 So many thanks to the organizers for inviting me to this very exciting workshop. I'm going to talk about some connections between the black hole, physics in the black hole interior, and the process of thermalization in a strongly coupled quantum theory. So is this working? Yeah. So in particular, I will try to talk about, I will present a natural class of non-equilibrium states, which should exist in any quantum chaotic system, and which may be interesting for two reasons. First of all, for the relevance regarding the describing the black hole interior, and second, they may have some role to play in statistical mechanics more generally. So we heard in a previous talks in this conference about the connections between black holes and statistical mechanics. It has been known for many years that the load of black hole dynamics can be related to the load of thermodynamics. Black holes have entropy. They can be used to study the dynamical properties of strongly-coupled plasma or condensed matter systems. So you can compute the transport coefficients. Many of these applications of black holes can be related to the fact that the geometry outside the horizon of black hole is universal. So after you form the black hole, the details of how you form it are not very important. You always end up with the same geometry, which captures a universal physics of the strongly-coupled dual theory. In that sense, the interior of the black hole is equally universal. The geometry just behind the horizon in a classical GR is as universal as the exterior geometry. But it's not very clear what the role of the interior geometry is from the point of view of thermalization in the Gears theory. So one connection which is not very clear is that correlation functions in the quantum field theory, which are computed along this contour, as Jan explained in his talk, where we shift to some of the operators by i beta over 2 along the imaginary axis can be related to correlators between the interior and the exterior of the black hole, but this connection is not fully clarified especially in the case where we're talking about the pure state. Another comment is that if we look at the process of thermalization from the point of view of the bulk, if we send the particle into the black hole, usually we think of it as in the dual quantum field theory, we think that the particle is falling with the coarculium plasma and thermalizes. But from the point of view of the black hole geometry, it seems that there are two phases in this process. One is the particle is traveling towards the horizon, then it enters the horizon, and then there's a period of time where it travels between the horizon and the singularity, and it's not very clear what the meaning of this region is from the point of view of thermalization in the quantum field theory. Finally, and this will be the main topic of my talk, we can think about states which have excitations just behind the horizon, and then the question is, how can we understand the states from the point of view of the quantum field theory? So as I mentioned already in my first slide, I'm going to describe a class of non-equilibrium states which should exist in general in chaotic quantum systems, and in the case of theories of holographic dual, these states would correspond to excitations behind the black hole horizon. And I will try to argue that the number of these states is in one to one correspondence with possible wave packets that you can write down behind the horizon in general activity. And even though I will, when I introduce these states, I will use some intermediate construction which is based on state dependent operators. I will try to emphasize that the existence of these states is logically independent of these operators. So you can write down those states using conventional operators in the quantum field theory. So in that sense, the existence of these states is quite robust. And the fact that these states exist in the Hilbert space of the quantum field theory, I think provides very strong evidence that the black hole, a big black hole in the s does have an interior and the CFT has the appropriate number of states to describe what's happening behind the horizon. So to proceed, I want to remind you a little bit what we know about describing physics behind the horizon in ADSFT, which is not very easy. So perhaps the simplest example is to consider the eternal black hole in the s, which is holographically dual to two non-interacting CFTs which are placed in a particular entangled state, which is called the thermophil state. And then you can sort of understand how the interior emerges by combining together the Hilbert space of the two theories. So excitations coming from this CFT and excitations coming from that CFT can be combined together to form the spacetime behind the horizon. What is more difficult to understand is how to describe the interior of a one sided black hole. So here we want to consider a state which we throw some matter in ADS. So it's a state which starts off as some sort of gas of glue balls which stabilizes and forms a quite long plasma. And after a long time, it settles down into black hole in ADS which is dual to a quark-gluon plasma ball on the boundary. And then the question is, how can we describe the interior of this black hole from the CFT point of view? Now, the reason that this is not very easy is related to the firewall paradox which was originally formulated for evaporating black holes in flat space. But as I will try to explain briefly, it has also been generalized and it can now be applied even to black holes in ADS which do not evaporate. So originally, the firewall paradox was developed as a refinement of the information paradox which had to do with the question of how does information behave under black hole evaporation. So the original question was how can a pure state, according to the computational hooking, a pure state seems to evolve into a mixed state which would violate unitarity or to phrase it more precisely if we look at the entanglement entropy of the hooking radiation, according to the computational hooking, it seems to keep increasing forever. On the other hand, if we want to preserve unitarity, then the entanglement entropy has to decrease after the page time and has to go to zero. And it has been understood since many decades that this can be explained by allowing explanation of small corrections in the outgoing radiation. So this part of the paradox, well, we have some idea of how it might be resolved but what has been pointed out more recently is that there's another aspect of this paradox which has to do with a quantum entanglement of fields near the horizon. And this was first emphasized by Mathur and more recently by Almeri Maroc Poliscik and Sally who noticed that we have two conditions. One is that when you have a hooking pair produced near the horizon, these two particles have to be highly entangled. At the same time, if we want the information to escape, this particle B has to be entangled with the early radiation and then you have a situation where a single particle has to be entangled with two different systems and this seems to violate the monogamy of entanglement. And the proposal of these people was that we have to give up the entanglement between these two particles which would introduce excitation near the horizon and then the horizon of the black hole would not be a smooth region of space time. So this formulation of the paradox has to do with evaporating black holes but a few years later, this paradox was generalized and it was developed in such a way that it can be applied to big black holes in the ADS. So a large black hole in the ADS is holographically dual to a quadrillion plasma microstate in the N equals four spin Young Mills. And this black hole is a thermal equilibrium with this hooking radiation so it does not evaporate. Nevertheless, as I will briefly mention, we have a version of the firewall paradox even for these black holes which seems to suggest that big black holes in the ADS do not have a smooth interior or they have a singular horizon. And I would like to emphasize that in some sense, this, I think this is the most precise formulation of the firewall paradox because it is phrased purely within the context of ADS-FT and it is applied to a big black hole in the ADS whose holographically dual we think we understand well. So the paradox can be understood in the following way. If we want to describe the space time in the presence of a black hole in the ADS, we need to have two sets of modes, some modes which are coming directly from the CFT which are left-moving modes on this diagram and some modes which are moving behind the horizon that they call B tilde. And if you look at what effective theory predicts for the space time, then you find that you have some conditions for these modes. In particular, you have the conditions that B and B tilde have to commute because they're spaced like separated and they also have to be entangled which is equivalent to the statement that this particular combination of B and B tilde has to annihilate the quantum state. It's the condition that an infalling observer does not detect any particles when crossing the horizon. Another question is how can we describe these operators in the CFT? So these operators B, it is well understood how to describe them from the CFT point of view, but these operators B tilde are a little bit more mysterious and the main question is whether the CFT contains operators of tilde which have the properties that we expect from effective theory in the bulk. Just to remind you the way that this correspondence works between B and the boundary is based on construction which has been developed by several people including people in the audience where you can express a local operator in the bulk in terms of a non-local integral over a boundary operator. So phi is the bulk field dual to the operator O and this kernel is a kernel which you can explicitly compute. And then the claim is that this object behaves like a local field in ADS in the larger limit at least. So we know how to describe this part of the space time. It's all very mysterious but what is more difficult is to understand this part and that's because we lack these operators B tilde which describe particles moving just behind the horizon. So the most straightforward thing you could imagine is to ask why don't we try to identify these particles by tracing them back and see where they came from. And this you can try to do in a collapsing black hole. So if you form a black hole by the gravitational collapse of a star in ADS you can indeed try to trace these guys back using effect field theory but this kind of construction has two problems. First of all, it suffers from what is called the transplankian problem. So if you want to find the moles at very late time and you trace them back you find that they get exponentially blue shifted. So when they collide with a star they have transplank and energy and the effect field breaks down. But another maybe more serious problem is that if you estimate how many states you can form by the collapse of a star in ADS they form a very small subset relative to the typical black hole microstate. So if you estimate the entropy of those states it's sub-leading relative to the biggest I hope in entropy which means you can form a very small subset of all possible black hole microstates by collapsing a gas of star or a gas of matter. So even if you were able to do this calculation it would only tell you how to describe the interior for black holes which are formed by collapse which is a tiny subset compared to all possible microstates in this theory. Finally there is the problem that I mentioned earlier which shows that no matter what you do I mean even if you were able to overcome all these problems there seems to be a mathematical paradox regarding the conditions that these operators have to obey. So it is an algebraic problem. So it is some inconsistency between the algebra of these operators and the spectrum of states in the CFT. So these modes B and B tilde each one of them has to obey the usual well ladder operator algebra. So B and B dagger behave like annihilation and creation operators. The same is true for B tilde. However it is also important to consider the commutator of these operators with Hamiltonian of the CFT. And you can do this calculation you can compute these commutators in effect field theory and what you find is that for the modes which are outside the horizon you get something which is not surprising. So B dagger is a creation operator and it increases the energy of the CFT. On the other hand for the modes which are behind the horizon you find that there's a minus sign relative to that one. And that's very important because this operator B tilde is B tilde dagger is a creation operator. It creates a particle behind the horizon but at the same time it lowers the energy of the CFT. And maybe you can imagine that this is an unstable situation where you can create particles and lower the energy and you can make it mathematically more precise by showing for example using this algebra that if you had an operator or tilde in the CFT dual to this bulk field B tilde it would, this algebra together with cyclist of the trace leads to the prediction that the trace the expectation value of the trace sort of the trace of this operator in the thermal state would be negative which is inconsistent given that this is a positive operator. Yeah. This is the nicest way to see the paradox which I've seen yet. How do you know this? I mean the only question is how do you see this H B dagger tilde is negative, is it obvious? Well, you have to... Well, I think the most intuitive way to think about it in my opinion is to look at a two-sided black hole and to notice that the killing isometry runs in the opposite way on the other sides which means that in some sense well the commutators of the operator of Hamiltonian will have an opposite sign relative to that one but if you want to do it more precisely you have to define these operators relationally with respect to this boundary so you have to consider some Wilson lines to define them in a different way and then you can compute this commutator within effect field theory. Yeah. Yes. That example shows that there is no history. Yeah, good, very good. So this example is not a very good example because this state, the eternal black hole corresponds to what is called the thermophil state which is just one state. While the argument, I didn't emphasize it, that we have this paradox if we insist that typical states have a smooth interior. So here at some point if you noticed I used this trace, right? I approximated some correlation function on a pure state by the trace which means I assumed that this algebra should be true for almost every state in theory. On the other hand, in this particular example what we know is that specifically the thermophil state which is just one, a typical state is dual to its black hole. Is this heuristically the thing that inside the black hole you have both positive and negative energy modes and so therefore if you could excite those you could have an arbitrarily large number of states not limited by the back and side hawking. Right. So if you believe back and side hawking you somehow have to eliminate a lot of these. Exactly, I think it's the same intuition, yes. So this argument shows that if you assume, if you demand that typical CFT states have a smooth horizon which means that you have operators with this algebra then you run into mathematical contradiction. Yeah. Can I ask a question related to Doug's question? Yes. Before, what was the standard counter argument to the argument you just paid for why black holes do have finite entropy since this is a standard argument Well, I think to make this argument precise from this point of view, from this point of view, it is important to notice that, so within effect field theory you don't see all different micro states. If you do the calculation within effect field theory you just see one state of the quantum field, the Hartley Hawking state for example, you don't see many different micro states and this paradox arises only if you take into account the fact that you have many different micro states for different energies. So a related way to phrase this problem is which was explained by Mark Poltsky is by noticing that these guys which are act like creation operators should act like creation operators lower the energy of the CFT. So they map a larger Hebrew space at energy E to a smaller Hebrew space at energy E minus omega which means that they should annihilate a fraction of the states if they're ordinary linear operators which would be inconsistent with the statement that they should not annihilate a typical state. So this type of argument, I don't think you can understand unless you use the fact that there are many different micro states in the CFT which are represented by the same bulk states in effect field theory. Okay, so another problem is that there seem to be a tension between, well, general intuition in statistical mechanics and the requirement that we must have specific entanglement between the smalls and near the horizon. And this tension is the following. Even if you look at Minkowski space written in the ringlet coordinates, you can write down the Minkowski vacuum in this entanglement form where all these phases are equal to zero. And then you can check that if you modify the quantum state by introducing some phases then the new state that you get is not the Minkowski vacuum in particular it has energy density on the horizon which means that if you want to have a smooth horizon it's not only true that you need to have entanglement between the two sides you actually need the details of the entanglement to be very specific. And then in the case of a black hole we have many different micro states and if we assume that this B and B tilde or O and O tilde are fixed linear operators it is very difficult to understand from the point of view of statistical mechanics why typical states would end up having the correct entanglement. If you write down any reasonable ensemble these phases would be randomly distributed. There's absolutely no mechanism you can imagine which will fix these phases from a statistical point of view. So it's very hard to understand how you can find fixed linear operators which will obey this equation for all choices of psi where psi is one of the black hole micro states one of the CFT micro states dual to the black hole. So there seems to be a tension between the requirement of having specific entanglement and the fact that if you have a bipartite system then the details of the entanglement between the two sides are well to some extent randomly distributed. Okay, so now let me briefly describe the proposal that I developed together with Suvat Raju over the last few years which seems to evade some of these problems. So the idea is that the starting point is that if we take a quantum field theory micro state psi of very high energy then we expect that it will look like a thermal state. What this means is that correlation functions of single trace operators on this state psi are going to be very close to thermal correlation functions where this temperature beta will be related to the energy of the state by the equation of state. Now this equation is true only under certain conditions. In particular, it's very important that the number of operators should not be too large. If you start looking at correlation functions where the number of operators is comparable to the entropy of the state then this approximation may break down in a very dramatic way. And that's fine because in general in statistical mechanics whenever we talk about thermalization of a pure state if we have an isolated system in a pure state and we try to talk about thermalization of the system what we usually mean is that we do not have we only have a small algebra with which we can probe the system and according to our measurements using the small algebra the system seems to be in a thermal state even though in reality it is in a pure state. Now in the largest theory it is a very natural way to define the small algebra by considering the light single trace operators of the theory and products of a small number of those single trace operators. So if a CFT has a holographic dual there's a gap in the spectrum between operators with spin less than two and higher spin operators so it's very natural to concentrate on the low lying sector where you take these light operators which are dual supergravity fields and they're small products and in that way you define some sort of small algebra and then the claim is that if you take a typical black hole state or CFT microstate correlation functions of these operators will be very close to thermal correlation functions. Now the point now is that even though we have an isolated system an isolated quantum system which is in a pure state if you concentrate your attention on a small algebra not the entire algebra but only a small sub algebra then this algebra probes the state as if it was an entangled state. One way to understand it is to remember that we just argued in the previous slide that correlation functions on a pure state are very close to thermal correlators and maybe you all know that thermal correlators can be written in a complete equivalent way by introducing a second copy of the same system and placing the original system and the second copy in this particular entangled state because if you take this kind of doubled system and you complete the reduced density matrix you recover the thermal density matrix. So all the algebraic properties of this small algebra when it's acting on the pure state psi would be very similar to the algebraic properties that you would get by acting on this doubled system. So this has been known but then the question is what is the physical meaning of this second copy in the case where we just have one CFT? So we want to work in a situation where we only have one theory one conformal field theory but we won't understand in what sense we can think about the second copy of the algebra. And the intuitive idea is that whenever you have this situation where you have a small algebra which probed the state and then the state seems to be thermal what's happening is that the degrees of freedom of the theory which you do not have access to in particular the order n squared degrees of freedom of the conformal field theory play the role of the heat path for the small algebra and this second copy in some sense represents the entanglement of the small algebra with the degrees of freedom of the heat path. So this gives us a very natural way to identify these operators of tilde as the operators with which the single trace operators are entangled when you have a pure state psi. So then the question is how do you identify these operators concretely? Now to proceed it is very important to introduce a concept which is natural and it is the following. You take the black hole microstate or the CFT microstate psi that you want to study and you consider all other states that you can get from it by acting with operators from this small algebra. So you take this CFT microstate you act with one single trace operator two, three, four, five and you take all possible linear combinations and this defines a linear subspace h of psi which is embedded in the big Hilbert space of the theory. And then the important statement is that all effective theory experiments in the bulk or simple effective theory experiments in the bulk will take place within the subspace. You don't need to know to use the entire Hilbert space of the theory if you just want to describe effective theory in the bulk. And let me just mention that this subspace is very similar to what was later called the code subspace. So it's basically the same idea. And as I will try to explain this interior operator of tilde are going to be defined only on the subspace. So they will not be defined globally. Now the important thing is that if psi is a black hole microstate then we have this important algebraic property that if you act on the state psi with elements of this algebra you cannot find any annihilation operators. Do you have a number bound on n? You mean this small n. Yeah, that's not a very precise we do not have a very precise bound. The intuitive idea is that small n must be significantly smaller than capital N where capital N is the rank of the gauge group of the theory but we cannot make it very precise. It's like when you want to ask when does large n factorization break down or does the large n expansion break down? The intuitive answer is that when the number of external operators is small relative to capital N so it's a very similar approximation. So the important statement is that you cannot find annihilation operators for a black hole microstate. That's very important. One way to see that is that if you do something very naive which is to take a lowering operator and you act on a black hole microstate you can consider this state and you can compute the norm of the state but then using the KMS condition you find that this is equal to e to the minus beta omega times psi omega o dagger psi and in general this object is non-zero which means that the left-hand side is non-zero so if you have a thermal state and you act on it with some kind of operator from this algebra which obeys the KMS condition then you cannot annihilate the state. So this pure state at high energies have very different algebraic properties than those of states which are near the vacuum and this is precisely the reason that we can define this second copy of the algebra. Now, since this is not the main topic of what I want to explain today I'm going to skip some of the details and I'll just give you a set of equations which describe how these operators can be defined on this subspace h of psi. So let me remind you we start with a black hole microstate we consider all possible states that you get by acting with elements of the algebra on this state psi this defines a subspace h of psi and on this subspace you can define these operators by a set of linear equations with a self-consistent and you can check that these operators of tilde have precisely the properties that we wanted in order to describe the interior in particular the most important property was that if you act with this particular combination on the state psi then you get that this is equal to zero. So this follows from the first equation and yeah. So the claim is that if you restrict your attention to the subspace h of psi then you can define a set of operators of tilde which play precisely the role of the operators behind the horizon and somehow these operators are able to avoid the argument that we started with. Remember there was this argument that if you assume that you have these operators with this algebra then you run to a contradiction. Now I'm telling you that you can define these operators with this algebra and you don't have this contradiction. So how can that be? The point is that these operators which we just defined have the property that they depend on the state psi which was the reference state of the black hole that we started with. So in particular if you look at the definition of these operators you can notice that just in the definition of these equations the state psi plays an important role. So another way to say it is that these operators have only been defined on this subspace h of psi which depends on the state psi and if you change the microstate of the black hole then you will get a different set of operators. So these are operators which are not conventional operators in quantum mechanics. They're not defined globally on the Hilbert space. They're defined only in certain subspaces which depend on the specific black or micro states that you're looking at. And this is why you're able to avoid the paradoxes that were introduced earlier because in these paradoxes for instance we use this trace which assumes that you approximate, well you're summing over all possible states using the same operator. Now what I'm telling you is that this operator or tilde is correlated somehow to the state on which you're acting. So this approximating the expectation value of this product by taking the trace is not correct anymore if the operator depends on the state. Of course having operators depending on the state is not very usual in quantum mechanics and well there has been a lot of discussion regarding the consistency of this construction. But today I want to emphasize that just start getting inspired by this construction. You can identify states in the safety which can be defined without any reference to state dependence and which represent black holes which have been excited in the interior. So let me say it again. I mean so using this previous construction I will identify some non-equilibrium states and would correspond to black holes excited behind the horizon. And as I already mentioned the number of those states will be given by the possible waste right down wave packets behind the horizon. And even though I will motivate the construction using this state dependent operators a little bit later I will explain that you can rewrite the same states without any reference to state dependence. So in that sense the existence of the states is robust and there should be no controversy about the existence of the states. And the fact that the CFT contains in this Hilbert space states which are naturally correspond to black holes with excitations behind the horizon is in my opinion a very strong evidence in favor of the statement that the black hole in ADS has a smooth interior which is described by general activity. So here is the intuitive story. We will be talking about equilibrium and non-equilibrium states. So I will define it more precisely in the following slides but roughly speaking an equilibrium state you can think of as a state which is time independent. So if you compute correlators on an equilibrium state the result will not depend on time. And then if you want to ask the question what is the bulk dual of an equilibrium state then the natural answer is that it is represented by a static black hole or at least the exterior of the static black hole. So this part of the geometry is rather uncontroversial. You can really compute correlators and show that an equilibrium state should be related to well at least this part of the geometry. There is some controversy about the interior but the exterior is totally uncontroversial. Then I will explain how we can look at standard non-equilibrium states. So these are states where you create a particle outside the horizon and those you can get by acting with unitary operators on an equilibrium state. And if you look at the history of those states what they correspond to are excitations which are spontaneously created in the past near the white hole region they fly out they reach some maximum distance in ADS and they fall back into the black hole. So these are states which used to be in equilibrium at early times they go out of equilibrium at some intermediate time and then they go back into equilibrium at late times. So these states are also very well understood and there's no controversy about these states. So the new states that I want to emphasize today are states of this type where you act with a unitary made out of these tilt operators on the same side. And I want to suggest that the national interpretation of the states are that we have some spontaneous fluctuation of the state but in this case the excitation is traveling behind the horizon and I will try to explain why this is the case and I will also try to emphasize that even though I introduced this state by using the tilt operators at the end of the day I can rewrite these states in a state independent way. Okay, are there any questions about this? Why are you missing the second problem? Because I want to study one single conformal theory. So yeah. But then this particular is where? Sorry, this one? Well. What is the meaning of that? Well, yeah. So the meaning of this would, you mean how would you, if you really want to verify the statement then the precise statement will be that if you jump into this black hole you would detect this particle. Yeah, so. Something like recent conditions in the horizon in the left part of the horizon, which correspond to? Well, I know how to describe these states from the CFT point of view. So I will just write down the quantum state just in the CFT and I will try to argue that the interpretation of the state from the bulk point of view is given by this diagram. All right, so let me now remind you how we define equilibrium states. One way to do that is by, so we're talking about pure states. So then the question is when is a pure state in equilibrium? And we define a pure state as being in equilibrium. If for a reasonable set of observables, for example, for observables in this small algebra A that I defined before, we demand that correlation functions are time independent. So an important thing to remember is that in statistical mechanics if you take a typical state then it's going to be an equilibrium state. What this means is that if you take a superposition of energy eigenstates with some coefficients which are randomly distributed then you can show that, well, for most of the states, the time derivative of any observable A is going to be equal to zero. On the other hand, if you have at typical states where this coefficient Ci are selected in some fine tuned way, then you will get time derivatives which are non-zero and the state will be out of equilibrium so things will be happening. But if you look at time evolution you can check that if you evolve the state for a very long time the state will equilibrate again. So the typical situation is that we have a state which is in equilibrium in the past. It goes out of equilibrium for a while and then it goes back into equilibrium. So to start these questions it is very useful to use a hypothesis which is called the eigenstate thermalization hypothesis which is, well, many people believe it's true in the chaotic quantum statistical systems. And the idea is that if we have an observable in this small algebra and we compute the matrix elements on the exact energy eigenstates then the matrix elements have this particular structure which has a diagonal piece and an off-diagonal piece. And the diagonal piece is characterized by a function f which is a smooth function of the energy. On the other hand, the off-diagonal piece has this exponentially suppressed factor e to the minus s over two and some function g which is a smooth function. And finally it has some phases, r, i, j, which are totally random which means that as you look at different microstates i and j if you change the microstates these phases can change in a completely uncorrelated way. So this is a very useful hypothesis which can explain many of the properties of chaotic systems and I will assume that this hypothesis is true for the n equals four spring angles so I'm going to use it. And let's now see how we can what we can say about typical states for example. So a typical state is a superposition of energy angle states and typically the sum over all elements in this superposition will run from one up to e to the s where s is the entropy. So each of these coefficients in this superposition is going to be a complex number with magnitude e to the minus s over two but the phase will be randomly distributed. So now we take this typical state and we compute the expectation value of an operator a and well to do that you need the matrix elements of this operator a so now we can use the assumption of the ETH which means that we get this diagonal term which has a ci squared absolute value squared times the smooth function of the energy plus the of diagonal terms. And now what I want to argue is that the diagonal term will dominate and moreover if this superposition of energy angle states has a property that the ci are highly picked around the given energy then this first term will be very close to the micro canonical average of this observable while the second term is going to be exponentially small. So to show that the second term is exponentially small you just have to do some estimate of this sum. So each of these two coefficients ci and suj has typical magnitude of the order of e to the minus s over two so all these three factors together give you a typical size of e to the minus three s over two but then you are summing over e to the two s elements because you are summing over both i and j and each one of them goes from one up to e to the s so you would get e to the two s terms. However, all these terms are multiplied by these random phases so if you want to estimate the size of this sum you have to take the square root and all in all if you put everything together you find that this off diagonal term is of the order of e to the minus s over two. So when we prove the statement it was very important to assume that this number ci were totally uncorrelated to the matrix elements Rij of the observable that we're studying which is true for a typical state. So what we have shown is that if you take a typical state then the expectation value of the observable a is going to be exponentially close to the micro canonical expectation value. Yeah. Yeah. The diagonal part of the ci square it's exponential minus s, no? If you think that the ci is exponential minus s by two. Yeah. So this guy is of the order of e to the minus s but then you are summing over e to the s elements. So this is order one. What you're saying that you're picked around the one ui? Yeah. So, well, when I say micro canonical, well, yeah, I mean that you specify some energy window, right? So this sum with delta e. And then inside this window you have a huge number of microstates which are of the order of e to the s times u, right? Yeah. Okay, so what we find is that for a typical state the expectation value of the observable a is exponentially close to micro canonical expectation value and from this it follows that typical states behave like equilibrium states. To make it a bit more precise we can try to look at this time derivative and, well, instead of computing this time derivative you can equivalently consider the Fourier transform and compute the matrix elements of an operator with no zero frequency on the same side. So I want to argue that for typical states this object is zero and that is very easy to prove using the ETH because this object a of omega is a Fourier mode of the operator with a definite frequency which means that it changes the energy by omega. It changes the energy by an exact amount of omega which means that all the diagonal elements will be killed, they will not contribute. You only have the off-diagonal elements and I just argue to you that the off-diagonal elements give you an exponentially small contribution which means that this time derivative is going to be exponentially small on a typical state. All right, so this is how we argue that if you select a state randomly it's going to be, it's going to look time independent and it will look like an equilibrium state. So now we want to start considering states that are out of equilibrium and so how can that happen? Well, we want to consider a state where the expectation value of an observable is not close to the thermal expectation value or the micro-canonical expectation value and this can only happen if you select the coefficient C i in such a way that the off-diagonal elements will give you some factor of order one if there's some conspiracy between these C i's and the R i j. So let me write down again the expectation value of an observable A on a state psi but now I will keep track of the time dependence and you notice that the off-diagonal term is time independent so the diagonal term is time independent but the off-diagonal terms depend on the time in this particular way which means that even if you select the this coefficient C i in a clever way so that at time t equals zero they click with R i j and give you a big signal if you wait, this phase will start rotating with a different frequency and they will decode here and after a certain amount of time this term will become exponentially smaller gain and the state will equilibrate. Of course if you wait a very long time then you will have prokaryotic currencies and then these terms will become important again but we do not want to start this phenomena in the rest of my talk. So the lesson is that whenever you if you take an equilibrium state well whether a state is in equilibrium or not is something which depends on time so you can have a state which seems to be in equilibrium for a certain amount of time but if you wait long enough it may go out of equilibrium and back into equilibrium. So in other words if you select this coefficient C i in a specific way it can be that the state will undergo spontaneous fluctuations out of equilibrium at some point in its lifetime. Now how can we produce these states? Well you can do it by starting with a typical equilibrium state like size zero and exciting it by a unitary. So you take the state psi and psi zero and then you act on it with a unitary which is made out of the operators of the algebra so it could be the exponential of some Hermitian combination of these operators and there are two ways to think about it. You can either think of it as an autonomous state so you just consider the state in the Hilbert space you don't care how you got it you just look at the state and try to find the physical interpretation of the state for all times or you can think about a state that you get by actively exciting the system by quench at time let's say t equals zero where you can couple the Hamiltonian of the system to this operator O and then the state before the quench would be psi zero and after the quench it would be you know 10 psi zero. But for the rest of my talk I'm going to think of them as autonomous states which means I will not try to explain how you can actively produce those states I will just think of them as states in the Hilbert space of the theory and I will try to give the bulk interpretation of those states. So to see that this type of states is in a time-dependent state which is out of equilibrium what you need to compute is the expectation value of this operator O with no zero frequency on the state psi prime which equivalently calculated in this way and then you can do a little calculation and expand this exponential to linear order theta and you find that the expectation value on this new state psi prime is given by the old expectation value plus the commutator evaluate on the state psi zero. Now this commutator we can estimate using the fact that this is an equilibrium state so it's very close to the one you get in the thermal ensemble and using the KMS condition you can verify that this term is no zero in fact it's order one which means that these states if you calculate the time derivative of operators as function of time then you find that the result is going to be no zero in particular it's be order one so you will get a very significant signal for these states. So the physical interpretation of these states as I already explained is this one so you have a state where at very early times it looks like a black hole in equilibrium then around time t equals zero where t equals zero corresponds to the time where this operator is inserted you get some signal which goes out of equilibrium and then if you wait a little bit longer the signal goes to the equilibrium value gain and the particle falls into the horizon so these are standard non-equilibrium states in the quantum field theory they're very straightforward there's no controversy about the interpretation of the states and what I want to consider now is the class of states where this particle is moving on the other side now I want to emphasize that you can also consider many excitations for instance you can act with many unitaries corresponding to many different particles localizing different regions in the bulk and particles of different type so in that sense you can kind of identify the set of possible excitations of the exterior of the black hole to the set of unitaries with which you can act on on the typical equilibrium state size zero so in that sense if we study the possible ways to excite an equilibrium state outside the horizon we have a one-to-one mapping from those excited states to the possible excitations in geometry so this gives us information about the geometry in the bulk tool and I will try to argue that the same is true for the region behind the horizon so these states that I want to concentrate now are states where you act with a unitary operator on the state size zero but now this unitary is made out of these operators of tilde so the main point now is that if you take those states so I will make a few points the first one is that if you take these states and you try to compute correlation functions of operators in the algebra A so correlation functions of single trace operators in the quantified theory you will find that these correlation functions are almost the same as the correlations you would get in an equilibrium state so if you look in the bulk from the point of view of this quantified theory using single trace operators you cannot detect anything there's no particle outside the horizon the state seems to be like the empty black hole with no particles outside it on the other hand, I will try to convince you, yeah what is the accuracy of that statement? it's true up to one over N corrections I know how to prove it up to one over N corrections there may be a way to prove it with higher accuracy however, so at this point you may think that these states are well, just equilibrium states however, I will try to argue that these are honest non-equilibrium states with time dependence which you can detect by computing correlation functions of the small algebra and together the Hamiltonian of the boundary theory the Hamiltonian is a little bit peculiar I already mentioned it because the Hamiltonian has a property that even operators which are behind the horizon have no zero commutators with the Hamiltonian so in that sense, correlators of the Hamiltonian are able to detect a certain amount of information about what's happening behind the horizon and I will try to show explicitly that there are certain correlation functions of the algebra A and the Hamiltonian which will give you a signal for the state with a property that it's equilibrium goes out of equilibrium and then goes back into equilibrium all right, so these two properties we can prove and then the next important statement is that even though I introduced these states by acting with so tilde I will try to argue that these states do exist in the CFT independent of the definition of the O tilde operators and you can rewrite those states using conventional operators in the quantum field theory without any state dependence which means there should be no controversy or no discussion about these properties because these are conventional states and moreover, the number of the states is in one correspondence with possible excitations you can write in this part of the geometry so as I already explained this, in my opinion, gives strong evidence that the N equals 4 or the quantum field theory contains in the Hilbert space a class of states which can be naturally identified with spacetimes which have been excited in this part of the geometry and I think this is a strong argument in favor of the smoothness of the interior of the black hole, yes Can you represent the limit of the utility from the score of letters to the quantum field? Can I recall? Yeah, yeah, I can you mean can I know which was that yeah, I can do that so okay, just let me very quickly explain how these statements work of course, the first one was that we want to check that these states if you look at them from the point of view of the algebra A they seem to be equilibrium that's very easy to prove because by construction, this operator O tilde has a property that commutes with O's and so this means if you consider any correlation function of the O's on the state psi prime you can write it in this way and this O and O tilde commutes you can take this unitary and pass it all the way to the left and it annihilates that you this would have been so these two guys will annihilate and give you one and you will get the correlator of the equilibrium state yeah they are required to commute to all the consoles yeah so they don't commute to all yeah, but this correlator is inside the console space so I'm allowed to use it moreover, in the second proof I will give you a little bit later I will prove the same statements without using O tilde's at all or without using a cold subspace it's just a statement for safety correlators without any state dependence so this, so you get the same result on the state psi prime as what you would have gotten an equilibrium state in particular the time derivative of this correlator is going to be zero and for all practical purposes the state will look like an equilibrium black hole from your point of view however, this operator this when we consider correlation functions where we include the Hamiltonian in the set of operators then this approximation that we used before that the correlator on the state psi prime is equal to state psi zero is no longer true the reason is that the Hamiltonian does not commute without tilde I already mentioned a few times this was the reason that we had a problem that the commutator of the Hamiltonian without tilde was non-zero, right? so in particular these two correlations, functions will be different and you can check that for any given unitary which excites the interior you can find an observable A in the small algebra with the property that the signal that you get in this correlator is time dependent and of order one so you do so now the natural interpretation of the state is that the corresponding expectations are moving in space time like that which means that if you try to compute correlators of the usual algebra you don't get anything you don't get any signal but if you compute correlators with Hamiltonian you can actually compute it and you find out this is what you get so you get a signal precisely at the time where you would expect the particle to be well present in space time away from the two singularities what's the intuition about how to find the particular A? I will give you an example in a little bit so here comes the main point that so far everything I said was based on using this operas of tilde and as I already mentioned there are some well there is some discussion about the validity of these operators but here is a formula which does not depend on the tilde at all so the claim is the statement is that if you look at the state u tilde this unitary is made out of the tilde and we know how the tilde operators act on a state size zero this was given by the definition of the tilde in this set of equations here so I wrote down these equations and now I want to have a unitary made out of the tilde acting as a psi so I can use these equations repeatedly many many many times to rewrite this state I'm sorry to rewrite this state in terms of conventional operators without any use of the tilde so you notice that we have this type of state we start with an equilibrium state we act with this object e to the beta h over two then we act with the huge dagger oh and then with this guy again now I want to claim that these states have all the nice properties that I listed in the previous slides in particular if you look at these states they seem to be equilibrium when we probe them by the small algebra however if you start probing them with Hamiltonian you get non-equilibrium correlators which are of this type and the number of the states can be naturally identified with all possible wave packets you can write down behind the horizon of the black hole so as I already mentioned this provides evidence that the Hilbert space of n equals 4 contains states which are naturally identified with black holes with excitations behind the horizon and from this point on you don't need to use this operator of tilde anymore so everything that follows is standard conventional quantum mechanics so an important comment is that this operator is not a unitary right? u is a unitary but when you conjugate it with these objects it's not a unitary I should have mentioned that this type size 0 I select this type size 0 to be in the micro canonical ensemble which means that the energy spread of this state is kind of limited which is important because this operator is highly unbounded you know if you act on a very highly excited state this will have a huge value but because this type size 0 is selected from the micro canonical ensemble there's no problem this state is in the domain of this operator so we can define this state properly now this operator is not a unitary so you may say what you mean then by acting on a state with an operator which is not unitary and writing down this new state so even though this guy is not a unitary I want to claim that if you look at the state which you get by acting size 0 with this operator you get a state which has a unit norm and well this is how you prove it you look at the norm of size prime it can be written in this way and at this point you notice that all of these operators inside these correlation functions are elements of the algebra I'm sorry I did not explain it very well but if you have an operator O in the algebra and you conjugate it from the left and the right by this object then this operator is also an element of the algebra the small algebra which means you can approximate correlators of these operators by the thermal trace so then you have this so you approximate this correlator why is this true? well you can expand this out and you get repeated commutators of the operator O with the Hamiltonian nested commutators and maybe I did not mention it but if you look at the commutator of these operators with Hamiltonian typically you get things like either minus omega or omega or if you have a dagger you get a plus so if you take commutators of the elements of the algebra of the Hamiltonian you get elements of the algebra and if you take repeated commutators you get elements you continue to get elements of the algebra but if you take products for example ace times or dagger or if you take the anti-commutator then it's no longer an element of this algebra so you have this you want to compute the norm of the state you approximate with thermal trace and then you have to use some cyclicity properties and you get that basically the norm of the state is 1 up to 1 over S corrections so even though this operator is not a unitary when it acts on an equilibrium state it gives you a state which has a unit norm alternatively you can produce the same state as I already mentioned by acting with a state-dependent unitary but for the reasons that I already explained we want to avoid using these guys for a while and we want to continue working with these states now similarly you can prove that if you have an arbitrary element of the algebra A inside this correlation function you can I'm sorry you can so this size 0 should have been all the way on the left you can and this should be yeah so if you want to prove that expectation values of elements of the algebra on the state-side prime are the same as thermal expectation values you basically repeat the same derivation the only difference is that you have an operator A in the middle and then you can check using the cyclicity of the trace that these factors cancel then this U dagger cancels that U and you're left with the expectation value of the operator A on the thermal trace so this shows yeah this shows that all elements of the algebra will have expectation values on the state which are close to the thermal correlation functions up to one of the rest corrections on the other hand if you insert if you try to compute correlation functions including the Hamiltonian then you find that this approximation does not work and in particular if you try to compute this type of object for example we can consider this operator A's prime which is the Hamiltonian minus the energy of the state and we can try to compute the expectation value of A times this operator on this excited state psi prime now to make the notation a little bit more condensed I introduce this object V which is defined as e to the minus beta is over 2 U e to the beta is over 2 so this object is not a unitary operator but okay that's what we use and so we want to evaluate this expectation value on this state and then you can use you can commute this A's prime through V you pick up the commutator and when A's prime acts on psi 0 sorry H hat on psi 0 gives you 0 because H hat was selected as the Hamiltonian minus the energy of the state or to be more precise it's approximately equal to 0 so all in all you find that this one-point function on this excited state is given by this thermal trace and then it's easy to check that you can select this operator A depending on what V is you can select A in such a way that this object is order one since I'm running out of time well I can you can do an explicit example by taking again a simply unitary of this type and then you can check that this particular case if you want to detect this unitary behind the horizon you have to measure this operator so this A is the same as that A anyway so you can prove that these states have this property that they exist in the CFT without any reference to state dependence and they have the property that the correlation functions that you get on the states are precisely those that you would calculate in effective theory for states which have expectations behind the horizon so I had a few more things to say about how you can understand this from the point of view of the ETH but I think I'm running out of time so let me try to summarize so I try to argue there's a canonical class of non-equilibrium states which can be written in this form so in this formula I use the fact that when e to the beta h over two acts on a state with energy psi is zero you just pick up e to the beta e zero divided by two so you can equivalently write the the states in this way so these type of states are non-equilibrium states in the theory which are parameterized by these unitaries you can combine many of them together to produce more than one-way wave packets so they seem to be related to possible ways to excite the region behind the horizon and the existence of the states in the quantum field theory is completely uncontroversial you don't need to talk about state dependence and these states cannot be detected by normal operators on the exterior which means that this suggests that there should be some constantly disconnected region of space time in the bulk which is naturally identified as the region behind the horizon and now to make contact with these state-dependent operators what the state-dependent operators do is that they give you transitions between states of this type so if you want to measure this state if you want to measure something using those states then the only way you can do it is using these tilde operators but the existence of the states is robust and does not depend on the tilde operators and this would exist in any chaotic statistical systems i didn't use anything very specific about the n equals four so it's been interesting to know if they have any role to play in statistical mechanics and the main point of my talk is that the existence of the states is evidence that a large ads black hole has a smooth interior thank you so suppose you wanted to repeat the construction for a small single-sided black hole in ads which does about that yeah and you wanted to work after the page time such that there's a large out-of-fortune radiation with which the black hole is entangled so these mirror operators the tilde would position them in the some disputes degrees of freedom in the radiation cloud yeah so come on but still the the sort of the expectations is described would not be distinguished by small products of single trace operators in the exterior so is there an interpretation that can give you an approach to this construction? yeah i think well i mean we have very poor understanding of this small evaporating black hole so my answer will not be very reliable i don't have any calculation to back it up but i think you would be able to create those operators and those states again which would have two possible interpretations either as excitations of this cloud of radiation in some very scrambled way or for the info observer they would correspond to localize excitations inside the horizon and the fact that you can have this simultaneous descriptions is again this idea of complementarity but i cannot do any calculation in this setting can you say something about beyond the horizon i mean singularity not about the singularity unfortunately i mean no i cannot say anything about singularity i would love to be able to say something but we're still trying to penetrate the horizon so which is more or less what i wanted to ask you what what describes something that's falling in terms of these things can you into the singularity or in the horizon what you mean how can you trace the passage something that falls to the horizon from the point of view of the safety yeah good so if you allow me to use these operators or tilda if you allow me to use these operators then i can write down a local bulk field using this smithing construction that i mentioned briefly where now you have to use the usual operators o and the otilda if you allow me to use the otildas and build this object then i can really keep track of the particle falling through the horizon i can see the signal of the particle crossing the horizon if you don't want to use these otildas then i just have to do it using these operators o and then the only information i have is that the signal from the particle will die exponentially right it will decay exponentially which seems to show that the particle disappeared somewhere but you cannot really keep track of the particle crossing the horizon unless you use these operators what measures the geometry well again if you allow me to use these operators i can probe the geometry uh if you don't allow me to use them then i can only probe the geometry indirectly in the sense i can do a scattering experiment where i send something into the black hole part of the wave will be scattered i measure what is scattered out and i can try to reconstruct the geometry these new non-equilibrium states you say are not typical but now if you restrict yourself just to those among those are do the typical non-equilibrium ones have firewalls or not have firewalls that's a good question i haven't thought about the ensemble of non-typical states i haven't thought about this it's a little bit tricky because um this is not a linear space i mean the space of non-equilibrium states or it's not a vector space because by superimposing a very large number of non-equilibrium states you can get an equilibrium state or the other way around so it will have to be a more refined analysis yeah