 Good morning, everyone. So yesterday, we tried to explain that if we phrase information paradox from the point of view of an observer sitting at infinity, the question is how can a thermal state, which is predicted by Hawking, be converted to a pure state by small corrections? And we argued that this is actually natural from the point of view of statistical mechanics and can be achieved with exponentially small corrections. On the other hand, we argue that what is more difficult is to reconcile this scenario with the smoothness of the horizon. And the reason for the difficulty is that in order for the horizon to be smooth, we need to have very specific entanglement of the quantum fields between the interior and the exterior, which was actually inconsistent with a previous statement that the radiation is unitary. And today, what we will try to do is to reformulate these questions into questions within the framework of the ADS-FT correspondence and we'll try to address them using the boundary conformal field theory. So before I go on, are there any questions about this? OK. So I suppose that you have some basic familiarity with the ADS-FT correspondence. So we will take the conformal field theory to be defined on a sphere cross time. So we take a CFT on SD-1 cross time. And some of the CFTs are supposed to be holographically dual to string theory on ADS-D plus 1 times some internal manifold. The internal manifold will not play any role in what I'm going to say from now on. So I will ignore it. And we will focus on the ADS part. And the idea is that everything that happens in the bulk is holographically encoded on the boundary. Now, before I go on, I want to remind you of something basic, which is that in ADS, in particular, in the backgrounds of this form, we can classify black holes into small black holes, those whose radius is much smaller than the ADS scale, and large black holes where the radius is larger than of the order of the ADS scale, let's say, and larger. And small black holes in ADS can evaporate. And we can consider the information paradox for evaporating black holes in ADS. While large black holes in ADS do not evaporate, in particular, so they emit Hawkin radiation, but the Hawkin radiation gets reflected by the potential barrier of ADS and the black hole reaches an equilibrium with Hawkin radiation. So large black holes do not evaporate, but as we will explain later, we can phrase similar paradoxes for large black holes. And the point is that small black holes in ADS are not very well understood from the point of view of the CFT, while large black holes are supposed to be dual to thermal states in the conformal field theory. And in particular, at the level of microstates, the black hole microstates of a large black hole in ADS are supposed to be dual to the coagulant plasma microstates of the deconfined theory at high temperatures. And in particular, this correspondence is very successful in explaining the entropy of black holes, because we think of the black hole entropy as being counted by the different microstates of the coagulant plasma. So now if we ask the question about the information paradox in the most basic formulation, in particular, whether a black hole evaporation is unitary, ADSFT predicts that the evaporation is manifestly unitary because the boundary conformal field theory is unitary. So you can imagine creating a black hole in ADS by the collision of two particles that you create on the CFT by some local operators, which will eventually, if the black hole is small enough, it will eventually evaporate into particles. And this process can be described within ADS. And it is equivalent to some process in the conformal field theory. So it is manifestly unitary. And then there's no longer any discussion about information loss. Now, so the question of whether information is preserved or not is settled just from the fact that gravity in ADS is equivalent to a conformal field theory on the boundary which is unitary. But as we explained yesterday, another question is what happens in the interior of the black hole behind the horizon of this black hole. And in particular, we would like to understand whether there is anything singular on the horizon which we discussed yesterday from the point of view of the CFT. So as I already said, we would like to understand whether these black holes in ADS have a smooth horizon. And in particular, we will phrase a question for large black holes in ADS. So at first, you might think that for large black holes in ADS, since those black holes do not evaporate, there is no paradox. There's nothing to worry about. However, in the next few slides, I will show that even for those black holes, large black holes, it is very difficult to reconcile the smoothness of the horizon with unitarity in the dual conformal field theory. And because these black holes are very well understood from the point of view of the boundary, this formulation of the paradox is the most precise from the mathematical point of view. Now, in order to address the question of the spacetime near the horizon of a black hole, we need to understand how to describe local physics in ADS because the question of the study is what happens to an observer crossing the horizon of a black hole. So it is a local question. It has to be phrased in terms of, for example, correlation functions that the infalling observer would measure when crossing the horizon. So the first step is to remind you how we can reconstruct local observables in ADS from the point of view of the boundary. Now, we will be working with a conformal field theory which has a gravity dual and we will always work in the larger limit and also the strong coupling limit. So you can think of the n equals four at large n and large lambda. And then there is a basic understanding of how to represent local fields in ADS, approximately local fields in ADS in terms of the so-called Hamilton-Cabat-Liffes Lore construction, the SQL construction, which is the following idea. Suppose we have a field phi in ADS, let's say a scalar field in ADS, which is dual to an operator O, a single-trace operator in the conformal field theory. Now, this field phi in ADS, in the larger limit, it would behave like a free field because all couplings in the theory are controlled by one over n. So in the larger limit, this field will obey the Klein Gordon equation in ADS with some mass which is related to the conformal dimension of the operator O. And moreover, we have a boundary condition for the field near the boundary of ADS, which is that as we approach the boundary of ADS, the bulk operator should be identified with a boundary operator at the same point x, provided that you multiply by some overall factor of z. So you have some field of being an equation in the interior of ADS, and you have some boundary condition. And this allows you to solve this problem and express the field in the interior in terms of the boundary values of the field, which is the dual operator O. So the most convenient way to solve this problem is by taking the operator of the boundary O and fully transforming it on the sphere. So we expand this field in spherical harmonics on the sphere and in fully emulsion time, thereby introducing these operators O, omega and m. So let me define them here. So these objects are defined as the integral of the boundary of the boundary, the local single trace operator, multiplied by some spherical harmonics. So these are the Fourier modes of the boundary single trace operator. And then the point is that if you take those modes and you multiply them by some wave functions which are solutions of the Klein Gordon equation in ADS, you can write down an operator in the CFT, which can play the role of the local bulk field in the larger limit. So as you can see, this object depends on the coordinates of the omega as well as the coordinates z with the regular direction ADS. So naively you might think that this is an operator in ADS, but the statement is that this is a CFT operator. You can see on the right hand side that we have written down this object in terms of CFT operators times some wave functions. And the point is that this object seems to depend on the coordinates of ADS, but it is a CFT operator. And in particular, you can verify that in the larger limit, if you take two points in ADS, P1 and P2, and if these two points are space-like with respect to the ADS metric, these operators, these two operators will commute. So in some sense, this operator, the CFT operator, can reproduce the locality of the higher dimensional spacetime, the causality structure of the higher dimensional spacetime. Let me emphasize that this, from a CFT point of view, this is a non-local operator because we have taken the local operator O and we have smeared it on the boundary, both in space and in time. So a little bit of, it's a peculiar object where we have to act with the operator at different times, but this allows us to reconstruct the point in ADS at a, a field in ADS at a particular point. Now, this construction is not exact. In particular, it is perturbative in the one-over-n expansion. So if you look at this commutator, and if you consider the one-over-n corrections to correlators on the boundary, you will find that this commutator becomes non-zero by factors of one-over-n, and then you have to go back and correct this definition by one-over-n corrections in order to restore locality in the bulk. So this entire construction is perturbative in one-over-n. So we don't know how to do it non-perturbatively, so we have to live with this construction for now. And also, I would like to emphasize that this construction is not entirely satisfactory because in order to define this object, we have to use the solutions of the Klein-Gorton equation in ADS with sort of assumes that we have a bulk dual with a particular geometry. And in that sense, it sort of uses the bulk equations of motion in order to write down this object. So one would hope that a more fundamental representation of these operators will be found one day where we don't have to put in by hand the fact that the dual geometry is ADS. But for now, and if we want to work in the large limit, this is a good starting point in order to explore local observables in ADS. Another way of thinking about this object is that, so by inserting these wave functions here and replacing this Fourier transform in terms of the local operator, you can change the order of the integrals and you can represent this local operator in ADS in terms of an integral over the boundary operator multiplied by some kernel, which we can write down explicitly in the case of ADS. And roughly speaking, you can see that as the point moves deeper into ADS, you can the support of this kernel of the boundary increases. So in this way, we can reproduce a local physics in empty ADS from the point of view of the boundary. And now we want to do the same thing in the presence of a black hole. Now, before I go on, I would like to say a few things about the types of black holes that we will be considering. So yesterday, there were a few questions about the ensemble, do we fix the temperature, do we fix the mass and so on and so forth? So let me make some comments about that. So one class of black holes that you can consider in ADS are black holes that you can form by gravitational collapse. For example, you throw in some matter in the CFT at some time t equals zero, it collapses into a black hole. If the amount of matter is large enough, it will be a large black hole and it will equilibrate and it will stay there. Now, I want to say that this type of black hole is not the most general state that you can have at a particular energy in the CFT. And if you remember yesterday, we introduced the notion of a typical black hole microstate which I will make a little bit more precise now. So we take all energy eigenstates of the conformal field theory which lie within a particular window of energy. So in the case of energy transport, this E naught would be some energy of order n squared. So we take some energy in the deconfined phase of the theory and this energy window delta E we can take to be something which is, let's say order one or we could even make it smaller but that is sufficient for our purposes. So we take a very small window of energy and there are many microstates, many eigenstates within that window. And then the typical black hole microstate will be a linear combination of those states with coefficient Ci which are randomly chosen. Now, the precise definition of randomness is in terms of the harm measure which I introduced yesterday. So remember we think of the Ci's as parameterizing points on the very large sphere. We define the uniform measure on the sphere and this defines the notion of randomness for the Ci's. So this guy here is what we will be calling from now on a typical black hole microstate in the conformal field theory. So the important thing to remember is that the energy of the state is sharply fixed, sharply defined. Okay, so now an important point is that these type of microstates the number of states that you can form by gravitational collapse is much smaller than the total number of typical microstates. This is a simple estimate you can do. You can consider all possible initial conditions, possibly initial microstates for a star of given mass M which can collapse and form a black hole of mass M and if you estimate that entropy which we can do perhaps during the discussion later today you will see that that entropy is parametrically smaller than the Decay-Stein-Hocken entropy. So in particular, the entropy, if we work in ADS-4 the entropy of a star of mass M scales like M to the three over two while the entropy of a black hole scales like the area which goes like M squared. So you notice that there are many more black hole microstates than the microstates that you can start with if you have a star that are going to collapse. So because of this difference, it's important to keep in mind that there is a distinction between black hole which are formed by gravitational collapse in ADS and typical black hole microstates and there are many more microstates of this type that those you can form by gravitational collapse. Some of the states- Kiriakos, can you repeat the question for everybody? In this story, in how many dimensions do we work? Do we work in ADS? Can we do any of this in ADS-2 or ADS-3? Well, ADS-2 and ADS-3 have special properties. So most, I mean, the statements I'm going to make are certainly true in ADS-4 and higher. Some of the statements may have to be refined for ADS-3 and ADS-2. Are there any other questions? Yeah, say that again, please. No, good, the question was, do we have a paced curve for this black hole, a paced curve? Well, these black holes do not evaporate. We have decided to look at large black holes in ADS. So these black holes do not evaporate. They are in a thermal equilibrium with their Hawking radiation, right? So the black hole is sitting there, it emits Hawking radiation, but as you know, in ADS, there is a gravitational potential which is pulling everything towards the center. So these Hawking particles will get reflected and they will fall back into the black hole and then the black hole reaches an equilibrium where it does not evaporate. So there's no paced curve. I'm sorry, could you maybe rephrase the question a bit? I did not understand. The question is, is this the ADS analog of the picture I was drawing yesterday for flat space? So in flat space, I drew two different pictures. I drew this one. This was a collapsing star. I also drew the one where we take evaporation into account, right? So this black hole does not evaporate, so it is the equivalent of this one. All right, so the rest of my talk will use this notion of a typical black hole microstate quite a lot, so I hope the definition from the CFT point of view is clear. And I want to emphasize an important property of these states. The important property is that if you select this coefficient C i randomly, this stays to be almost time independent. How do we see that? Well, you calculate the time derivative of some operator A, or the state psi, the state psi, and you estimate the size of this time derivative provided that these C i's are selected randomly. Now, let me explain this estimate. So we have this operator A that we will normalize so that we will normalize A so that the expectation value of A squared on the state is of order one. So we take some operator whose two point function on the state psi is of order one. Now, if you take this relationship and you estimate the size of the matrix element of the off diagonal matrix elements of this guy, you can show that these are exponentially small by a factor of e to the minus s over two. The idea is that if you square this guy, you get e to the minus s, and then there are e to the s states that contribute because there are that many eigenstates, and then in order for this to be order one, then the individual matrix elements must be of the order of e to the minus s over two. I will explain this in more detail later if you want. So these matrix elements are of the order of e to the minus s over two, and the coefficients c i are of the order of e to the minus s over two as well because we have e to the s coefficients in that sum and sum of c i squared must be equal to one because the state must have unit norm. So this means that each one of these guys must have magnitude of the order of e to the minus s over two, and since these guys have been chosen randomly and they're complex numbers, the phase of this object will be randomly distributed. So then we have this sum over c i, c i star c j times i j, and we want to estimate the size of the sum. So we get a factor of e to the minus s over two from this coefficient, another factor of e to the s over two from this guy times e to the minus s over two. But then we are summing over i and j, so you would normally think that we get the factor of e to the two s, and then if you multiply all of these guys together, you get something which is of the order of e to the plus s over two. However, you have to take into account that these numbers are complex numbers with random phases, so if you sum e to the two s random phases, the typical size of that object is the square root of e to the two s, so in that way we lose one power of s and the whole thing scales like e to the minus s over two. So this shows that if you take a random superposition of energy eigenstates in this quantum field theory, the states will appear to be time-independent, up to exponentially small corrections. So in other words, what we will be calling typical states can be identified what we would naturally call an equilibrium state in the conformal field theory, which we define by states where the time derivatives of observables are close to zero. Yeah. Hermeser is what I defined yesterday. You think of the CIs as parameterizing points on a sphere, this is an equation for a sphere. You define the uniform measure on the sphere, that is the Hermeser, yeah. Okay, so now let's start thinking of, so these seem to be black holes which are in equilibrium, so if you think about the Penrose diagram in ADS, it will have the following form, it will look like this, at least we will start talking about the exterior and then we'll discuss what happens behind the horizon. So for now, do not pay attention on this part of the diagram, we just want to study this one. So these states seem to be in equilibrium, so we expected the geometry, here will be the ADS-Roschild geometry, then the equation we'll address later is what happens behind the horizon, but this part is expected to be ADS-Roschild. And we want to consider local operators or fields in this region of space-time, and we want to reconstruct them from a boundary. So the starting point will be to analyze this problem in effective theory in gravity and then we will see how to reproduce the same result from the CFT. Now, if I give you a black hole in ADS, which is in equilibrium, so it is a static black hole in ADS, you can quantize the field on the background of a black hole and if you demand that the field is in equilibrium, you find the following results. You take the field and you expand it in some modes B times some wave functions, which are modes in ADS. So these Fs are the solutions of the Clang-Orton equation on the background of an ADS-Roschild black hole. So you expand the field in a base of modes in the bulk and then you introduce some operators B, which will obey the usual algebra. And as you can imagine, what you find if you do the calculation in effective theory, so it's similar to the calculation of Hawking, is that these modes are thermally populated with a temperature beta, which is determined by the master black hole. So in the bulk we have this mode B, we seem to be thermally populated and they can be used to build up the field phi, describing a Clang-Orton field in the exterior of an ADS black hole. So now we want to see how we can reconstruct this from the CFT point of view. The equation was the spread of energy delta E, how does it enter? The answer is it does not enter. So this statement here is about effective theory in the bulk. So in the next slide I will talk about the boundary CFT where this spread of energy would enter. But let me say that for this calculation that we're doing, it doesn't matter what the spread of the energy is, it can be quite small. And in fact, if the N equals four obeys the Eigenstate Thermalization Hypothesis, which I think it is generally expected that it should, then you can take the spread to be almost zero. So you can even take an energy Eigenstate. Yeah, yeah, so this is what is called the ADS Hartle-Hulking vacuum because we assume that the black hole is in equilibrium. Which follows from the previous statement, right? This typical states, the time independent up to exponentially small corrections. So it is reasonable to assume that the field in the bulk is going to be in an equilibrium state, which is given by the ADS Hartle-Hulking state. Okay, now let's move on to the boundary. So we have this microstate psi, which has energy over the N squared. And then we need to use a property of single trace operators on this state, which is that if you consider a endpoint function, small n, of single trace operators on this typical black hole microstate, then this correlation function factorizes into products of two point functions. Now, is the statement obvious to everyone or should I explain why that is the case? So one way of motivating this expansion is the following. First of all, we consider these objects here. It is a correlation function on a typical state. So we have psi, o, o, o, psi. And as a first step, we can use the results that we presented yesterday. So I wrote down a theorem that any correlation function of this form is going to be very close to the micro-canonical correlation function up to exponentially small corrections. This was a theorem I mentioned yesterday, where rho m is a micro-canonical density matrix centered around the energy of the state psi. Then you can also approximate the micro-canonical ensemble with the canonical ensemble. So this is going to be equal to the trace of rho of e to the minus beta a's over z times o, o, o plus one over s corrections. So this shows that the correlation function of single trace operators on a pure state are going to be equal to the correlation functions on the thermal states plus one over s corrections. Now, these guys, we can argue that they do factorize. For instance, you can think of it in terms of double line diagrams. This thermal correlators you can formulate in Euclidean signature by compactifying the thermal circle and then you can think about it perturbatively in double line diagrams. And then the usual top expansion holds so we expect that this correlated factorize. And then by this reasoning, we conclude that even the correlators on a typical pure state are going to factorize at large n into products of two point functions. This is the usual large n factorization applied to a pure state, a pure heavy state. However, the two point function in which they factorize, so this object here is not the two point function of the operator in the ground state. Rather, it is the two point function of the operator on a thermal state. So this big guy, this correlator factorizes the products of two point functions but each of these objects is actually very complicated and it's not easy to compute it. It is a thermal two point function. So however, even though we cannot really compute it, we can say a few things about it. For instance, we can verify that it obeys the so-called KMS condition which is a statement that if you take the two point function of this operator O at finite temperature and you analytically continue the time argument up to minus high beta, then you get back the same correlator with the order of the operators interchanged which is the result we get this minus two and minus six. So proving this is very simple and we can discuss if you have questions but it's a very standard property which you can think of basically as periodistic Euclidean time. So this is an exact statement even though we cannot calculate this correlator, we know that this has to be true and we will use it in what follows. So as before, we define this boundary Fourier modes of the field. This is what I wrote down there and the KMS condition can be translated into Fourier language and then what it says is that if you calculate O dagger O, then you get this factor into the minus beta omega times O dagger. All these correlators are correlated with finite temperature. So this is a statement about finite temperature correlators in the N equals four. Now, remember we had this expansion in the bulk. This was the field in the bulk which was expanded in terms of this mode B which played the role of creation and inhalation operators of particles around the black hole. And now we will try to identify this B with the boundary operators O, the Fourier modes of the field. Now, when we do this identification we also have to take into account of the normalization of the operators and in particular we will divide this boundary Fourier mode O by the square root of the expectation value of the commutator. And the reason we are doing that is in order to ensure that the commutation relations between the B and the dagger are the canonical ones. The reason that we have to do it is if you define the Fourier mode of the boundary field in this way, then it is not true that O omega, O omega dagger is equal to, well, let's do it precisely. It is not true that this is proportional to delta. There's some function here, G of omega which is not trivial and we cannot calculate it. Because the theory is totally coupled. But by dividing by this factor we get rid of that factor and we can define operators which have canonical commutation relations and can be identified with the creation and inhalation operators in the bulk. Yeah. Yes. Well, here we are working with the primaries which are dual to, so the question was, does this equation hold for all primaries? So here we want to reconstruct the fields in the bulk that you can see at the level of spragravity. So we want to concentrate on primaries whose conformal dimension does not scale with n or with lambda. So the conformal dimension of this guy is taken to be something of order one. Finally, it is a simple algebra then to check that if you define this operators B by identifying them with O then and using the KMS condition you reproduce the expected thermal occupation level for these guys. So remember somebody asked me yesterday whether we are working in the canonical or the micro canonical ensemble? Well, here we see that even though the state is self psi, the state psi is a state in the micro canonical ensemble if you consider the quantum fields in the exterior of the black hole, they seem to be thermally populated with a temperature beta where beta has to be related to the energy of the state in the micro canonical ensemble by the usual relationship between micro canonical and canonical which allows you to translate a particular temperature to a particular energy. Well, so now we identify this B with O which allows us to express the local field in the exterior of the black hole. Sorry, I will repeat the question in a second. So we identify this O with a B's and now we can write down an operator in the CFT by using these O's multiplied by some wave functions and then you can check that this field phi, the correlation function of this phi on the pure microstate reproduce the correlation function that you would calculate in effective field theory in the background of a black hole where you take the quantum state of the field to be the Hartley-Holkman state, right. So did I answer your question? Yes, I'm a massive correlators. I'm trying to reproduce the correlators of the bulk by calculation of the boundary. Yeah, so the question is, is this supposed to be an exact identification? So this identification is supposed to be truly at large N at the level of operators. Stating in please. So the question is, are we making an operator identification or simply the fact that the correlators agree? The answer is that, as I said before, even in empty ADS, when we write down these equations, even without a black hole, even in empty ADS, unfortunately we cannot do something stronger than mass of the correlators. So it's not a full identification of operators because we do not know how to extend this construction to all orders, not even to all orders in all of our N or even what would be even harder to do it non-perturbatively in N. So at this point, the reconstruction of the bulk is perturbative in one of our N. So I can identify the operators only at a particular order in one of our N. But that is okay because the paradox can be perfectly formulated, even working with a leading order at large N. So even to a leading order at large N, there is a paradox that has to be resolved. So the question you're asking is something even more difficult, right? Would be to find the exact operator identification between the boundary and the bulk, which is not something we are able to do right now. So in this way, we can define the CFTO operator which seems to reproduce the correlation functions of the bulk operator around in the exterior of an ADS black hole, evaluated on a typical black hole microstate. Okay, now, so everything is fine in the exterior. I just want to make one small comment that if you follow this reasoning, you can derive some bounds about the variance of this object among different pure states following arguments similar to those that I explained yesterday. And this impose very strong restrictions about the possibility that the geometry outside the horizon is modified on pure microstates. So sometimes there are proposals that the geometry of a given microstate of different black hole microstates may differ depending on the microstate. But this construction impose some restrictions because if different microstates have different geometry, then these correlators will have to have a very large variance among different pure states, which is excluded by some of the arguments that I presented yesterday. Okay, so this was everything I want to say about the exterior of the black hole. And then the question is, how can we move? What happens if we try to move behind the horizon? Yeah, so I think the question, let me repeat the question. The question was that this microstate side that we're talking about is a superposition of many different energy eigenstates. So the question was whether each of the elements of the superposition could be very different from the average, was that the question? Yes, that's the possibility that would contradict what is called the eigenstate thermalization hypothesis which postulates that energy eigenstates look very similar to each other, basically, and to the thermal state. And as I mentioned yesterday, your question is related to something I mentioned yesterday which is that I argued yesterday that typical states look identical. That's what I argued, right? However, it is possible to find a basis of atypical state. So you can find the full basis of the Hilbert space of states where each of the elements of the basis is atypical, that is possible. I can give you an example in terms of a spin chain if you want. It's very simple to see that. So what I wanted to say here is not that you cannot find some atypical states whose geometry will be different, there could even be a basis of states. What I wanted to say is that if you take a random superposition of those states, the geometry will always look like the ADS-Schwartz geometry. Take it, please. I'm sorry, I cannot hear you. Can you speak a bit louder? Yes, it can be completely orthonormal. Okay, let me give you an example. Take a spin chain. So you take n spins, right? So the spins can be up or down, right? So we have n spins, and we want to write down a typical state of this spin chain. The typical state of the spin chain will be a superposition of these states where we have definite spin. Let's call them i. So i is a state where each of the spins is either up or down. So these i's are eigenstates of the spins. And a general state is a superposition of all of those where this i goes from one after two to the n, right? This is a typical state. If you calculate the expectation value of a spin of this state, you will find that it's zero, right? And it's exponentially close to zero based on the theorem that I mentioned earlier. You can really show that if you just randomly select these i's and you compute the expectation value of the first spin, it's going to be exponentially small. However, these guys, they are an orthonormal basis. In each one of those, the spin is either plus one or minus, plus half or minus half. So it's very far from the average in each of those states. So this is a basis of eight typical states for this particular example. Okay, yeah. The question is, do I use regularity at the horizon? Yeah, so this F beta WM, they are constructed in some ADS-Schwarzschild black hole geometry. Using some regularity at the horizon? Yeah, so these were the modes that you would get assuming the naive ADS-Schwarzschild geometry where you quantize the field in the exterior and you get a continuous spectrum, right? In particular, you do not impose any boundary conditions at the horizon. You just impose normalizable boundary conditions at infinity and you get a continuous set of modes. Which I'm using here. So no regularity condition at the horizon? Well, this is a usual calculation. Near the horizon, the field is a linear combination of in-going and out-going modes. I don't know if that answers your question. So the modes, well, each of these modes, if you take a particular frequency, it's highly oscillating near the horizon, but that always happens. Even in regular space, if you expand in base of regular modes, you get these highly oscillating modes, which you can then superimpose to form smooth wave packets that can just go through the horizon without any singularity. So here I am assuming, what I am assuming here is that the classical background is the ADS-Schwarzschild geometry. And these F betas are more like the Rindler modes? Exactly, exactly. Okay, yeah. The question is, is it possible to go beyond semi-classical picture? If you are talking about including one-over-end corrections, it is possible, but it's a little bit difficult because when you start going to sublimiting order in one-over-end, there are two different effects that interfere. One effect is that you have to take into account interactions in the bulk. But at the same time, this difference between the micro-canonical and the canonical ensemble gives you corrections of the order of powers of one-over-end which will mix with the other types of corrections. So there are two sources of corrections. And in general, it's difficult to disentangle the two and to control, in particular, it's not very clear how to control these corrections in a strongly coupled theory. The one-over-end corrections, when you convert from the canonical into the micro-canonical. But as I already said, that is okay for us because the problem is already visible to leading order at large end. So before we even go to the sublimiting orders, we have to resolve a problem. Okay, so now we want to move behind the horizon. And as we discussed yesterday in the context of Rindler space, if you want to be able to cross the horizon smoothly, you need to have modes on the two sides of the horizon which must satisfy particular properties. So these guys which were the Rindler-like modes we were discussing before have to obey the commutation relations that I already wrote. And then we need a new set of modes, B tilde, which must obey similar commutation relations and they have to commute with a Bs because they're space-like separated. So B tilde and B are independent modes so their commutators should be zero. And so in particular, if we want to have a smooth horizon we need to impose, we need to identify some operators B tilde in the conformal field theory which will obey commutation relations of this type. Please notice when I write one, well, to be more precise this should have been a delta function of omega minus omega prime and you can also write down the angular momentum indices M but they don't play any important role so I will skip them from now on. I will just write down the frequency omega. Now these are some of the conditions that we need. In addition to those, we need the condition that both the Bs and the B tilde's have to be thermally populated in order for the horizon to be smooth. Now if you don't understand how to derive these relations the best thing you can do is to go back to Rindler space and work out the similar structure there and you will see that in the Minkowski vacuum these conditions are satisfied of course and these are also satisfied where for Rindler space beta would be equal to two pi so the temperature has a particular value. And you can also verify in the toy model of Rindler space that if you modify any of these conditions you start to see excitations on the horizon. That's what I mentioned yesterday that if you modify, sorry I forgot one more equation. We also want that B omega, B omega tilde to be e to the minus beta omega over two omega, B omega dagger. So we also need the most to be entangled in a very specific way. And if you violate any of these conditions you will generate a stress test on the horizon which will indicate that the horizon is no longer smooth. So the question now is can we identify CFT operators which satisfy these conditions? Yeah. Yes. Yes, yes, yes, yes. Yes, right. We will discuss this point a little bit later. So the question is in Rindler space we drew the diagram yesterday we had the B's defined on this side and the B tilde's on this side and then we said that these B tilde's will propagate in the future like that. So an infalling observer will detect the B tilde's coming from the left side and the B's coming from the right side and they have to be entangled in a specific way for the horizon to be smooth. So the question was can we think of these B tilde's as coming from the analog of the left side? That's a very good question. We will come back to this in maybe in the next lecture. So I will talk about this. Can you say it again please? What do you mean by similar equation? Yeah, yeah, so the question is do we assume that these B tilde's will have similar equations as the ones that I was writing in the previous slides? We're trying to get there. For now what I'm writing down is the minimal set of conditions that we need in order for the horizon to be smooth. So I'm not making any statement at this moment about how this B tilde is realized in the CFT. I'm not saying that this B tilde is related to some single trace operator. I'm just saying that if we want the horizon to be smooth there must be an operator B tilde which has these properties. And we'll try to identify this operator in the next slides. Okay, so now let me make a comment about where these operators B tilde are coming from which is actually related to your question. So if you consider a black hole from bi-gravitational collapse in ADS these B tilde's are nothing else but the guys that you can, as we discussed yesterday, you can trace them back through the collapsing star and see where they came from on the CFT. So they will get reflected on R equals zero and then these modes will go back and hit the boundary. So if you have a black hole from bi-gravitational collapse then you could try to identify these B tilde operators by tracing them back and identifying them by some operators in the CFT at a point which is actually before the moment where the collapse shell was injected into the CFT. However, this type of, this strategy to identify the B tilde's has some problems. The first problem is the so-called transplantion problem which I already discussed yesterday and the problem is that if you want to find these B tilde's at very late times and if you trace them back you find that they get blue shifted so then they collide with a star at very high energies and then it's not clear anymore whether the one of an expansion will be reliable. But apart from this problem, the other issue is that as I explained before the number of states that you can generate bi-gravitational collapse is a small subset of the number of states which is counted by the Beggarstein Hawking Entropy. So these are not typical states. The states you can form bi-gravitational collapse are not typical black hole microstates. So if we want to identify the B tilde's for a typical black hole microstate we cannot do it by starting a black hole which is formed by gravitational collapse. Okay, so let us proceed now and explain what is, explain why it is difficult to identify these operators. Does not, the question was is the collapse matter null matter? It does not have to be null, it could be time like. Okay, so let me now try to explain what is the problem. Now I'll give you a more general argument against the existence of these operators which will make clear why there is a paradox. So before we wrote down the algebra of the B's and the B tilde's, but what I did not write down in the previous slide was the commutator of this guy's B and B tilde with a Hamiltonian. And now I will introduce that commutator for the modes which are outside the horizon. Just by using effective theory you can verify that the commutator of the Hamiltonian with a creation operator B Dagger is equal to omega times B Dagger. That makes sense. This object is a creation operator. It adds one particle outside the horizon and the energy of the CFT increases. That is intuitive, that makes sense. On the other hand, if you look at the commutator between the CFT Hamiltonian and the B tilde Dagger, you find that there's a minus sign relative to that one. So this means that this B tilde Dagger operators have the property that when you act with a creation operator of B tilde Dagger, you're actually lowering the energy of the CFT. So if you add a particle here, you're lowering the energy. If you add two particles, you're lowering the energy even more, so and so forth. So as you can imagine, that is an unstable situation because then you would imagine that the system would thermally produce pairs of particles, B and B tilde without, there's no energy cost in the production of these pairs, but you would get some entropy. So it sounds like an unstable situation that the fact that we can add particles and lower the energy and as I will explain, there's a precise mathematical contradiction in this algebra that I will explain. But just to give you some intuition why we get this minus sign, you can think of the Hamiltonian as being the generator of the killing isometry of the solution. So the killing isometry is time-like in the exterior. So it's time translation in the exterior, but in the interior it becomes space-like. So the killing vector field in the interior of the black hole is space-like, which means that these eigenvalues of the operators with respect to the vector field play the role of momentum so they can be positive or negative from the point of effect theory. And that is the reason that you can find some modes in the interior which have effectively negative momentum with respect to the killing vector field, but when you translate back on the CFT, it becomes negative energy. Okay, so now let me explain why this algebra is inconsistent. So we will try to calculate the expectation value of the number operator for modes behind the horizon using this algebra. And we will take the trace because we want to calculate the average of this quantity over all possible microstates. Now let me call this object a small n. It's the expectation value of this number operator. Now using this algebra, if I use the condition, the commutator is between the dagger is equal to minus omega, the dagger. From this one, we get that e to the minus beta h between the dagger is equal to e to the beta omega times between the dagger e to the minus beta h. Okay, so now I will apply this relation inside the trace. So we will change the order of these two guys which will give us e to the beta omega times trace e to the minus beta h, beta tilde. Now we can use a cyclist of the trace to rewrite this as e to the beta omega trace minus beta h between the dagger. And now I can use the other commutation relation, this one, to rewrite it as e to the beta omega times. Trace e to the minus beta h be tilde dagger, be tilde plus one. Okay, but this quantity here is precisely what we call n in the beginning, right? That's the thing we want to calculate. So looking at this equation, what we found was that n is equal to e to the beta omega n plus one. Now we can solve for n and we find that n is equal to minus e to the beta omega divided by e to the beta omega minus one, which is negative. So we started with an operator which was positive definite because it is, sorry, it's non-negative. It is of the form bit dagger b, bit tilde dagger bit tilde. So this is a non-negative operator. And we're trying to calculate the trace of this operator multiplied by density matrix which has positive eigenvalues. And we get something negative. So this is inconsistent, right? So this shows that postulating this algebra, I mean, this algebra that we wrote down between the dagger is equal to one, h between the dagger is minus omega between the dagger has some sort of inconsistency if you try to evaluate the expectation value of this trace. Yeah, is that stop? Yeah, so just two minutes, right? So we get this inconsistency, which means that that algebra has some issue. So are there any questions about this? Yeah. This Hamiltonian is the safety Hamiltonian. Well, okay. The question is, is there another problem here because the Hamiltonian seems to be unbounded from below? Yeah, so somebody could say that that is not maybe a very big problem because these equations are supposed to hold only at the level of effect field theory, which means that you're allowed to lower the energy only a certain number of times which does not scale with capital N. And remember that the black hole that we started with had energy of order n squared. So we start with the black hole of energy of order n squared. So in order to get to zero, to zero energy, we would have to lower this energy of the order of n squared times, which would not be allowed at the level of effect field theory. But this problem here is present even let's say low point function of this B tilde dagger. So even if you have a small number of times, you run into contradiction with this, because of this argument. Now, you could ask me a question which is you claim that you derive these equations using effect field theory. And now you claim that there is a contradiction that you get this negative trace which is mathematically inconsistent. So this algebra must somehow be problematic. And if everything we have done so far is effect field theory, you could ask why hasn't this problem been noticed many decades ago? Because this problem was actually pointed out by these authors very recently, I mean a few years ago. So you could ask why wasn't this noticed let's say in Geryll and Davis, right? When they analyzed the quantum fields on the background of a black hole. The point is that this inconsistency, we show this inconsistency by using the trace. Which means we try to demand that this algebra is true on a very large number of states which allowed us to replace the typical state by a trace. In Geryll and Davis, there would have been only one state psi of the exterior, for example, the Hartz-Hocken state. And then one of the steps that they used which was a cyclist of the trace would not have gone through. And you would not have been able to derive any inconsistency. So to summarize, this algebra is inconsistent if you demand that the algebra is true on a very large number of states. If you only have one state or a small number of states, there is no intrinsic inconsistency with the algebra. Okay, so to close, the fact that we have this inconsistency suggests that it is impossible to find operators be tilde in the CFT which have the desired algebra for typical states. And hence, typical states will not have a smooth interior because you cannot find these operators in the CFT or perhaps the CFT is not able to describe a black hole interior. So in the next hour today, we will explain some ways of trying to avoid this problem and identify these operators be tilde and we'll explain how this, what's the loophole in this argument. Thanks.