 We have lots of people of CISA, Saffone Baroni, for example, who knows everything, so that's perfect. And yes, correlation. Now, the correlation is a tough problem. And actually, this is the source of self-interaction problem, because if we are to fuck, everything is just clear. You have two potentials, one minus the other, and you know that those two will compensate each other. If you do everything self-consistently, it's beautiful. But then if you start messing up things, introducing correlation, and blah, blah, blah, blah, and then breaking the harmony between states and energies, then you don't know. Even if many bodies, there's a long history, as all theory I can tell you, but I don't know. Good. Now, to introduce many bodies, actually, and diagrams, and not only documents, but the methods that are formally introduced to solve the many body problem exactly. So beyond archivoc, we need to do a step forward in the definition of operators in quantum mechanics, introducing fields. Why are fields so important? Because in second quantization, you define the state of the system as an object that can be mathematically manipulated as a foc state. That's fine. Everything is really compact compared to formulation in terms of wave functions. Everything is very compact, but you cannot describe the time evolution of the state. So in the case of many body, and the original formulation of the quantum many body problem, was written in terms of paths. Indeed, there is this very much famous sentence by Dyson talking about Feynman and saying that it was crazy in thinking that you can rewrite the complexity of the interaction in terms of paths. But actually, this is the truth. You can format rewrite all complexity if you are able to take into account all possible trajectories of the electrons. This equivalence between correctional trajectories is the origin of the many body problem. How do you do that? First of all, you introduce those field operators. They are just an expansion of the state of the system at a certain point of time and space in a complete basis. This complete basis can be either a collision basis, a single particle basis, or an autonomous basis. Because of the normality, this field satisfies a commutation or anticommutation. It depends on the fermions or boson relations. And there are good objects to rewrite your problem. Indeed, you can formally see that through the states, fields, you can rewrite the Hamiltonian in this form where you have an explicit depends on the fields. And indeed, you have single particle objects where you have only two fields. This is one field entering, one field exiting, while the two bodies' interactions have four fields. This dependence on four fields is the source of complexity of the many body problem. Now, what you can do is to, in the ground state, zero-temperature approach, there is a very subtle but key approach to connect the state of your independent particle part to the full state. This is the so-called Gellman and Lohd theorem. This Gellman and Lohd theorem is very much intuitive, but at the same time, it is very difficult to accept physically, I tell you. In the case of the Gellman and Lohd theorem, they introduce a pre-factor of the interaction, the four body term, is a pre-factor made in such a way that it goes to zero at minus infinity, plus infinity, and goes to one in the internal region of time. Then the theorem states that if the limit of this eta that goes to zero exists, then you can connect the free state to the full interacting state by switching on adiabatically slowly the interaction. So your system goes adiabatically from a free state, non-interacting states, to a full interacting state. I mean, this was a corner in quantum money by theory, because it writes the problem of correlation in terms of a time evolution. So you do a time evolution of this Hamiltonian with the other bodies, which is non. And then adiabatically you go from the non-interacting to the full interacting. Great. So this means that if you expand the time-dependent ground state in powers of the interaction and you write it explicitly, the time evolution, then you realize easily that this evolution can be made in such a way to connect minus infinity, where it's free, to time t, where it's full interacting. This expansion is formally written in terms of t products. It's just a reason. It's just due to the fact that every interaction will appear in a concatenate form. And this concatenate form can be written formally as a t product. So this is just the evolution operator of the system switched on with the adiabatic switch on. So you have, again, adiabatic switched on from the free state to the interacting state, then you rewrite this time dynamics as an evolution operator. These evolution operators will introduce the interaction. Again, the interaction is this part. It's the Columbian interaction. You introduce this interaction in a power expansion with the time order with this t product. OK, this is your evolution operator and that defines the time order product. That is just a time product of two operators ordered depending on the time relative order of the two arguments. And then the last step you need to introduce the last tool you need to introduce the methodology is the Green's function. This is a key object in our many body approaches. Why did I do this very complicated and super messy definition of the t product? That is one of the most difficult things to introduce in one slide, the t product, because it forces the definition of the Green's function. So the Green's function physically is just the propagation amplitude of a fermion. It's just the probability, the amplitude for an electron to go from time t to time, oh, sorry, it is t prime. It's x prime t prime, so it is a mistake, x prime t prime. So it's just the probability amplitude for going from t to t prime from x to x prime. The point is that this trajectory can be in one direction or the other. So in one direction you say that is a fermion in election going forward in time. In the other it is a hole going forward in time or an electron going backward in time. The time order just imposes these two different orderings to have a minus. And this is technically due because of the expansion of the evolution operators. Unfortunately, there is no simple physical intuition for that. There is no physical intuition for that. And I think this is one of the drawbacks of using zero-temperature approach. But anyway, that's it. The important thing is that, yes, it's the evolution operator going in one direction or for an electron going one direction or the other. The important thing is that you can expand. This Green's function is actually an object that contains some exact information about your system. Because if you expand it within the completeness relation for any states of your system, then you can realize that this Green's function will have this expansion terms of poles. And the energies, the poles of the Green's function are the excitation energy of the system, exact excitation energy of the system. So the Green's function will be made in such a way that the spectral, the imaginary part of the Green's function. So the spectral function, so if you manage to calculate this Green's function, then you calculate the imaginary part. And the imaginary part of the Green's function is the spectral function will just show peaks at the exact excitation energy of your system. Exact, exact period, excitation energy of your system. This, you get it from here. This is the ground state. You expand, you see, with the excited states of the system when you are creating or annihilating an election, so here you have to include the excited states with n plus 1 elections. You plug them here, and then you define those functions as sandwiches. And just, and exponential, you get those differences. It's an easy math. It's a lemma representation. I mean, it's math, you need to do some passages. I know. The physics is just that if you accept this t-product, and then you define the Green's function with a t-product, then if you calculate exactly, you get the exact excitation energy of your system. That's great. Then how do I get this t? How do I calculate it? OK, again, the problem is of calculating the time evolution of this t-product with this given Hamiltonian. So actually, you have, where is it? Yes. One possible way is just to apply that actually is in a way to approach the problem, is to calculate the time evolution of the field operator. If you calculate the time evolution of the field operator, you can just plug it inside this definition, because if you do time derivation of this g, there's time derivation of this psi. So I get time derivation of this psi, and then I have to calculate this commutator. This is grant phi, but it produces this object. Edt of the Green's function is the single particle Hamiltonian plus this object. This is v, the Bercouple interaction. Then you see here a t-product of 1, 2, 3, 4 field operators. I say, wow, what are you talking about? I'm talking about a simple thing. Now, you have your t c-dog, right? This is 1 and 2, 0.1 and 0.2, OK? In the free world, this evolution is just, that's it, by free. No way. Then the column interaction. The column interaction, how does it look like? Yes, this thing. That thing is something like, OK, the integral v, r, r prime, and then there are c-dog r, c-dog r prime, c-r, c-r prime. Now, if you draw an interaction diagram for this, what do you get? You get r, r prime. Then you have something that is created in r, annihilated in r, created in r prime, and annihilated in r prime. Great. Now, this is the interaction. Now, this is the free. This is the free part, right? Now, instead of this, we start from 1. We need to arrive in 2. But yet, we have that interaction term. So this interaction term will appear suddenly in the middle because it's the potential. This will mean that we will have processes where this potential will just affect the dynamics of an electron moving from 1 to 2. This is a typical interaction term. I mean, they will be scattered through this potential. But if you look inside in the middle, you see that for suddenly the electron from being alone becomes a pair. Because the column of interaction is a two-body. So they will scatter. And then suddenly, the second body of the column of interaction will appear. I mean, the column of interaction is something. OK, let's put it another way. So the column of interaction is made in such a way that I'm interacting with them. That's fine. So when I run, then I get the interaction. I get modified. If we are two, I get repelled. So I move this way. But then I get repelled as well. And then we start moving. So in this time, suddenly, the dynamics becomes two-body dynamics. So before I was alone, suddenly I became two-body dynamics. And this is what the theory is saying. Look, that if you want to calculate the evolution of a single-body dynamics, suddenly I must get a second body involved. And then I have a two-body Green's function. Is it closed? No, it's not. Because then I will continue traveling. I will interact with another body. And then it will become a four-body. And then it will move again. So if you do the math, if you study the math, and if you continue the math, you will see that the order of this Green's function will increase. It's a problem of hierarchy of n-particle Green's functions. So if you rewrite exactly the time evolution, you will see that at the end, after we are 55, after 55 iterations, the Green's function will be a 55-body Green's function. So no way to solve it. OK, there are ways. I mean, people have been thinking also how to solve the problem using this way, the truncation and terminating and so on and so forth. But the many-body problem for a continuous system or a solid, no way, you have to find a way out. How do you do it? Well, you have two ways now. The point is, how do I rewrite this four-field monster in a human-readable, human and computer-calculable object? Then I have two methods. One is through phamandagrams and the other is through the Schringer approach. I mean, they are both complicated. No way that I do it like in 10 minutes. No way, no way, no way, no way. So I just give you a snapshot of what you get at the end. If you want to get a math, we sit together and we do it. Actually, if I have to teach you those things, I will not use this approach, but I will use what are other approaches, but we can do it. So with the, yes, you get this. It's a pentagon. It's a dean pentagon. This is what you get from the Schringer approach. You manage to write those four-field operator in terms of objects. You define a new quantity of this. And specifically, you define the screen interaction and the vertex functions. Then you get for free the dyes in a question. That is an equation for the g, for the greens function. And then you get also an equation for the vertex. It is an object, mathematical thing, is a mathematical object. And then you get also an equation for the polarization function. OK, every corner of this is something physical. So this is the suffanagy, so levels. This is, again, levels. Those two are the bitters of Peter. They lead to the bitters of Peter equation. And this leads to the definition of W and, for example, plasmons. So that's physical objects. So the point is that physics is always like that, right? DFT is the same. You have this exchange code. This is an on-embroke code theorem that you write everything in terms of an on-embroke code energy functional. You give a name to fix that. You cannot calculate. So you write this energy functional in terms of what? Of an exchange or correlation part and not return. So you move the things you don't know from the functional to the exchange correlation. And turn on exchange correlation to the function derivative. This is physics. When you don't know how to calculate, you rename it. And here is the same. I mean, you are just introducing objects. And more are the objects, more is the complexity. So you have to find approximation for those objects. And this is what is done when you really go to computers. So in Yambo, for example, there is the GW approximation. That is when the vertex is neglected. This is something you will use and you will calculate. OK, this is one approach. And actually, it's the one I don't fully like. Because it's pretty rigid. So you cannot move too much. So if you want to describe some, what I would say is that this approach is practically dialectic-based. There is another approach that follows from the Gellman and Lothiorem that is by introducing diagrams. This is what I've tried to draw here. So maybe you do understand anything. For me, it's trivial because it's diagrams. But I mean, of course, this is borrowed from the diagrammatic theory. For endograms, instead, you rewrite all orders of perturbation theory graphically. So you draw lines and scatterings and kinematics or electrons moving around with arrows and lines and so on and so forth. It's a geometric approach. And that is based on the theorem by an Italian, the frontrieste Giancarlovic. So in general, when you diagrammatics and you rewrite correlation with this fancy graphical way, you have something that, I mean, in the edin's approximation, you rename everything. In the diagrammatic approach, everything is exposed. So you need to calculate it. So in general, what you do, you choose class of diagrams, class of processes driven by physical intuition. What I can show you, if you are interested, I can show you how you get actually for free at the very low level, a very low order of the diagrammatic theory. Actually, the diagrammatic theory is very powerful. It's very powerful tool. Indeed, archifog is what you get for free at the beginning, really. When you warm up, you get archifog. And this is indeed archifog. OK, so I need to get the potential again. But at the end, what you must be aware of is that there is no approximations that is valid everywhere. I mean, there is no exact solution of the many-body problem at the moment. And this depends on the physical problem you are interested in. So this is something that is not clear in edin's approximation, in edin pentagon. The edin pentagon is a compact and easy way to introduce the GDAP approximation. And the bit is a peter, but it's a dielectric base. If you want to get a more wide view, you have to use diagrams. Because in diagrams, you would be at a certain point at the corner, and you will have like 1,000 different roads, depending on the kind of diagrams you will include. And there, you will see that the physics is very much important, and it takes the kind of approximations. So you have, for example, the textbooks, known approximations in specific cases, whether it's a short-range interaction, low-density regime. And then if you want to be conserving, so do you want to be self-interaction at any order? Do you want to conserve the energy at any order? So in that case, there are some rules that tell you how to do conserving approximations. Also, the die-density regime. So this events very much on the physics. And this is the phases of people. This is Kubo, Larsedim, Kohlumbine. This is Elvin, Salpeter, Bithe, Vick, Kadenoff. This is the name of the guy, Rashba. I'm always confused by the two. OK, this is Lundwist, Feynman, Dyson. This is Rashba. And this is, I don't remember, I would tell you. I like the people of the phases of people. That's it. That's all. OK, now we have the break. Then there will be the lecture by Stefano Baroni. And then we will meet again this afternoon with the lecture about linear response. OK? Yes? Oh, and then there is a sorry, Paolo after Baroni.