 amb el laborat que hem fet les últimes hores sobre normalitzar-ho. I, doncs, la meva motivació, de totes aquestes tècniques, és a provar les solucions numèriques que donen quantitatives prediccions per la màxima regime de qfts. I, per mi, de la màxima motivació, és a atacar interessants qfts de qfts. I, d'acord, la gràcia és que s'ha de solucionar qfts eficientment i numèrically. I, doncs, aquestes tècniques, de tècniques americàries, són més o menys desenvolupades que les dades de l'Atlés Montegarlo, que s'han desenvolupades ara per a dècades. Per tant, són complementaris en moltes maneres. Per exemple, es pot estudiar les formacions d'interactes en fixats. També es pot adreçar real-time obstacles, com el que vam veure a l'Àfrica. També, el que vam veure a l'Àfrica, els exemples d'exemples d'exemples amb real-time obstacles. I, d'acord, tenim accessi directament al espectrum, i es pot adreçar quantitatives de coses, quantitatives de coses, com si hi ha potència simbètica, i també, a l'espectrum de la funció del volum, tenim accessi als elements esmàtics, i tal. Ok, doncs, vull fer un xoc, però no sé què és un xoc. Ah, ok, sí. Gràcies. Ok, doncs, el plan del xoc és que vaig començar amb una molt bàsica introducció, una invitació a intentar fer alguna sort d'anterior normalització. I, d'acord, vaig fer alguna invitació a les metodes. I, després, vaig explicar el que és... m'agradaria explicar el que és el estat de l'art de l'anormalització, en l'anormalització de l'anormalització, i, després, concluiré amb algunes altres al·lícules possibles, d'opens problemes en aquest problema. Bé, doncs, per començar, molt simple, em pregunto per a tu, perquè aquestes al·lícules són molt breus, que aquest d'anormalització de l'anormalització que estem pensant, és una generalització directa amb la qft de l'enganalment de Quanto Mechanics, l'art de l'art de l'anormalització. El quanto mechanical s'aprèn... Quan tenim una hameltonia, podem fer una part solable i alguna interacció, doncs podem pensar... per exemple, podem pensar que l'A0, per exemple, és l'armonic oscillator. i per B hem de pensar que tenim una interacció harmonica o una x per la interacció. I l'idea és que hem de construir aquest Hamiltonian com una màtica a la base de l'estat de l'hermonic oscillator. Això serà una màtica amb alguna cosa com aquesta, B i J. Ara constructem aquesta màtica a aquesta base i després trunquem i analitzem. Respectem com es veu una funció de trunquació i energia. En particular, per aquesta potència en quantum mecanismes, és probable que sigui el millor way to solve numericament. I pot ser que... Et posem... Pots ser que la spectra que es va trobar per aquesta màtica converteix exponencialment a la real expecta. Aquesta proposta de la sort de la interacció local. En particular, per exemple, només amb una size de l'espai 15, hi ha una molt bona converteix. És 0.1 i és 10 a la mena de 9. Aquest és per l'estat de l'estat de l'especte. Bé. Per veure les seves idees de mecanismes quantum a l'Efield Theory, era el que va fer... A l'estat original era el que va fer a l'Eurovandromològic, en el 90. I li posaré una altra referència. Aquí, en un cop de papers. So they borrowed these ideas of quantum mechanics to quantum field theory, and two of the prerequisites to apply these ideas into quantum field theory are the same. Namely that we split the Hamiltonian into a solvable part, plus an interaction, and the requirement that we know the matrix elements of interaction in terms of the solvable one. And so the solvable interaction can be a free theory or a CFT or a more generic interval model. And then for some particular cases of interval models we also know these matrix elements. Then for this talk, what I will assume, but maybe the assumption can be relaxed, but to be in the same ground what I will be assuming is that I mean canonical quantization in a series of fixed times. And then to keep the spectrum discreet what I need to consider is finite volumes. So I will be in finite volumes all the time. And then we'll inspect extrapolations with the volume of the spectrum. The theory of extrapolation of these energy states with the volume was developed or was initiated by Lusser in the 86. So these were known. And so one of the original theories where these ideas were applied was the theory that was being studied yesterday, which is the icing perturbation. So let me write the matrix. So the matrix is going to look something like this, of course. So we are going to have e n's, e1, no? b1, 2, in the diagonal, no? b1, 1, sorry. b1, 2, b2, 1, b2, b2, no? So it's something very basic i a large final size. So one of the original QFTs where this was applied was the icing in two dimensions with a magnetic perturbation. And it is a very special theory for a number of reasons which make it very suitable to be studied with the Klamathian techniques. So icing plus deformation. So this theory is an interval flow. So interval. So these allow them to compare with... So they trunque these matrix and these allow them to compare with exact known eigenvalues. And another property which makes it special is that in this case the matrix elements are known. So bi, j are known in closed form, exactly. And something very peculiar of this is that being this... the image interaction, very local de fermions fields, has a very low... It's a regular interaction which is very low, okay? The dimension. So to show... Just to show how spectacular the method works for this kind of theory. And they computed the eigenvalues with this kind of approach and what they found is percent or even better precision by only having a given space for 30 or 34 eigenstates. So they found... They could compare with exact knowledge and they found something like this, okay? And for the third excited state they got something like this. So... So this is very spectacular and I would like to... to borrow this approach to RQFT and so on. So the reason why it works so nice in this case is not about interoperability but it has to do about the fact that this interaction is strongly relevant, okay? I intuitively the reason why it works so well is because for a strongly relevant perturbation if you think about the Hamiltonian matrix what happens is that the interaction turns off in the UV, namely the eigenvalues of the free Hamiltonian are the same as the in the Hamiltonian so we have very low mixing in the UV. Well, it gets very large in the IR, okay? So for this reason keeping finite truncations we are close to the truth, no? So it's not because we have a mass gap the spectrum is discrete but dense but it's because there is this smooth transition, okay? And... Okay, so what I wanted to say with this is that this kind of techniques work very well when the interaction is strongly relevant, okay? The idea at the same time is a limitation of the method because as we go to less relevant deformations the convergence is worse of this method, okay? And in particular we can set a limitation to this method and for this talk I will be considering anomalous dimension of the formations which are less than the helps debing the spacetime dimensions, okay? Past this point you get already UV divergences en termes d'observations, starting by the vacuum. Then you get further UV divergences if you keep making it more relevant interaction, okay? This can be seen, for instance, by looking at corrections to the vacuum with the deformation, okay? So this starts diverging starting at the hubs. Sorry, is this a firm result? Is that something you can argue only perturbatively near the starting point or non perturbatively as well? So these are non perturbatively the stemming, yeah? You get this, what do you mean? These scaling dimensions make sense when you are nearly scaling the starting point. I see, I see. Yeah, I think it's a perturbative stemming in the sense that those anomalous dimensions are measured in the fixed point and the only thing I'm saying is that deforming the theory infinitesimally by those deformations you get to UV divergences. It could be that naively you have a UV divergence but non perturbatively the theory still makes sense and it's finite. Even for irrelevant deformations you would have irrelevant deformations which give you... So for instance, we should write this down we only get a divergence at the hubs, okay? But if we keep pushing it, we will get further and further divergences. The limit that you go to irrelevant operators will get infinite number of counter terms. But still non perturbatively the theory could make sense in the sense that you may have a finite spectrum and well defined and so on. Yeah. So this is a limitation of the method that may be overcome at some point but for this talk I would be... I will consider that I'm below this bound, okay? And another limitation of the method is that generically, or for many examples that we care the size of the hybrid space grows exponentially both with the truncation energy and with the space time dimension, okay? In some cases it can be computed in closed form but the more generically the size of the truncated hybrid space goes something like this, see? L is the volume of the system et is the truncation energy, okay? So c is going to be a third dependent constant, okay? So these are limitations of the method, okay? And so motivated by this growth of the hybrid space what we would like to do is that we would like to keep finite truncations which are not too large and do some sort of analytical normalization into the Hamiltonian in order to reduce the error, okay? This is the main motivation of the... of setting up a normalization problem. So, okay, so this was an introduction and now I will try to review which is the theory of trying to normalize this matrix, okay? Of course, the Hopi's tend to push it to learn relevant deformations and to higher dimensions. Maybe I can use other values. Okay, so this normalization is pretty generic but to keep things concrete in the back of my mind I will be thinking of some local theory like this one so we will be thinking in 5, 4, 3, 2 dimensions, okay? These are relevant deformations so it fulfills one of the requisites I was demanding for the method to apply, okay? So let me write directly the Hamiltonian. So the Hamiltonian, the 8, 0, is going to be the harmonic theory, okay? And then... and the deformation is going to be, say, lambda, 5, 4, okay? And we're going to build the Hamiltonian in the focus space of the free theory, okay? And we can study many sectors but in the process I will show you later I will be reciting myself to the sector of 0 total moment. It is not important. So this theory is very simple looking but it has a number of interesting features which makes it a nice to study with these methods. So the dynamics of the theory are interesting because so at weak coupling the theory has to symmetry and as many of you know this symmetry is spontaneously broken at strong coupling, okay? Then at the strong coupling there appear king states and boundary states of king states. So these are things that can be measured with the method of Hamiltonian location. So it makes it an interesting theory. The deformation is relevant. Okay, there is also a weaker strong duality for this theory that allows you to compare with analytical results also at a strong coupling. We have a weak coupled theory. So there is some other motivation while we are thinking of this theory the result I will be explaining about normalizations are more general. Okay, so how do we normalize? Let me derive it for you. So again let me write the Hamiltonian like this. Okay, and so this is the full Hamiltonian I haven't introduced any runcation yet and what we are going to do is we are going to divide the full space into a low energy part and a high energy part, okay? So by projecting into these two super spaces. These two super spaces of low and high are defined with respect to the free Hamiltonian, okay? So the states in the low energy part are going to be those with an energy EI of the free theory smaller than certain cut of energy. Okay? Now what we do is that we integrate out the high energy part of the Hamiltonian. So the again value equation is going to look something like this. So see it's an again vector that we project into a high and low energy part. So this is an equation. So let me write it super explicitly, okay? So this is a system of two equations and the trick to derive an effective Hamiltonian is to eliminate from this equation C-H in favor of C-L, okay? The high energy component of the again vector. So getting C-H from this equation, what you get? And this is a key equation that will appear later on. So just to make sure that the notation is clear that I'm using. When I put sup index like L-H, okay? Sup index I-J in this equation. What they mean is that I'm projecting B with... P-I-P-J with I-J being the low or higher Hilbert space, okay? So then you can eliminate C-H in favor of C-L and plug it back to the first equation, okay? And what you get is this exact effective Hamiltonian. So H-L is what I was calling before the truncated Hamiltonian, okay? This is a finite matrix that lives in the low energy Hilbert space. And delta H is the correction that takes into account the effect of all the high energy states that I'm not including, okay? So delta H is equal to B L-H or 1 L. This is, of course, a well-known equation. So the point I'm writing in this equation is to make you notice that if we knew exactly delta H, okay? The problem of finding the again values would be very simple, okay? And, of course, we don't know for interesting, well, for threshold we care, we don't know delta H exactly just because the inverse matrix elements of delta H-H are unknown, okay? So the program of normalization in this context I'm explaining consists in finding a systematic and precise approximations to this operator, okay? Again, because if we knew delta H exactly, we could keep very small truncations and decrease the error, okay? Also notice that the error, which is controlled by delta H, is even by this mixing that I was referring before between light and high energy modes, okay? So now I will explain to you some of the approximations that one can do to this delta H operator. Maybe you want to explain, like, you have an epsilon dependence or E dependence of delta H. Yes, thanks for the question. So, let me put back the epsilon. So, in this equation epsilon, the effect of Hamiltonian depends on the imbalance that you are considering. This find numerically what we are going to do to solve these equations that we will solve it recursively, okay? So we are going to plug some good answers for epsilon. It might be the epsilon that you get from Hamiltonian truncation alone by setting delta H20, and then you record it back. The comparison with respect to this procedure of fitting the epsilon to this equation is extremely fast, so this is not an issue for the method. But if the cutoff is high enough, you can just ignore epsilon. If the cutoff is high enough, you can ignore epsilon, that's right. I can't ignore delta H, you know. Yeah. What you could... No, but... What you could ignore is very fine calculation of epsilon in the denominator. That's what you meant, I guess. So, epsilon is going to have an error, but you can plug in just the truncated one and it's going to suffice. Okay. So, why don't you take delta H equal to zero from the beginning? I mean, what's wrong with that? Delta H equal to zero? Yeah, you just... Okay, okay. And there's nothing wrong with that. If you neglect delta H exactly, you are committing an error, right? Which is given by the fact that you are not considering the full Hilbert space. This is fine if your interaction is very relevant, because you are only dominated by the low energy states. Okay? And it's also fine if your conversion is very fast with DT. If you want to push it to higher values of DT, you may be obstructed by the fact that the size of the Hilbert space grows exponentially with DT. So now you want to keep small values of the Hilbert space. Okay? And the only way to reduce the error if your Hilbert space is small is by including this kind of corrections. Because notice that this equation is exact for epsilon. Okay? Even for very small truncation of the Hamiltonian, this equation gives for you the exact value of the spectrum. That's the answer to your question. It's a very important question, because it's the motivation of the talk. In fact, you are not going to solve the exact problem, right? I mean, you are... No, but I'm going to find much better conversion with respect to DT. So by including this operator and refinements of this operator, you are going to find that when you look at the spectrum as a function of the truncation energy, the conversion is much faster and smoother. So that's the motivation. The operator, I will show it in the next minutes, is analytical and gives zero cost computationally. So it's always worth it to include some sort of correction like this. I will show it now. So let me give you some reference before I forgot. That's okay. People will be interested they will find the reference. Okay. Okay. So I hope I motivated well enough that we want to compute this operator, especially if we want to go to higher dimensions where the equilibrium space explodes in size. Okay. So the first approximation that you may try is to tell or expand the matrix elements of delta H, to perform some sort of tell or expansion, okay? This is the first thing that one can try. So just imagine the formal expansion where delta H looks like this. Some of delta H ends where delta H ends. Okay. So here in the numerator H is the full Hamiltonian, which is H0 plus B, and I'm expanding in powers of B, okay? So I have this formal expansion of this operator, okay? And the first thing that we may try is to approximate delta H by the first term of this expansion, okay? So the first term in this expansion is what I'm calling here delta H2, which looks like this. So... And a good idea to do for this operator is to approximate it by a sum of local operators straight away, okay? So we approximate this by a sum of some local operators. So, intuitive, yeah? So that depends on how big V is, right? I mean, I'm just asking, like... Yeah, I'm going to comment now about, obviously, this. Let me go for one more minute and then you ask me again if I don't answer. So the reason why this approximation might be justified, of course it depends on how big is, but for relevant perturbations, the states that we mostly care about are the low-lying states, okay? The states we are much below the cutoff. For those kind of states, local approximation should apply because the energy exchange in this kind of interaction, here H0, HH, is ET or higher, while the energy of external states is much lower than ET, okay? So this is the intuition of why this is a good approximation. Alternatively, another way to thinking, so this assembles very much to quantum mechanical perturbation theory, okay? With some small difference that here we have an epsilon dependent instead of the free-theory energy dependent on the quantum mechanics, but nevertheless, this also admits a parameterization of this operator in terms of integrals and you can apply then a Feynman diarum, so what I want to say is that you can look this as a very simple Feynman diarum, so you have some external state coming in, say, a guy with three particles that interact with, say, lambda-5-4, let's say 2, okay, 2 particles, then you have some loop and some exchange of energy and the energy exchange between these 2 points is much higher than the energy of external states, okay? Then this interaction can be very well, this is a usual approximation that we do in FTF-30, just that here it has some different looking, but this can be approximated a simple interaction with one single vertex and some local operator, okay? So this is a kind of approximation that you can do for delta H2 already, okay? So for instance, for lambda-5-4 you are going to have a bunch of diarums, this would be one of the diarums that you get, then you get also a diarum that renormalizes the vacuum and a diarum that renormalizes the mass, okay? But all in all you have 3 new local operators that you need to add to your trigonade Hamiltonian in this first approximation. So, let me write maybe... You are already there. Sorry? They are not new operators, you just add corrections to the greater... Right, right, right. But, okay, in general, by performing this expansion we may get new operators that are not already there. So, I don't know if this is a good idea. So, by performing this kind of local approximation and what it was proposed is, instead of considering the trigonade Hamiltonian, okay, alone, we are going to add this delta H2 local, okay? Which is inexpensive and computational, okay? So, we are going to have here the H0 plus and then we are going to have the terms that renormalize the bare coupling plus some corrections, and then we are going to have the terms that are not already there for the 5,430 plus corrections to the vacuum and corrections to the mass that is fitting here, okay? So, let me show you some plots on how this improves by a lot the rotuncation. Can I get... Okay, so these are a couple of plots of a paper by Slava and Lorenzo Vitale. So, in the first line here it's for lambda 5,4. Okay? The first line involves... I think the last point involves about one million states, I believe, and this is simply the rotuncation, okay? And... for the vacuum and for the mass. Then the dashed line here, the one that approaches from below, this is the correction from the local... by including these local interactions, okay? And then let me check the numbers, because I forgot. So, as I remember, the point finishing here, it's about... it's about some few thousand states, so about 40,000 states or so on, so much lower. So for hybrid spaces, much lower we get much better precision because this line is already kind of flat, which for both observables, the mass and the vacuum, which gives evidence that you have already converged to the exact eigenvalues, okay? While the truncated... the truncated curve still has a long way to go to converge to the proper eigenvalues. Okay, so... including the local correction is a good idea, they are... is it include and improve a lot the method. Excuse me? Is there a way of interpreting this as a change of basis on some property gets more localized? So this new Hamiltonian shares the low energy spectrum with the old one. Yeah, it's a... Yeah, it's a good question. In fact, it can be directly think about a change of basis in the sense that another way of seeing this effective Hamiltonians is that you do a canonical transformation where a unitary transformation where you bring the Hamiltonian diagonal and then you can expand so I'm trying to say is that it's indeed a change of basis what you are doing. You can think of it as a change of basis. But it's an expansion then you need to integrate something here. Yeah, so the precise connection that you would do with a change of basis is that imagine that this is our original Hamiltonian in the free basis, okay? And you do some unitary transformation, okay? Like this. That brings it diagonal, no? No, no, no. I still can try to play. It still a productive game, but the only way you can try to play is to expand these unitary matrixes to bring H diagonal. And these are completely equivalent approaches. They're going to be some small differences in the finite terms of delta H2, but the UV divergencies are going to be the same in this Hamiltonian approach. So it is a change of basis. Okay. Just to show that we access to non-perturative information. This is a plot by a paper by Bainoque and Lager. These are for very... So the theory will change the sign of the master. We get to Bacchio. There is spontaneous symmetry breaking and we can study kings and bound states of these kings. And this plot is showing these... So these solid lines are the two first excited kings. Then on red and green are the two breeders that at some point become unstable because its mass is higher than two times the king mass and they can decay. So this is just an example to show that this information is accessed. And to perform these plots, they also perform some analytical normalisation of this kind of the local. Okay, can I turn now? I think in this paper they did not normalise, they just performed some feat. So they performed some feat, but to perform that feat, they used the formula, the parametric behaviour that you expect for the convergence, taking into account the corrections from the local. Good. Okay, so... So at least to me these results are very nice because it shows you that there is some way to go by doing analytical corrections and improving a lot your Hamiltonians without going to high energy truncations. So we would like to go beyond these, okay? We would like to improve to find which is the theory of computing these corrections more generally, okay? So the first thing that we may try to do is to approximate the delta H by including higher terms in expansion, okay? So we don't get it here delta H2, but we may think, okay, let's go on and let's compute the higher order terms, which we can compute by similar means by doing these quantum mechanical diaries, okay? So this would be a... and a strategy that seems a good one, but it fails, okay? And the reason why it fails, to recognise why it fails, we should go back on how is the calculation of delta H2 done, okay? The reason being that if we want to compute, to introduce delta H3 corrections, okay? We have to go beyond the local approximation of delta H2, okay? Because local approximation of delta H3 is going to be more precise at the leading order calculations that I was introducing here, okay? But what it turns out is that if you include delta H2 exactly, okay? So imagine that instead of introducing only delta H2 local, we just compute delta H2 exactly, okay? This could be done numerically or analytically. Surprisingly, the Hamiltonian that you obtain, it's very, very valid behave for a reason related to the question that he was asking, and the reason is that, even though the local approximation is a very good idea for the states that are much below the cut-off, for the states that are very close to the cut-off, this approximation fails very valid, okay? Just because it's not a good expansion for the matrix elements close to the cut-off. So, when we do the local approximation, we are committing an error, which is order one for the states that are just below the cut-off. And this order one error, the couples when we perform the analysation. Instead, when we compute delta H2 exactly, what is going to happen is that we introduced a correction in the states just below the cut-off, which is order of magnitudes away from the true result of delta H, okay? So, to appreciate this, imagine that we take an state, N, which is just below the cut-off, and it's a state with N particles at rest, okay? Then, you can estimate how big is this correction. Let me see. So, imagine we take a state with a large occupation number like this, okay? N on delta H2, okay? So, H0 is case like N, because it's case like the location energy, so it's going to be like NM, which is proportional to ET. Then, B is case like one... B on this state is case like one coupling times N squared times LM squared. This scaling, we can simply read off by performing the expansion in modes on lambda 5-4, okay? Lambda... So, we introduced expansion on lambda 5-4. We sandwiched with this state with large occupation number and you simply find this scaling, okay? And with this scaling, what you see is that delta H2, very close to the cut-off, has a very bad behavior in terms of the occupation numbers. You get something like G cubed, G squared N cubed, L squared M to the 5, okay? So, with all this, what I want to show you is that if we go beyond delta H2 and delta H2 local and we compute it exactly, what we find is that delta H2, introducing delta H2 exactly, performs worse than the truncation, the reason being that we have large effects close to the cut-off that spoils the low energy spectrum, okay? And the problem goes on, because now if you consider our operator, which is B times denominator, B times denominator, BN, just keep accumulating in occupation numbers and you're going to get further divergencies, okay? So, the way to go beyond the leading coordinate calculation of the local is not by introducing this expansion like this, it just fails for this kind of theory, okay? So, we are led with this conundrum that want to make sense of this original expansion that I was showing you for delta H, we want to make sense of this expansion, okay? And in order to reduce the error for delta Hamilton. So, in the last few minutes, I would like to sketch very quickly an idea on how to improve the Hamiltonian beyond the leading coordinate. So, how to make sense of this series, okay? So, in order to make sense of this series, the idea is to go back to this equation that we are considering at the beginning, okay? When we truncate the Hamiltonian in the first place, we are forgetting about the high energy states, okay? So, somehow, we should consider a basis, which is not simply the free basis, but takes into account the effects of the high energy states, okay? So, in particular, instead of considering simply the free basis, let's consider, we can consider a more clever basis, okay? So, let's consider now the following basis, okay? We are going to consider a truncated clever space, where we are going to have the lower energy states that we are considering at the beginning, plus some states that its aim is to try to reproduce this behavior of the high energy states of the wave function, okay? So, this is going to be spanned by the free theory states, the low clever space, and then we are going to introduce some states that we call Taylor states, okay? Which, of course, if we could compute those exactly, we would solve the problem, we cannot compute these states exactly, but instead, what we do is that we approximate the denominator of the full Hamiltonian by the free one, okay? So, we consider this kind of a state. So, this is the main idea, to consider this kind of basis, okay? And now the Hamiltonian, in terms of this basis, it's going to be a finite Hamiltonian, because notice that the span of these states is finite, there are, because this space is truncated to some energy, while this space, the tail clever space, is also finite. So, let me give you the same notation. Because it is spanned by a number of Taylor states given by the size of the low energy clever space, okay? So, now the Hamiltonian, so, they can value problem in this super space, it looks like this, so, we are going to have... So, borrowing the notation I was using before, I will have HLL, the low clever space, HLT, HTL, HDD, okay? But again, this is the low clever space, and now the high clever space is not the space of... the high energy state of the free theory, but it's the space spanned by the Taylor states, okay? Whose aim is to reproduce the way functions correctly. So, now, we can look at this finite matrix now, where we need to introduce a gram matrix to take into account that these Taylor states do not normalize to one. And now, the first thing to notice is that, of course, this approach is guaranteed to perform better than the pure truncation approach, just because we're expanding our basis, okay? Being this operational method, since our space is bigger, we are just going to perform better, okay? This is guaranteed. But then, what is interesting is to compare also this approach that I'm proposing here with this kind of basis, with what we are doing in the Hamiltonian truncation alone, okay? And for that, what we can do is that we integrate out now this part of the matrix, okay? So, if I integrate out this part of the matrix, I'm performing some sort of manipulation I did before, and is it clear what I'm doing? I'm not sure. One question, the diagonal term. This is the identity. Yeah, that one below. This one is a gram matrix just taking into account that I haven't normalized these states in the first place. So, this is how they can value problem looks on this set of non-normalized states. So, your approximation is where, right? You have two approximations, the silver space of HL plus HT is still finite dimensional, so you've still thrown away a lot of high-energy states? No. This was not the first approximation. The first approximation was substituting the full Hamiltonian by H0. This is an important point. Notice that, had I considered the state with exact Hamiltonian here, it would be a finite silver space which would still recover the exact again value equations. I mean, this wouldn't be surprised in the sense that before I introduced you an operator which was delta HL plus delta H exact, which was finite dimensional and was recovering the exact again values. So, the only approximation I'm doing is that you're ready to compute these state states, I need to know these inverse matrix elements. So, here is where I do the approximation, right? Maybe it would be clear if I write for you which are these matrix elements. So, let me write them down. So, this matrix element here is just the sandwich between the interaction v and these two states, okay? So, HLT is simply I, b, psi j, this type of state. And it's simply delta H2. And this is the nice thing about this approach. And then delta H is equal to minus delta H2 plus delta H3 plus G, ij. And G is the matrix which is just the overlap between the two states. Thanks, yeah. H, t, t, yeah, hi. This matrix here, yeah. Okay, so, what I was saying is that by contrasting this matrix is going to work better than the pure truncation which consists of considering only this operator. But now, what this has to do with the, with what I was saying before about approximation of delta H, okay? And to see which is the connection between this approach and approximations of delta H, what we can do now is that we integrate out this part of the Hilbert space, okay? So, now I'm going to do the same procedure with before. Before what we did is that we split the full Hilbert space between the low and the very high Hilbert space, okay? Which span up to infinity. Now, I divide it again with the same low plus this funny looking Hilbert space by details. But I can still integrate out this finite dimensional Hilbert space, okay? So, if you do so, you're going to get a similar equation as before just that you have a different delta H, okay? So, you're going to get HLL plus some delta H operator, okay? I put a tilde because it's just a different operator. Which, again, of course, depends on the epsilon. And, okay, this is going to act on some low energy state, epsilon i, okay? And this delta H tilde operator, it should be more precise because I already perform an approximation, this is not going to be the exact eigenvalues we started with. But, so, this delta H tilde operator looks like this. Delta H2 times delta H2 of 1 over delta H2 minus delta H3, okay? So, all this whole approach of introducing these new basis of states, the point is that the operator that we get, okay? It's equivalent to the original expansion that I was doing to approximate delta H up to higher order terms. Because, if now we expand these interactions of delta H in powers of the coupling, what we get from here is delta H2 plus delta H3 plus dot, dot, dot. Of course, the difference is crucial in these high energy terms, which is going to make this operator a good approximation to delta H. But, it still is an operator that includes the next two leading coordinate corrections, okay? And, so, so, briefly the point is that, so, originally we had a series that we couldn't make sense of, okay, which was the series expansion of delta H, okay? Because, any finite truncation of this series where we computed delta H and exactly, led to Hamiltonian that was very bad behave because it stays close to the cut-off, give very big contributions to the Hamiltonian. This was the original problem. The way we solve this problem is that, we're considering another ananzat, such that when we iterate part of theanzats, we recover the same series up to higher energy, higher order terms. So, in a sense, what we are doing is that we are connecting the good old fashioned perturbation theory with a variational approach that they agree up to higher order terms. So, in this way we are guaranteeing that including these higher order corrections, the results that we are going to get are better that we have not included them, okay? So, in principle, if you are doing... I was just wondering that this approach of actually introducing these tail states where these amounts actually have two cut-offs, like the first energy cut-off between the two low energy space and the tail space and the tail and the rest. Let me see if I understand what you are saying. So, in principle, for the tail states, here, I have the freedom to include as many tail states as I want up to the truncation energy. So, this index i, which is the same as this index here, spans in the low-hielbert space, okay? This is finite-hielbert space, which is spanned by these states. In principle, I could dial how big I want this hielbert space up to the maximum size, which was the size of HL. I, in principle, I could play with the idea, for instance, of considering a lower number of tail states and only including those that are more relevant for the physics problem that you are considering. For instance, including only those tail states, which have a big overlap with the vacuum, and not all of them. This is the cut-off that you were talking about? No, I think we need to discuss it, but I somehow thought that these tail states are the ones above the cut-off, but I think that's not what you're doing. It's very different. You cannot think about them as being above the cut-off. They reproduce the effects of the states above the cut-off. That's the... There are linear combinations of states on the cut-off. Yeah. There's some particular linear combinations. But it's not correct to somehow look that to somehow have a sandwich situation where you have a lower energy space in which you want to formulate your Hamiltonian. Then you have an intermediate window where somehow you treat the interaction between the intermediate window and the low energy one, a bit more refined one, and then you have the stuff which you only integrate, which you have completely integrated out. Is that not... So, I'm not sure... I guess you're referring about... OK, let's go later. OK, so this is the main idea in order to include a higher order correction into the Hamiltonian, tot i que, a més, es触a els perconstans. I'll just add it to the next l Carry-In Gordon. And... Let me see if I have some more comments. So, let me show you some plots about these corrections. So, what's the idea that you're going to... You have this delta H tilde now? Are you going to substitute the local approximation to go to the delta H? OK. per compre these delta h tilde. So I'm going to include these delta h tilde, exactly. And by exactly what I mean is the following. I'm not going to perform any, I'm going to compute as follows. So the point is how do I compute delta h2 exactly, no? I guess it is the question. Yeah, delta h2. Yeah, delta h2, delta hc, all these operators. El llit que fem per comparar aquests operatius exactament sense introduir algun error al cost del cut-off, és que... So, let's restore the indices, okay? So, this is going to look like this, no? I, some index H. So, of course, for a stage, as I already said, for a... I'm just dropping the epsilon because it doesn't matter for what I want to say. Let's restore it. So, for a stage below the cut-off, I can just do the local approximation. But what we can do is we split this sum on H between two chunks, okay? So, I write this sum as some H to Hmax, or BHH, plus the same from Hmax to infinity, okay? Now, if this Hmax is moderately above your cut-off, all the... of the original cut-off ET, all the states, even those that are close to the cut-off, are going to have energies much below Hmax, okay? So, say, Hmax decides that... let's call this something like EL, okay? This is the scale where local approximation applies to all the entries of your Hamiltonian. So, we're going to have... we're going to have some hierarchical ideas, okay? ET, yeah, okay? So, now, having introduced this second scale, I can approximate... I can approximate all the matrix elements of this operator by a local approximation. Now, this I cannot... I cannot approximate all matrix elements of this operator by a local approximation. However, this operator involves a finite number of sums, so thus I can perform them analytically still. So, the only approximation I'm going to do is that I'm going to perform a local approximation, but this local approximation does not start at the scale of ET, but a much higher scale. So, in this way, I can reduce the error arbitrarily small. And, same trick applies to delta H3. So, you've come from a delta H3, delta H tilde? Yes. Well, this is a pretty technical point. Maybe, since we've done... let's see the plots and then we can continue this in the discussion section. Okay. Just summarize the points that... The only approximation I'm going to perform here can be reduced to be very insignificant, yeah. So, virtually I'm going to introduce exact delta H tilde. But it's key that the large matrix element can be killed by the denominator, is that it? That's right, that's right. Yeah. Or made order one? Or made order one, yeah. That's the point, exactly. So, ensure you... It's saying that you found a good way to resum. The... right? Yeah. So, you can phrase it in these terms, if you want. This is a resumation which has a variational interpretation. Which is good, because then you know that you are just going to do better. So, I will show a couple of more plots. To stress it, why, yeah, these kind of corrections are... They behave very nice, so... I don't know where... Okay, so, first the plot I was showing you before. So, something I didn't comment, but let me mention it now, is that as you can see, this is a truncated Hamiltonian line, okay? Again, for the vacuum and for the... for the first excited state, okay? Now, the local correction that I was advertising before, it behaves very nice because it converges very quickly. However, note that we have lost the variational interpretation from below, instead of coming from above. And then this last line here is computed by performing this next-to-reading normalisation, okay? And then you see again that this line approaches from above, okay? For both the... well, for the vacuum and for the higher excited state, okay? This is a nicer way to see the convergence. So, here I'm plotting as a function of 1 over e t, cube and 1 over e t squared. So, in the left, it's for these NL corrections, with this L that is still there. And in the right, it's just performing the local approximation. And some other nice feature of this approximation that includes all finite next-to-reading order correction is that the extrapolation up to infinite energy behaves much more... So, these lines behave much more smoothly, so it allows them to perform some nice feed that can be separated much easier, while the lines for the local behave much worse. We tried many powers and there was no... there was no nice way to see about a smooth behaviour for the local approximation. Even if there are some oscillations. Yeah. There are some oscillations here, like, for instance, these red dots here do not behave smoothly while. You can compare these dots with respect to these ones here. So, this is good because if you want to perform the extrapolations with respect to the volume, it's good that, first, you can extrapolate with respect to the energy very smoothly and then study the physics of the finite size effects. So, what we did is that, using these corrections, we performed these extrapolations to infinite energy and then having the extrapolation up to infinite energy, we also extrapolated up to infinite volume using Lusier formulas, which give you exponential behaviour when you are away from the mass gap, but getting close enough to the mass gap but not on top, where I will also extrapolate the behaviour of the mass as a function of the coupling and see how the mass closes to zero and to measure this critical exponent and to match it with the expected critical exponent. And then, we can compare this with other methodologies that are in the literature and we get comparatively precise and in agreement with other methods. I need just to show you some of the precision calculations that we can do with this method. So, okay, some of these blah, blah, blah conclusions. And... I should start finishing. Maybe you can... So, tomorrow we will have a talk by Marco Sironio who will be basing his work on this plot. Yeah, I was going to comment now. No, no. Marco, do we need to discuss this plot in more detail or will you show it in... I will show the similar plot to the rivet that you seek to bashing till. Okay. Shall we compare? Shall we compare after your talk? Yeah, maybe, yes. Compare me, more or less, it's the same with some bigger error. Anyway, this is an impressive plot. I'm surprised I don't say more about it, because... Advertis more than Joana. Okay. Em... Okay. What's funny? Em... Okay. So, some of the conclusions I wanted to comment. Instead, of course, this problem of normalizing has some limitations that we would like to overcome. And... Okay. One of the obvious directions for these methods is to go to less relevant perturbations. In particular, this means, for instance, to consider to go to higher dimensions. I think that would be a very nice set of calculations. Em... In the direction of less relevant, one may also... stay still in two dimensions and consider less relevant perturbations when we have to care about the divergences and so on. But I think that would be one of the nicest things to do next. Other... More speculative ideas that one can try is to analytically continue the coupling to complex values and study, for instance, this plot I was showing you about the mass to show how it behaves for complex values of the coupling. This is an obstacle that we can directly access with this kind of approach. And this might be interesting physically, but also it may allow for a better extrapolation to the critical points by studying the behavior of these functions in the complex plane. And... Well, I think I will leave it here.