowed to move one. These are deposition. These are called usually deposition models. The LA, however, says that let go of a particle at infinity, then it moves on a random walk. It doesn't come down directly. And whenever it hits an existing structure, it sticks. These things which I showed, just these two rules. Exactly. So the probability distribution of this particle here is harmonic, which is why it's called diffusion, diffusion-limited aggregation. It's an aggregation, so growing the process is diffusive, yes. The result of these two are different? Yes. Yes, the particle is coming from somewhere random, but still the result is different, yes. Okay, thank you very much. Let's go through that. And then I suppose I press these two and then these two again to go back to my presentation, I hope. No. You told me how to start it, but you didn't tell me how to finish it. Thank you. All right. Okay, that was a little intermission. Now, this. In your previous slide, did you mention that you have the fractal dimension of 2D image and you want to calculate the fractal dimension of 2D neurons and reconstruct it? Not from the fractal dimension of the 2D image, from the 2D image. The input is the 2D image. The 2D image is, if in a black and white version, is a number of pixels, is an image with a number of pixels. For the fractal dimension, essentially, you can take a row here, either 0 or 1, which is the density of this pixel. And you make a rule for yourself as what to do. That is, you may take it below half 0 above half 1. And that is enough, usually. It gives you a very good answer for the fractal dimension. However, in a real image, I'll show you these have a range, a wide range for row, gray scales, a wide range of gray scales. You can then interpret this if you take a row between 100 and 0. You now can interpret these gray scales as distance of the image to the plate, photography plate. Then create a three-dimensional entity. From that, you calculate a fractal dimension of the 3D object, which created the image. Usually not used for people, but when you have a cluster of, for example, atoms or molecules growing on the surface, they form fractals, like the one which we're looking at was on our palladium crystals on glass. They are 3D, it's a very bad sentence, but they were 3D fractals, not 2D fractals. You're used to thinking of 2D fractals, but fractals of volume can also happen. Okay, we want to construct a conformal field theory. How do we do that? I'm going to try and be detailed about this topic, and I'll be happy to be stopped whenever you learn to ask a question. So what do we usually do in a quantum field theory? For a quantum field theory, we need two sets of things. One is a Hilbert space, and the other one is a set of operators. So if FC is in the Hilbert space, the operators act on FC and move you from one state to the other. So I'm going to be trying to construct these components to construct my conformal field theory. Now the point about conformal field theory is that it has to have, it's more than quantum field theory because it has to have conformal invariance, or I should say conformal symmetry, which means that this Verosora algebra has to somehow be taken account of. And I will do so in the construction of the conformal field theory. So by construction, this symmetry will exist. So I need a number of field operators, these O's. So I need a number of operators, which I call phi hh bar. They are functions of z and z bar are defined over the complex plane. And under z going to w of z, this will go to phi of w, w bar, which is related to phi of z, z bar through these derivatives. What this does is it immediately tells me how my fields are supposed to transform under conformal transformations. So a simple conformal transformation is w equals to lambda of z. This is the simplest, perhaps, transformation I can take, which is a scale transformation. I've just scaled my coordinates by lambda. Hence this is easily solved. Now since w bar transforms to z bar, it has to become lambda bar if lambda is complex. Hence we can rewrite this as lambda lambda bar to the power plus h bar lambda over lambda bar to the power of h minus h bar. I think I need a half. Therefore if I take h and h bar as some of two components, then I have here lambda lambda bar to the power delta. A little louder please. This line? Yes. Does this help? You still need an explanation. It helps. Okay, thank you very much. And lambda over lambda bar to the power of s. Excuse me. This is not a scalar field. Yes, it's a field with a spin. And the spin is reflected in the difference between h and h bar. If h equals h bar, then it's a scalar field, which means that it has no spin response to rotation. But if h is taken on equal to h bar, then it becomes a spin. It has a spin-full field. However, it's true that in most CFD calculations we only use scalars because that so happens. More than one component. Okay. If you look at it as a representation of the Poincare group. But actually we will do that. We will have, for example, two fields for a spin-half particle. But I will treat them as two independent fields. These which respond to rotation in a particular way. But my theory to be complete under Poincare group has to have the required number of such fields to transform accordingly. Now, so you see that this is actually something like e to the power i theta. And this is something like the length of lambda to the power delta. So in this representation, I have the complete behavior of phi as it is necessary for it to rotate. So by this form, I will have the complete necessary form for all the fields I want to do. I want to do it. I know you can construct a field with very little symmetries such as lambda phi 4. But if you are happy with that, then you are happy with that. You cannot do everything you want with it. What do we have to? We don't have to. You don't have to do anything. Where I want to construct a conformal field theory, so I must have a field with all these symmetries. I don't understand the question. Conformal symmetries. That I need these symmetries. And then I should make this field theory. Yes. I have spent the last two days trying to impress that. That your theory, as it is usually, has rotation and translation symmetries. Usual theory. We assume it's correct in this room. But on the certain circumstances, and those circumstances are when you are close to a critical point, the theory develops extra symmetry. And this extra symmetry I have been arguing is conformal symmetry in any dimension. However, in two dimension, it becomes even bigger. Because in two dimensions, not only you have the conformal group by the entire holomorphic functions, and then you have the entire holomorphic functions available to you, you have to construct it such that it is symmetric under Vilosaural algebra. Yes. There was a guy who was telling a story about Leighley and Maginot. And at the end, somebody asked, is Leighley a woman or a man? This entire course is about why conformal symmetry is important, and it comes about at the critical point. At the critical point, you have a scale invariance and a scale invariance induces conformal invariance. Therefore, I'm going to use that conformal invariance to solve some of the problems which you would do otherwise. For example, calculation of fractal dimensions, for example, characterization of the domains, boundaries of clusters, probably loop distribution functions, all sorts of things can be done with it. But the way things are, of course, is that you don't have to. There are other ways to do it. There are other ways to do it, but I think this method is more powerful and prettier. Sorry, did I explain? Yes. So, I mean, right now, we are just combining some additional symmetry with the quantum field theory and we want to focus on one point, which is a so-called critical point. Yes. So, can we do the similar thing with the classical fields, like something like, and combining this conformal invariance idea, or you have to go to quantum field theory because of this complex signification? First of all, because you emphasized it, I should say that there is nothing quantum mechanical here. So, I will use the methods of quantum field theory to construct this theory, but it's going to be actually applied like a statistical field theory. So, I don't have quantum effects. I'm talking about, at the moment, I'm talking about classical thermodynamics. Complex value, yes. To be complex value doesn't mean that it's quantum. In the end, when I want to do some calculation which has to do with physics, everything has to become real. It has to be real numbers. So, I construct a conformal field theory by the methods of quantum field theory. But in the end, I will use this conformal field theory in its classical meaning in the sense that a statistical field theory is used. And the same is true about Landau-Ginsburg. Landau-Ginsburg is usually not used in statistical mechanics as a quantum field theory. It is used as a classical field theory. And then, also, yes. Landau-Ginsburg theory itself is a conformal field theory, a very simple one. So, that's also... So, exactly... You are exactly right in that Landau-Ginsburg will also be part of this construction. As you know, Landau-Ginsburg theory becomes scale-free at the limit of g tends into zero, the coefficient of phi-4, and that is then a conformal field theory. So, already we have that as an example. It's a wider class. So, any more questions? Yes. Yes. Yes. You can understand what? Yes. Yes. Yes. It's a good question, but that's exactly what is happening. It's the beauty of it and the importance of it. Otherwise, I wouldn't have been saying anything exciting. So, let's go through this again. There seems to be a lot of questions here. I may have... So, I have these two variances provided you don't have any impurity or frustrating class effects. These symmetries hold. Yeah? For example? No, both phases. All the way through... Continuously, yes. You have... You have a piece of material, let's say a steel from... which you heat up to above the curry temperature and then you cool down to below curry temperature. This will always hold. Then you come to the critical temperature. Now, at TC, I observe that all fluctuations become important. Fluctuations at all scales become important. This is observed in water, for instance, through critical opalescence. Light is absorbed at all scales. So, to this, I add a scaling variance. So, I have an enlargement of my symmetry, at least by this. Now, something magical happens. This is, although mathematically correct, but physically, it is forced to accept an extra symmetry, which is the special conformal transformation, which is just a vector, of course, just added to this. A vector, I mean that in d-dimensions, it's a d-dimensional operator. And the sum of this will be a conformal symmetry. That this is added of its own accord is important. Don't put it in as an input. It jumps in. The proof that it jumps in is, on the one hand, is rather simple. On the other hand, it is difficult. It's simple because of this argument, which I gave you, that if you go to complex variables, then the energy momentum tensor will look like this. All complex quantities, but overall only three independent members. And when you look at the conservation of energy momentum, you find that this is what it looks like. If theta vanishes, if this vanishes, then you have just d bar t is equal to zero. That means that t is only a function of z. If t is a function of z, it means that I have a totally holomorphic theory. I can make any transformation of z to g of z, and it's still an acceptable energy momentum tensor. This is something which is reminiscent of what happens in a harmonic equation. You have probably seen this, that if I want to solve this problem in two dimensions, I want to solve the harmonic equation in two dimensions, then I can rewrite this operator as d bar d. Therefore, v has a solution in this form, a function of z and another function of z bar. If I want v to be real, this has to be the complex conjugate of this. But if I remove that restriction, this can be any function of z bar, plus any function of z is a solution. So a solution has been found, but not in the same way that we do by hard work and extracting. So any function is a solution because of holomorphic transformations. So the same is true here. Any function is a conserved energy momentum tensor, provided we have a scale invariance. Hence, conformal invariance jumps in. I don't put it in. It jumps in. It's like going to a lunch with your friend, and the guy you don't like says, I'm coming as well. This comes to lunch. Why are we interested in? For any physical theory, I have an energy momentum tensor. I must, because this is what gives me the conservation of energy and momentum. This is the structure which gives me the conservation of energy. If I don't have it, I don't have conservation of energy and momentum. Since in physics, we must admit that energy and momentum are conserved, and energy momentum tensor is a must. It's part of our theory. These questions are good. If you have more, I am willing to hear. It probably clarifies things for a lot of people. Ask any question you have. I told him he can ask any question. It's not real. Okay, go ahead. T is equal to zero and so on. What happens that causes tetravanish? What causes tetravanish? When you have a scale invariance. It's a little difficult to prove now, but you can show that there is a proof that if you have a scale invariance in any theory, that's in any theory, right? Then that theory will have a traceless energy momentum tensor. I think Hermann Weil proved that. Traceless means theta is zero. It does. Theta is the trace. He will not show up in the trace at all now. No, I think theta is the trace. In the complexified form, it's off diagonal, but in the real form, it's the trace. Theta is T00 plus T11. It's the trace. Or minus T11. If you have a pseudo-remanium. Sorry, I didn't hear that. In the X and T variable, the trace of T is something. But in the Z, Z bar representation... It's off diagonal, yes. It's off diagonal in Z, Z bar representation, but in the X and T representation, it's the trace of T. So this proof by Hermann Weil says that if you have a scaling variance, you have traceless T and if you have traceless T, then you have just two components in your energy momentum tensor in 2D and those two components have to be TZZ and TZZ bar. Okay? Yes, go ahead. This is the metric of spacetime. What do you mean? What kind of? Yes. It's the usual g mu nu, yes. A little louder. It is the symmetry of your spacetime. It can be Minkowski and if you have a pseudo-remanium geometry, it can be just the Riemannian metric if you have a positive definite metric. Doesn't matter. A little louder, please. No, I don't care what the theory is. You are adding it. I'm saying any theory. Any theory which has a metric, yes. And a metric is necessary in physics. You must be able to measure distances. Your theory will have a metric and that metric under variation of a scale behaves in a certain way and then you can show that that leads to traceless energy momentum tensor for that theory. Whether it is relativistic or not, any theory you have. So in this way, you have conformal invariance and to arrive at conformal invariance, I need to have fields which are these guys and they behave in this certain way under conformal transformations and the main current I have is the energy momentum tensor which I can then expand. I can expand it in a Lorentz series. Lorentz series expansion is possible over complex plane only and it's just the same idea as Fourier expansion except that it goes from minus infinity to plus infinity and the coefficients are given by counter integrals around the origin in this case or around the point of expansion. Now, the claim I make which has a lengthy proof and I don't really want to give it and I don't really expect you to go into conformal field theory and learn it in depth in such that you have all such proofs. If you are interested, I can point you to where to see it. The thing that then happens is that these LNs generate the video sorrow algebra which is saying a lot. It is saying that all your conserved currents come out of this one conserved current which is the energy momentum tensor. In the simplest case which is this sort of simple naked conformal field theory with no extra symmetries, this is the only conserved current you have, yes. It is operator value. That's why LN is also an operator, yes and that's why the proof is a little hard why LN satisfies the commutator relations. So this is what LN satisfies and exactly as was pointed out because LNs are operators now you have this extra term here which you didn't have in the Witt algebra. The Witt algebra was the symmetry of just simple functions over the complex plane. Just any function of Z. But when they are operator valued LNs are the operator valued versions of the Witt algebra so they satisfy the Witt algebra which is the first term but they also have this central term. It's called central because it's a term which commutes with the entire algebra. It's just a number. It's in commutation with all LN and in this central term you observe an undetermined constant which is called C. Universally everybody calls it C. I think Virasoro himself was responsible for calling it C and it's a number, a real number which determines which conformal field theory you are dealing with. So if you set C equal to zero you have just an empty conformal field theory a null theory with nothing in it just a vacuum. But then if you set it to other values you get other conformal field theories and one of the things that those who work on conformal field theory have to do is work out the value of C which is consistent with all the unitarity, etc. conditions and correspond to the physical content that you are interested in. Yes, but there is no physics. Actually this is a little sticky point. I haven't said anything about it in my slides but because of your question you can see in this algebra that there is at the L2L-2 stage so if you take N equal to 2 M equal to minus 2 then this central term becomes non-vanishing because delta is satisfied and you get a factor here which I guess is 6. Now I take this between two vacuum states. If I take it between two vacuum states the opposite combination vanishes. This vanishes because it's just 1L if I either case the vacuum on the right-hand side or the left-hand side. So the central charge is the only thing that is left behind. So you get C over 2 but this is what I can take as the norm of this state. So if the theory is unitary this means that C has to be positive and it equals 0 only when L-2 annihilates the vacuum. And then I can show that if L-2 annihilates the vacuum then it implies that all LN annihilate the vacuum. This is three lines of proof but let me not give it here afterwards if you want. If all LN annihilate the vacuum then there is nothing in the theory except the vacuum. You cannot construct any state more complex than vacuum out of it. Having said that a very important theory corresponds to C equal to 0. So I just claimed that the C equal to 0 theory is null yet C equals to 0 gives you a very important theory. Percolation is a C equal to 0 here and it's a sticking point. You don't understand it very well. I don't understand it. So it works for C negative. It doesn't work for C negative. If C is negative then the theory is non-unitary. However, since we are using it as a statistical theory it's not necessarily bad but you need an understanding of it. What do you exactly mean? It has to be explained. So C negative means that there are no negative norms so norms cannot be interpreted as probability anymore. Again, please? No, it is null. If it is not null, it's non-unitary. Yes. Percolation is not non-unitary. My conformal filter to explain percolation is non-unitary. Yes? Yes, please? It means that the Hilbert space that you have constructed for it has norms which can be negative. If it has negative norms then you cannot interpret the norm as probability. Okay, so we want to construct a quantum field theory with conformal invariance. This means that the fields and the Hilbert space have to carry a representation of the Virussouro algebra. That's what we mean when we say a theory is something symmetric. Now, how do I construct something which is a representation of the Virussouro algebra? This is how. This is a little bit like the construction of the angular momentum states or the construction of harmonic oscillator states. So I start with one state which I call a high-state state and it has the property that if I apply Ln to it, it vanishes. What is this? This state itself is an eigenstate of L0 and if you apply negative Ls to it, it doesn't vanish but it gives a state with n plus h. That's easy to check. Just look at L0 L minus n and this can be written as L0 L minus n commutator h plus h L minus n h and this is just n L minus n on h because of the Virussouro algebra. So it is equal to n plus h times L minus n times h. So because of similarity with angular momentum, they have called this the highest rate state. It's actually the lowest rate state because all these other states have a higher rate and you just go down into infinity by applying many L minus ones. You can also have things like this. You don't need only one of them. Instead of going higher, you go lower, yes. But you call it going higher. So after I do this process, I find that I have a mountain. I heard Bellowin himself giving a talk about this and he called this the grandfather and his children and the children come below. At a level like this, I have the h plus n state and the h plus n state is not just one of it. There are many of it because I can reach there by L minus n on h or I can reach there by L minus one to the power n. So there are many states here which are not identical but they all have the same weight. But they are not identical because they are reached by these different methods. And there are how many states are there here? These are just the partition of n. Partition of n is how many ways you can write n? So you can write n as just some of n ones or you can write it as a two plus. These things and you can just write it as n ones. It's a very famous problem, by the way. What is the general expression for partition of n? I think Ramanujan derived that if I'm right. All right, now I said all this for h. I could have also said it for h bar. So there is in fact a state h h bar with similar mountains under them and each of these is called the verma module. So this is verma module of h and that is verma module of h bar. H exists. That's the next stage of this discussion that I have to now prove when h exists and if it exists what value h takes. And once you do that the entire problem is solved. You have the complete conformal field. And in fact it has not been solved. It's an unsolved problem to find all possible h's. But for a very small section of the spectrum which is called the minimal series all possible values of h have been found. And in fact that goes to is part of the proof of existence of a conformal filter. A conformal filter exists if h exists and all possible h's will come into a family. I will give an exposition of it but I won't prove it. It's very difficult proof. So once I have h and h bar I can add them up. So h is multiplied by h bar because the complex section is separate from the complex conjugate. It's independent. It's a direct product. And I can add them all up. All these are added up and they form the Hilbert space of my theory. So first task is complete. I have my Hilbert space. Second task is to find the field operators and that is done by a simple trick which is called the state operator correspondence and a little like harmonic oscillator for every operator there must be a state and these are in one-to-one correspondence. And the way that this state are operated is that I say that all states can be arrived at by action of an operator on the vacuum. So these states which were my the ones that I created my Hilbert space they are created from the vacuum by the action of some field h bar at origin. And in this way all my operators come into existence. So I have all operators and my Hilbert space. There is a very simple lemma here that the operator one acting on the vacuum will give back the 00 state. So the proof is complete or the statement is completed in this way. Sorry today we went without a break so 15 minutes more I let you go. It just so happened that we didn't get a break. Okay. So here I have my conformal field theory. However, as you know a field theory is nothing if you don't have the correlators. It's nothing in the sense that you cannot do any calculation with it with all these nice structures if you don't have the correlators. So we are going to need to go and calculate the correlators. Correlators are calculated in quantum electrodynamics or QCD by complex Feynman-Graf calculations. Really hateful calculations. And no calculation I hate more than Feynman-Graf calculations. But here it is a very simple way to calculate the two-point functions. Very, very simple. And that's by symmetry. So let's calculate the two-point function by symmetry and the others we won't do. Let's just do the two-point function. So we want the two-point function of phi of Z1 H1 phi of H2 Z2. And I have not written the complex component to simplify things because so there must be a Z1 bar, H1 bar, Z2 bar, H2 bar as well but I haven't written it just to make the notation easier. How do I do it? I ask this to respect the symmetries of the theory. So whatever it is on the right-hand side it has to be a function of Z1 minus Z2 for translation invariance and it has to be a function of the magnitude of Z1 minus Z2 for rotation invariance. So to make things simpler I just set that to zero and this becomes a function of magnitude of Z. Now I ask it to have a scale invariance. Sorry, I set this to zero too soon. Better not. Now I ask it to have a scale invariance. That means that I put in a lambda here and hence a lambda here. That we know has to behave like this with lambda. So actually lambda to the power minus Z1 minus Z2. Right-hand side for symmetry to be acting right-hand side must have a power form as well. So it's if has actually no choice than being Z1 to minus Z2 to power minus H1 minus H2 and therefore lambda appears and is cancelled. If can have a constant term as well which depends on H1 and H2. That's all. If I have a scaling here lambda comes out which gives you this term. The other side has to equal that. Therefore lambda has to come out in this form and for it to come out in this form there is no choice other than having a power form for F. So the only thing which is not determined here is a multiplicative constant C depending on H1 and H2. So lambda can be a complex entity but I can always take it real. For it to be a complex entity I have to be careful and write the complex component as well. Since I haven't I keep it real to be able to do it. Otherwise you have to write the H1 bar, H2 bar, etc. Okay finally I have an inversion also possible. That means that this term, this form has to be invariant under Z1 going to minus over Z1. A little more complex calculation but I promise to you it works with this form. The only thing is that it forces you to have H1 equal to H2. And then the constant C I can absorb into normalization of free. So this guy just disappears. And now I have the two-point function merely by symmetry arguments. In order to conserve the shape of our coordinates. So all this theory is to be able to write the field with this transformation. However life does become a little harder when you go to higher point functions. This is the two-point function can be completely derived. When you go to the three-point function as you see below C123 remains undetermined. That number remains undetermined. And when you go to four-point function it becomes a little harder and so on. Five-point function even harder. But sometimes life just needs two-point functions. You don't have to worry about the higher ones. Yes, yes. So in fact as I said in the construction of the states and fields this is done at origin. But in fact then I have to say what happens at other values of Z. The other values of Z are connected with that by a translation. And the translation is L minus 1. So in fact you take phi H h bar of 0 e to the power minus Z L minus 1 Z L minus 1 and this is the phi at Z. Once you have phi at Z then you can make an inversion and go to phi of 1 over Z and if Z tends to 0 that will go to infinity. The whole point about this theory is that it's easy. You can solve these sort of problems by these kind of transformations. And it's a completely integrable solvable model of a quantum field theory. That's why it's so interesting. However, that's not what we are here for. A lot of people are working on conformal field theory as a quantum field theory and using it in various ways. What we are interested in is that because it corresponds to critical phenomena. That is I will give you a table maybe tomorrow because I'm running out of time now of all the conformal field theories which correspond to critical phenomena. And in this way once you have that correspondence you can analyze that critical phenomenon using that particular conformal field theory. So okay, the three point function is written here and similar techniques have been used to derive it. So first of all it only depends on the differences of distances. X12 is Z1 minus Z2. That guarantees translation and rotation invariance. Then a scaling variance is guaranteed by this particular form that if you now scale Z1, Z2, Z3 by lambda, right-hand side will transform accordingly. Finally these rules as to what exactly A is in terms of H and etc. is imposed by inversion. Compared to the two point function the coefficient up there remains unknown. And you then have to tell me exactly which conformal field theory you are working with so that I can extract Z1, Z2, Z3. In the first one you don't need to tell me. This is completely determined. To maintain completeness we can go and look at the four point function, play the same game with the four point function. So four fields are inserted. These powers come out and a scale invariance is maintained. Everything works except that there is something very interesting at four point function. When you have four points on the complex plane which is Z1, Z2, Z3, Z4, you can make this combination which is called the cross ratio. The cross ratio has been known to be invariant under conformal transformations since the 19th century. And first conformal symmetry was discovered. And because it is invariant this means that I can have one function, undetermined function f of eta which is just a function of cross ratio and it's invariant. In previous correlators this was a constant so I was happy. But now it's a function. I have a dependence on the space time through this function and so long as I don't determine f I don't have a final determination of the four point function. So that gives me the four point function up to f. If you go higher this problem becomes more difficult. So a five point function will have two cross ratios and so on. So you can really not go by symmetry any further. This is the limit of the symmetry. There is a technique for determining f which is also relevant to our work. I cannot bypass it. So I have to talk about it but perhaps not today which is the question of null states. I will perhaps tell you about null states tomorrow. We use null states to determine f of eta. And that's about the most difficult thing about CFD which I want. So don't worry about CFD. It will be finished tomorrow. Thank you.