 I will repeat some things which I said last time to give a general framework. First of all, I'm discussing problems which interest me now, but one of the features of them is that they are not all prepared and packaged in a well-defined way. They are loose ends and there are problems which we have already seen and some more that we are going to address as we go along. The general theme which interested me here is what does semi-classical geometry capture, what type of phenomena and what is the fate of singularities of various types in string theory. The general tool one uses is the ADS CFT framework where in this duality type arrangement actually the responsible adult is the field theory and we try to extract as much information as we can from the quantum field theory in order to be able to extract various interesting qualities about what happens on the string theory side description. But in a sense the QFT is a definition or the non-perturbative definition of the string theory side up to maybe some loopholes which appear, but this is a general point of view. And I will repeat that most of the work I described here was done with Chose Barbon from Madrid. Okay, so we spent quite a lot of time discussing field theory and correlations in field theory at large time. And I didn't enter, I will now, but I did not enter into the details of what precise two-point correlation functions I am calculating in the field theory and it will turn out that the result depends very importantly on what these things, what these operators are. Now in retrospect it will come in a few slides. We will see that for chaotic operators and I will try to define later what I mean by the chaotic operators. If one calculates a two-point function in a field theory which has in it a discrete gap spectrum and is unitary, then there are some general features one can say. The first feature is that depending on which precise operators are chosen there is either a power like or generically an exponential decay in the beginning. However, a certain correlation function which we defined which was the time average of the string written above integral time average taking that value squared normalizing it by its value at zero has a property that while it starts up at order one by definition here it reaches a certain lower bound around which it can oscillate a little. So I would say it's not a precise lower bound but it is it gives you a sorry it is precise for the average of the square it's not precise lower bound for the thing before being squared but before being squared it would fluctuate around that average lower bound because it would have to reproduce it. So one has a decay one begins to feel the fact that the spectrum is not continuous after a time which we call the Heisenberg time which is order of the exponent of the entropy of this system and then after a time of exponent of exponent of s one approaches which is called the Poincare time one approaches the original values to whatever precision you actually want. Now the fact that we took this particular operator and this Thibault asked me later okay so you took this operator it's an operator it's a local operator on the boundary theory what does this correspond to in the bulk. So the general idea but it was never proven it is part of the let's say dogma in the field which people are trying to probe and understand is that the behavior of this chaotic operator corresponds to the behavior of a local two-point function in a background of anti the thermal anti desicter and in the presence of a black hole in anti desicter. So the claim is that the way one captures the let's say black-holish or near-black-holish properties of the system comes by this operator being chaotic and the precise details of how locality is matched are not known and are not taken into account here. Now we found out that if one looks at this correlation function which is the correct answer calculated on the boundary and compares it with the bulk result at a temperature in which is a temperature where a black hole dominates let's call it the statistical ensemble that is technically called beyond the hooking page transition which I described last time in some detail how to construct it out to Maxwell construct it etc. Then there is a paradox because in the presence of the black hole calculating such a thing in a bulk would not show a lower bound but this and the average quantity would go to zero instead of being bounded from below. Now the remedy for that for the average was to suggest that one has to remain one has to recall the other stationary point in the problem which was the ADS thermal ADS background and I explained why in calculating in a background which is thermal ADS the correlation the average correlation function behaved exactly as it was expected with the same numerical values as what one got from the field theory. So one is relieved that the ADS CFT correspondence worked and one learned on passon that you have to take in the bulk theory at least two different topologies which was always a question does one take the same topology are you there is super selection rule which sticks you is one topology or should you take several ones and they also remarked that actually before this this is due to Maldesena there was in the at a look at the higher level it was shown that if you don't take the black hole when you take thermal ADS as a background and you forget the black hole which has a non-zero sorry which has a 0 pi 1 unlike thermal ADS that you would again get a paradox. So these are two places independently where you needed to define the rules of the field theory so that you should at least take all fields which are solutions of equations of motion which have the same conformal boundary even if the topology when you look at the whole bulk is different. So these were the various lessons that we learned and then we learned that if one asks what happened at any particular time t we totally miss the picture namely we cannot reproduce by the thermal ADS which does reproduce the average it does not reproduce the more exclusive property which is how the correlation function behaves at each time. So we learned that geometry can produce very small e to the minus s effects when they are averaged in one particular example and on the other hand we have we have seen that it did fail to get more exclusive properties. The simple geometry of just taking the two solutions didn't work we discussed what happens if one takes tries to take more solutions I won't repeat it now. No because gamma is totally not universal gamma really depends on the details of the operator the e to the s and the e to the s to the s so on these are universal they only depend on the fact that this is a chaotic operator on the other hand the particular value of gamma does depend on which operator one puts on and as one does not know the exact correspondence between the bulk and the boundary this exercise I mean is you can do it try and do it case by case but you don't have a general picture like you have here and people look at some specific cases but the picture is not general and we ended up with a question which I will answer today was the failure of the thermodynamically dominant contribution to reproduce the average we defined quantum noise was the average value of the square of this operator was this an accident or is this something which would occur in general in a certain I mean in general under certain assumptions as usual so again the conclusions was what I wrote before and that at that stage the burden of proof that the well-defined information paradox exists shift to the claimer that topological diversity is required and string theory is quite a formidable bastion of consistency because we did succeed to extract e to the minus s effects which are not very easy to extract so this is a summary of what we did in the in the last talk now the burden of the fact that the burden of the proof shifted back to the claimer didn't mean that somebody won't lift the ping-pong ball and shoot it again and indeed after this set of works have been done another attempt to claim that the theory isn't well formulated came by through claims that for various reasons which would require a set of talk on its own which I will not do here is that actually whoever volunteered many years ago to parachute into a black hole just out to see how he crosses or she crosses the horizon and enjoys the experience it would actually never enjoy the experience would be boiled and it would meet when they reach the area of the horizon would reach a firewall where the temperature would be ill-defined would be infinite and the object would just boil this has led to many lots of exchanges in the literature and from my point of view the problem has not yet been formulated sharply enough to be able to decide one way or another even though there are many people which would not share this opinion but this is my own opinion so I will concentrate on one particular round in this fight is there a firewall is the horizon something special or is the horizon like we would think from gr something that you could just cross using the equivalence principle as your insurance policy and nothing would happen or is this fallacious and I will take one particular aspect which follows the same which we discussed here what does geometry capture and what does geometry miss and this came goes under the name er equal EPR er is Einstein Rosen EPR Podolsky is added to this and the claim is that the various paradoxes several of the paradoxes which were brought up which led some to conclude that the simplest solution would be a firewall other solutions would be more radical even gets the problem gets resolved because entanglement issues can be related to the existence of Einstein Rosen bridges again it's very nice to discuss it I won't do it here I will just discuss the particular element in the controversy which relates to the same we are discussing so there was a paper by Maldesena and saskine which said the following let's take the arguments that a group of people I'm hurry Marov Polchinsky and maybe the younger guys would remind me I'm missing somebody starting with M okay or s okay Sally maybe a paradox is is is just not there so they say let's consider an eternal black hole in ADS and it turns there's they're very good indications that an eternal black hole in ADS and its dynamics can be described by a product of two disconnected conformal field theories each living on one boundary and writing a maximally entangled state which would reproduce the geometry and the excitations of the geometry of the eternal black hole and the state would be the following you here by the way in field theory one the Hamiltonian and time they have the same Hamiltonians in one time flows this direction and the other times flows in that direction and each state here will correspond to some story in the bulk so they state they take here is the maximally entangled states they diagonalize each Hamiltonian then they take the product of eigenstate of the Hamiltonian n of field theory a which is identical to eigenstate n and field theory B and you multiply it by e to the minus the energy level appropriate for the state n and you divide it by some temperature t where this temperature t will define what is the temperature of the eternal black hole so if you want to build a black hole of term of temperature t this is where you put the parameter t and the claim is that all the physics here would be the physics here and in particular if you look at this object it has no firewall so this is a situation which in principle would obey would be one of the examples used by people who claim their firewalls and they say look here is a counter example so the debate did not follow about the counter example but about the generosity of the counter example the claim was that this is a very atypical case and then Marov and Polchinsky suggested okay here I just remind you that if you look at a one-sided black hole that is you integrate over the other field theory you will get a density matrix which will be appropriately describing the the black hole it won't play now any more roles so the arguments of Polchinsky and Marov and Polchinsky was the following say okay you took a very special state which was the maximally entangled state why don't you take a state where you have a general nm structure here with some general matrix g and m and not insisting that this is a delta function look at this state and now calculate the two-point correlation function which we discussed before and then they said okay we learned in the past how the two-point function behaves in the presence of geometry and this is what I described to you it starts being of order one it decays and then it reaches its average and jumps around the average was from time to time big jumps and mostly just very small fluctuations and they said that if you calculate the noise let's call it in this configuration you will not get this behavior which we saw before that was the claim you will not and the emphasis was here they claim that you will get that the ever the height of this correlation function is of the order e to the minus s you never reach a one so it's not something which at least is known to be reached by a geometry because when you look in the either black hole or thermal ideas you get a totally different behavior this will give you a different behavior so for a generic state in the quantum field theory you will not have a geometry and therefore you cannot argue that that everything is smooth you just don't know what is going on there you cannot ascribe it some particular configuration and say look here is a very unsingular near the horizon geometric configuration because the noise calculated in on the quantum field theory side does not correspond to anything that we know about geometry and maybe there is something new about geometry but then show it but it's not that our experiences in geometry okay so this led to a second round of looking in more detail in all the questions of the behavior of the longtime behavior of the correlation functions and in particular a point was maybe was known to people but let's say when I spoke about it here many years ago definitely was not appreciated enough by by me is that the observables you choose and the way they rep how representative they are for the dynamics has a great influence on the results so the things I showed you before and I emphasis emphasized work for chaotic operators are not true in general one should really revisit this and studies this for each type of operators now in particular as I told you the common wisdom from what would represent successfully and yet give you features of black hole physics in the bulk the claim was our chaotic operators in on the boundary and the chaotic operators were defined in the following way they are supposed to be the representative operators again doesn't mean they're not other operators but they're supposed to be not representative so how are the chaotic operators defined so this is taken from things which were done many many years ago in let's say fundamental quantum mechanics and statistical physics it went under the eigen eigenvalue to a thermalization hypothesis eigen sorry eigenvalue to my eigen yeah eigenstate thermalization hypothesis and the argument goes the following they say that a general operator which would be some Hermitian operator of does not commute with the Hamiltonian and moreover if you look at the eigen functions of this operator they are highly uncorrelated with those of the Hamiltonian H and these are the operators which you should measure their time dependence and to be more specific the claim was that if you look at that unitary matrix which takes the operator the generic operator B in the basis of energy eigenstates and you diagonalize it so B is in capital B is in the eigen states of energy basis B small b is being diagonalized into the basis where its own eigen functions that the matrix which does it is a pseudo random matrix so if you do this if you make this assumption so it's a strong assumption where you say you have a feeling on what is the right physics and as I said among the open the loose ends and people are working it from many points of view is to give more and more basis for this foundation I will two foundations to this claim I will now just take it and see show you just what are the consequences on the structure of a representative matrix B from this definition from its definition so the claim is and I will show it that a typical matrix or a representative matrix has some diagonal form this is again in written in the basis of energy eigenstates on which there is nothing special to say and then it has an off diagonal firm form where you have some smooth number small b which multiplies it this this be it will be relate of course the eigen values you have this is written between an eigen state of energy en and em and this how you define e bar and and omega so this is some number which has no special characteristics then you get that the typical lengths or magnitude is of the order of e to the minus entropy over 2 which is a strong statement so these are very small of diagonal matrix elements but they exist everywhere because this object here is supposed to be a random matrix of absolute value 1 so it's some e to the i theta where theta is random so this type of matrices are supposed to be the ones where one does the calculation so how does one get this result so in a very crude manner let's look the matrix u must be unitary and by the assumptions that it's essentially all eigen functions of e participate in one eigen function of b you get that each matrix element there e to the s matrix elements so each matrix elements must be of the order e to the minus s over 2 so this is now the u b the off diagonal part of b is you some number b u bar so this would be a sum of e to the s elements of some random because of their character their product would be e to the minus s because each is e to the minus s over 2 but then you don't get e to the s of them really summing together coherently but because of the random nature you it's some random walk and you get e to the s over 2 so that's how you end with this b being of the order e to the minus s over 2 and multiplied by some other stuff of order one which is random so this is the picture one has and as I told you this were ideas borrowed by Asher Peres and the Deutsch Srednicki for many many years ago so now one does given that this is the matrix which is characteristic for a black hole what you should ask what you should use when you ask about systems where black hole should be there then or for the system of n equal 4 that's a claim you do the calculations with that now there are many type of calculations you can do and I will I won't do all here I won't show you all there are some where you do the noise when it's one-sided the EPR noise is of course when you have the you're looking at noise as I showed you of states and m with a matrix g and m between them what what noise do you get there then you have what happens we'll discuss when there are several bands and also when you have a system which is quasi integrable okay this is just to show you now that there are you can do various things you get various results which depend on many considerations I'm not I'm not going to enter into all the details I'm just going to give you the results that we are interested here but I just want to know to show you because it was a lesson for us that there is a lot of structure involved in which operator you choose to take and what state do you take as a state you are interested in the system which is a conformal feed theory is it diagonal is it untangled is it pure is it mixed all these things change and each one gives different results for the noise so now let's go back to the original claim so Maldesena Saskind took the diagonal state they said that is the state you can go ahead and calculate the noise in that state and of course here there is no horizon at all sorry no a singularity if no sign of singularity on the horizon and you get the same result as I discussed last lecture so this all looked very geometrical and very non problematic so we have the value of the correlator itself before being normalized is of order one it characterized by the geometry and the noise it gives is of the order e to the minus s times some numbers which I didn't discuss anyhow now let's take the state which is non diagonal and ask ask what happens there so you can ask what is the value at t equals 0 or in general what is the maximum value that you can expect this to give to give but taking the lessons of last lecture you can also ask what is the average noise that this produces so if one looks at the first question the first question was the one which it was addressed by Marov and Pochinsky look at the correlation function and try to identify its maximum when the matrix were the ones that I we have discussed and I remind you that our M M N as these random the random there was e to the minus s over 2 which was multiplied by the order one but random phase which appeared so when you look at this correlation function you get this normalization structure then you have because you're summing over n and m you have these number of terms you have these which have phases and you have this torus of let's say e to the s dimensional torus which we can build from e n and e m now it is clear that each of the phases here is ergodic so you can try and use part of it to remove the phase structure the random part of our M N which you have for the general state however there is only an exponent s of such independent phases you can use and you need in order to make this of the order e to the s that is to make all this sum coherent you need e to the 2 s terms and you can do it so this won't help you and what you end up is that when you look at a random walk structure here you have only e to the s elements which means that the maximum that this will ever give you is e to the minus s and therefore for a generic state in the in the conformal field theory if you try and calculate what does it give you don't get the geometrical picture which is at least somewhere this was of order one and it repeated itself this order one many times very sparsely but it was there here you cannot get it because again the argument you have a retaurus of phases but these are not enough phases to give to get rid of the random structure of r n m however if you calculate the noise that is the average value of the correlator you do get exactly the same result as you got from the diagonal maximally entangled state so this is another example of geometry in this case let's say eternal ADS which was really corresponding only to the maximum entangled state giving you the correct the exact and correct sorry a giving you the correct average quantity the average quantity was calculated by the geometry very good the moment you want to change and look into more detailed structure like what is the maximum this has at a certain t what I mean is there a t for which this will be order one the answer is no indeed there marofel ponchitsky stated correctly this doesn't look like a geometry when you ask a very exclusive question but if you ask an integrated question you get the right result from the geometry so this is example number two and as I said you we show that it does not depend the noise the average noise does not depend on the entanglement then this is just a remark to people which were more concerned with this problem and aspects of it that if you want to get an order one amplitude which I showed you is not what you would get you would get into the minus s is a maximum you need to have a state dependent conditions and indeed some solutions of the problem of the firewall came through the direction of changing a little bit how the rules of the game are and say that some of the their particular things you measure are dependent the state where you are doing the job you're doing the measurement so indeed never mind what I mean you actually know that this is what would diagonalize infield theory a this what would diagnose infield theory b this is the random part on matrices a and b if you have a particular relation between the matrix elements then for this specific operators which is again type of question you asked me Tbo before but these are operators which are let's say more interesting because they are space dependent state dependent because they depend on the matrices omega you needed to use for the diagonalization if you take if that is what happens then you can arrange for coherence but that's not the generic situation so to conclude what we got here up to now we had two examples where geometry gave a number which was very small the noise it was e to the minus s it's non-perturbative from the point of view of this gravity theory should be zero to all orders in perturbation theory and nevertheless geometry can capture it but if you ask a more exclusive question like how does it behave at any time t geometry fails cannot doesn't give the right answer maybe eventually a way will be found but as of now it's we don't know and the question could be so murky I was using as an analogy when people first thought that you you could solve confinement by instantons in four dimensions didn't work and then people began to look at various configurations one of them were called may runs and one started to invent more and more rules to try and get the result you want but it ended up as a very murky subject which actually did not lead to any concrete conclusions and it could well be that they attempt to get exclusive information out of detailed semi classical geometry will also end up being very murky maybe some nice structure will emerge and I mentioned let's say the issue of the brick wall or the stretched horizon which may be by some way can effectively describe the sum over all kind of things which are murky but that's where the where the status is today averages of very subtle and small quantities are given by geometry more details geometry doesn't know how to give them but you know the result from the field theory but you don't know how to reproduce it from the bulk at least there are no clear there's not a clear calculation which produces them now I want to go back to two more details about this in particular I'm interested in the detail was it accidental that the low energy band that was thermal ADS dominated the behavior of the system and gave us the result we wanted so here we take a matrix which has to totally let's say different structures in terms of number of states which is characteristic of n equal 4 I showed you that the actually four bends let's start just a moment just look at two bends so there is one where there is a lot of states very high and there is one band where the number of states is very small and the claim is that generically it will be that the low energy band will dominate and how does this work so assumes are a few bends okay this is a system it could be that the system doesn't have a taller band structure and breaking it into pieces doesn't make sense I tried to argue last time and I will show again the graph in a few minutes that in the system at hand it does make sense to look at several bands so the calculation I'm giving here the estimates I'm giving here are in the case when there it makes sense to look at the band structure so if there are several bands there is a probability that you will land in that band and then there is the type of noise that you will produce if you are in this band for example when there were two when we go back not to the field theory with the bands but we think about the bulk picture there were two configurations there was a black hole with a certain temperature and there was thermal ADS and each one was given with the weight of the free energy corresponding to it's to the Gibbons Hawking calculation of the free energy of that system so for a general object correlation function you will give the probability that you are there multiplied by what happens if you are in that band what type of behavior you will get and when you do this analysis you expect as I said the probabilities will be related to the free energies and the noise we estimated how noise should behave and noise goes like e to the minus s for the time being we are looking about chaotic operators so if we look at that system we see that actually if we take the noise and we multiply it by the probability which is a free energy we get exactly the Boltzmann distribution which will tell you that the low energy band will always dictate the level of the noise the average noise so this it's a simple argument that was missed before and it explains that you will the fact that the thermodynamic dominating band did not give you the noise is not accidental it's always the lowest energy band which will give you the the maximal weight for the noise that statement number one now let's look at the case where the system is quasi-integral if the system is quasi-integral the relevant operators to calculate are not these ETH operators but in that case they are clearly operators which are a few field excitations that means there's some combination of a and a dagger if you have some quasi-particle picture for that so you can calculate what is the noise not of an ETH operator but an operator which has this structure where the coefficients here are given by what you would expect the both the distribution in a system of which is let's say quasi-particle excitations and the result which let's forget here the calculations are important for me now the result is the following you would get that the noise is much larger it goes like one of a square root of the entropy so it's not the e to the minus s or s over 2 effect it's just one over a square root having a much larger noise so this is again to realize that when you do an ETH system and you look at it's like long-time behavior when that is a characteristic operator you get a very small noise but if you are in a system which is nearly integrable or has lots of conservation laws where a quasi-particle a free quasi-particle picture makes sense you will get a much larger amount of fluctuations a much larger noise and that goes like one of a square root of s okay one can do the same for the EPR noise and I emphasize again that the dynamic the representative dynamics picks the representative observables and has a big change on the how the result looks like okay I will come back to it here I can just say this is the type of field theory that we discussed in the ADS CFT so this is on the CFT side there are several bands I told you these are gravitons 10-dimensional these are strings these are Schwarzschild black holes in flat space and these are Schwarzschild black holes in ADS and there is this four bands that are expected from the bulk side okay would be nice to get it of course from the boundary side and here is what we discussed depends ETH or black holes in strings you take the ETH and for the gas you take operators of the form that you discussed which look like quasi-particle so now you can do the calculations in these four bands and here I will just give the final result so if you look at the noise at a temperature which is much larger than the Hawking page transition so where the black hole dominates the thermodynamics each of the bands gives a certain answer from here you get from the graviton gas you get the one of a square root of s and the you get the various pieces and you find that again which is the leading one and you get the same type of answers that we got before it's a low energy which determines it in general you can say that the average noise is determined by the lowest band the fast variations are also determined by the low energy band on the other hand the height and the long-time variations so which I show here the long-time variations and the total height they are actually determined by the leading thermodynamical configuration so if you would be listening to ADS you are part of NSA you listen to ADS you know which are the pieces which you can associate with the low energy part which dictates the average noise and which are associated with the various time scales in the problem so the conclusion of that of that part is geometry produces correctly the average property which I would say now we have two average properties given correctly by geometry it reproduced a very small non-perturbative result it does not reproduce final details of the non-perturbative behavior of the time-dependent correlators so that's where we stand here slogans I repeat diversity counts geometry captures what I said I think this is just repeating itself and just making the slogans clear okay now you want to do you think a ten minute break would be good now or I go on what do you say okay so from now on your your your responsibilities that we go on okay so now I want to take a different diagnostic tool on this for the system before I have a discussed long-time correlation functions as a diagnostic tool now I want to discuss complexity now I will define complexity I will discuss classical quantum and I will calculate various properties of complexity of singularities we will try from them to learn what do singularities behave like in particular I'm telling you already now that some singularities will have a property which reminds you of think which to ball like to work that is of systems where when you approach the singularity you find a decoupling of different points in space-time and you actually have like a strong coupling lettuce system that is a very low degree of entanglement and you just have each object obeying its own quantum mechanics and similar things will or ideas looks like that will come as a result of this analysis but first I would I need to define the complexity what's are its expected properties and then what happens in the presence of singularities now I emphasized in the beginning of the first talk and of today's talk that there has to be a responsible adult in in the analysis namely that one needs to understand how the quantum field theory works and one has to have precise quantities and well-defined quantities in the quantum field theory for the long-time behavior this was the case here this will not be the case that is I would even the quantities defined on the quantum field theory side need have loose ends and there things one doesn't really understand in the definition of of this complexity and it will become a parent where where the loose ends are and hopefully eventually these will be closed I think it's a very interesting line of sorts is what developed I don't know by let's say Dennis by Daniel Harlow by Lenny Saskind by Steve Schenker many people let's say in the Stanford group at one time or another spend a lot of time to push this this thing forward they they concentrate it on analysis of black holes here I'm concentrating on I will concentrate on the analysis of singularities and we will have various type of singularities to analyze so let's start with a definition how would one how does one classically define complexity so here is a definition and I think there are no precise when we go to the quantum case there are no precise data on how universal the statements are because I'm going to that statement will depend on various inputs and the it could be that some features of the result depend on the input I will tell you what is not expected to depend but I don't know if the if there are theorems which validate that so first you're supposed to start with a simple initial state a simple initial state means not highly entangled state so you take some physical state and this is a reference state for you and one again the question how universal the answers will be and how the match they will depend on what you picked is I think to some extent open then comes the notion of simple operations I will give examples of simple operations we are we are in we are in some we are doing some classical system in the beginning and then it will the examples will become precise you will see how to go on so simple operations I will give an example of a simple operation simple operations are supposed to be if you have a state it's given to you then you make operations which are changes either at one point of this which defines the state or it's a small neighborhood of that point and you have a small number of these simple operations could be one could be four but a small number much smaller than the number of degrees of freedom you have in the problem and then you define the complexity of the state as a minimal number of operations you need to construct that state from the initial state so here is a concrete example let's say that a state is defined by an it if it has their S points in the state and it's defined by being zero or one and if you want identify you can have a overall Z2 identification or not is up to you a simple operation is that at one moment you take either zero to one or one to zero and you have S sites okay is that S will be like the entropy because you have essentially two to the S different states in this Hilbert space so the log of it goes like S so it's essentially S is here you should think of the entropy now the maximal complex classical complexity for this particular problem where this was the initial state this were the operations this was the system is proportional to S the number of sites is that okay too boy is that precise enough so now okay so this is the classical entropy of the system when you do quantum mechanics things become much more complex because the state now is a sum with complex coefficients of states this is now the most general state which of which you want to calculate the complexity a general state in the Hilbert space and remember we want now to give each state in the Hilbert space a number which will be its complexity is two to the S or to the K or whatever number you want lattice points you have here and you have you have to tell all coefficients alpha i okay so you can be happy that there is one phase and one normalization but this is irrelevant relative to the large number of information you have to give about the system in order to know what it is at all so the manifold of pure states requires ab initio but in the definition requires an epsilon resolution because this is a continuous manifold this is the manifold of possible states so there's no no meaning of what of a minimal number of states because that's a discrete notion so what you need is first to already ab initio on the field theory side impose a resolution e and then you can ask what is the minimal number of steps and you will define what steps are they can be let's say if it's a spin system this could be four types of sigma matrices that you allow sigma plus to act on let's say two sides sigma minus to act on another two sides sigma three just to measure you pick a set of operators and you move on the initial state and you ask what is the minimal number of operations you have to do in order to bring it into a region epsilon of their state that you want to determine its complexity and that's the best you can do from the start so that's why I told you that the problem of definition starts in the beginning but people do it that's that's people try to do this game and the answer is and I'll give some intuition to how you get it is that there is a bound on the complexity the comp this was shown a bike it by Kitayev and the collaborator whose name now escape me maybe you know who was no but okay these are two or it's a two also paper and and the the claim is that the complexity goes like e to the s times log of one over epsilon and this is important that the resolution epsilon factorizes out so you can identify a piece here which is epsilon independent which is e to the entropy so you the claim is if the total number of states of the system let's say is of the order e to the s this is the maximal complexity a state can have and we will see it's not as true I mean the answer may look trivial but the consequences are not so are not so trivial and remember it's very important to remember that this is not let's I mean we all think of the Hamiltonian as some of the transfer matrix as something which takes us takes a state in time when it's not an eigenstate and moves it in the Hilbert space and tries to cover the whole Hilbert space in this motion but that would not necessarily give the minimal number the time it would take for a Hamiltonian of a system to get to the state we want to measure the complexity of is not necessarily the best shortcut and this is one of the issues here are for systems what is the difference between the minimal number of steps what is the time it will take to the Hamiltonian to do it is there another Hamiltonian where the time to get to reach the state minimally would actually be that time it takes a state from tour to go from initial to final and this structure of the of of this is not is studied the please the tensor product of n copies of plus minus for example it could be an ising model let's make a model the dimension of Hilbert space is 2 to the n then what is s s would be n and still okay is still no not the Hilbert space this would be no the dimension is the number again the dimension of the Hilbert space is the dimension of this object here because you have all the remember I don't know how it's called a block space I think sorry you should remember it has all the alpha i's okay and that's what dimension so it's a dimension so that so the dimension you should take here this is e to the s times e to the s minus one over two yeah yeah indeed but okay but the projective gives you I mean okay so it reduces by one that's because as we know there is a total phase which is not important and then there is a normalization which is the other one which comes here but basically the the large part comes from this this this cannot change that this is e to the s times e to the s over two this is really the enormous dimensionality of that space okay so if we have we look at that space and we put cells then this is the number of if you wish that's an estimate to help you what you were asking we have one over epsilon e to the s pieces cells in that space we have it each step of the order s choices to make so in n steps one can reach s to the n states I act once twice so these are the number of states assuming no repetitions I don't get stuck at a fixed point or something of that nature so the complexity is bound from this you see that the complexity is bound by e to the s and that is the result that one gets here okay now let's relate this is the the statements which I told you are not sharp enough because we don't know the dependence on the initial state we don't know how universal it is the hope is that if the state is simple then all most simple states would give a very similar definition to a complexity of a state it's not clear the exactly which operations I would do and when how many operations I'm allowed to do simultaneously how large if I have the state on how many elements should I work simultaneously in in quantum computing these are called gates and they are various gates of the system again the belief is that there is if the state is simple enough that is has a low original the original state is low entanglement and the operators that you act are local operators and their number is much smaller than the number of sites of this lattice then the results are more or less universal this this is the feeling but as I told you it's not much sharper than that now how would complexity go for a black hole and I will discuss and show we'll discuss that so the idea was like this if you have a black hole it's complexity in the beginning grows now unlike the case we discussed before actually here nobody knows in the bulk why the complexity stops growing but we know there is a bound so the part as a part I will show you the pictures yeah so this let me reach there I will show you the geometry here let me state the results okay and then I will show you the geometry which corresponds you calculate various maximal volumes and I will show you what that you will see exactly what it corresponds to so in the black hole case I there is a period of linear increase of complexity is a belief it reaches with because of the bound which you know from the boundary it should reach a bound at that stage it's not clear how actually what objects that exactly will be because we don't know the mechanism which will give that bound and then because of the Poincare recurrences of the system the system will repeat itself it will go down go up again go down and go up so I didn't show you the intuition for this result I will show in a moment the time scales here are similar to the ones we discussed last time it reaches a maximum time of the order of exponent of s because the assumption is that more or less the complexity goes linearly with time so the time it takes for this would to reach such a situation is e to the s and therefore the complexity this would fit the bound and then it would go down again now this reminds us very much to the thing we had before but in the inverse before what we had is we had something which dropped down here we have something which increases it reached a bound here it reaches an upper bound there it reached a lower bound and then it repeated itself here we more or less understand how the lower bound what is the source of the lower bound and what is the source of the continuing structure in the case of complexity at least for me up till now I don't I'm not I don't know what the mechanisms are supposed to be now from what I understand from quantum information theory it is expected that there are regions where such a system would indeed as I said the DC DT should be constant and C should increase linear with time this T's temperature it's not the time and this is more or less given by dimensional arguments and the claim is and this is now a claim which one has to check the claim is that the way to calculate the complexity is by taking the maximum volume and I will show you off of the system and I will show you what is the maximum volume how you define it at a given time so here is now related to your question so you start with the black hole in ADS this is all all this thing otherwise there's no connection to the CFT so it's a large black hole at this stage a large black hole okay this is large black holes and the temperature is above Hawking page okay always so it's a it's a temp there are no small effects which occur at temperatures which are lower than the curvature of the system you this will it's complicated enough like this so you take it where the system is supposed to be simpler that is look at the temperature above the Hawking page then you start this is the diagram which describes the hole and you define the complexity in the following way you take time on both boundaries to be the same you could take them different that it doesn't matter you could keep one one of them fixed and let the other increase that's up wouldn't change the characteristic of the results so given let's say the same time here and there even though the calculations are done also for independent times so you could have a TB here and the TA here so this is anchored now calculate the volume of the various sections which are anchored on these two points so there are many of them choose the maximal one that will give you the complexity this is the claim of of what is the right thing to do and the reason it goes to infinity yes that's a problem so I told you one does not understand how on unlike the case that we had before where we understood the lower bound and its source in this case at least as far as I'm concerned maybe some other people figured it out it's not clear how a semi-classical picture will describe the saturation now why did this come how did this come about what were the motives of saskine that are in proposing in looking into that and proposing it because it's a proposal it's it's not something that I mean would be great if one could really prove that and do the exact mapping from the boundary to the bulk no you always remove okay the inf that's why you asked about ads it's you always remove the the infinity and you are interested in the increase you you are you are measuring the increase so you take the reference volume you throw it away sorry I should have said that okay so you all you throw away the reference volume and then there is something which increases and that's what you want to focus on you're not interested in the constant you're interested in the first derivative and it comes from in in the bulk from the fact that as the time moves up the extremal surface becomes larger and larger so here it was like that the opposite okay so what were the motivations where did they come from why associate this geometrical object which you know anyhow at a certain stage you will violate the bound so it can't work all the way but why I did why do that there were several arguments one argument came from the ideas of using tensor networks to build space time and when you use which is another subject on which actually I'm not an expert at all if you if you do these you see that the time progression of building increases the complexity of the system in relation to its volume so this was one of the motivations of why people wanted to associate this with that with the with the increase with sorry with the complexity the other reason is this is more or less here when you cut it this is the length of the Einstein-Rosen bridge and people wanted to understand how do the Einstein-Rosen bridge increase in time it's originally not its volume but people thought about the lengths now it was clear that the time scales involved here were totally different than the thermo dynamical time scales because as we saw thermalization is of order s in the system or maybe even log s but definitely not higher than s the page time which we described while the times associated with Einstein-Rosen bridge increasing are e to the s so that was another motivation to try and check if the volume will follow that and the volume did follow that now I don't want to I cannot argue except this intuitive claim and see what follows up from it why is this dictum is the is correct especially because the authors several months ago it changed the dictum and they said that instead of having here a volume they want to calculate the action on the classical what they called the wheel or the wit wedge of the classical action which had the same actually for black holes had the same very similar properties as the volume itself so as I told you we are here on territory for which we don't have the established and clean definition on the field theory side because we don't know all details of universality of the definition we don't know where for example the quantity epsilon or again I don't know right now where the quantity epsilon would appear here exactly in the calculation the epsilon was crucial in order to make the cells on the field theory side where this epsilon appears in the bulk is is not not very clear but remember that the whole point of tie of bound was that the entropy part can be isolated independent of the epsilon so it's probably sitting in some coefficient multiplying it but it doesn't it's not important from the point of view of the s dependence the s dependence was like e to the s undefined field theory side we are talking about thermal states we are well we are calling about the complexity and we have a temperature and this temperature take to the black hole yeah but we have no time dependence I mean complexity should not depend on any motion of time it does it does it does that's okay okay sorry maybe I jumped here one I jumped no let me you asked the question I give you the answer so forget now the black hole think of the field theory okay so now the field theory we want to map the whole Hilbert space there are many ways you can map the whole Hilbert space you can go and pick your trail and go in it but one way to map the Hilbert space is let the Hamiltonian develop the system so take a state which is a simple state and that each time t if a bigger button not if you I don't want an eigenstate of the Hamiltonian if it's an eigenstate of the Hamiltonian I won't get it's not a good way to be a tourist yes pure but not eigenstate of the Hamiltonian I get a well-defined state I let it evolve in time first I can afterwards do a thermal average of it that's not the point the point is I let it develop in time as I develop in time I get I reach every time in your state I ask what is the complexity of that state okay that so that's that's where you get this this thing but you have to remember that the complexity of the state could reflect a shorter now and a smaller number of operations than the Hamiltonian that took you there so you let the Hamiltonian evolve it so your first jump would like to say ah the complexity just related to the time after I change the units appropriately it's the time it took me to reach it but it's not because that is not necessarily the minimal number of steps that you have you would have to do to reach the state and the minimum number of steps is not defined by the Hamiltonian it's defined by some other quantity which is simpler than the Hamiltonian so this is how you how the question of time dependence of complexity comes in and this is why you know there is a bound because eventually once you cover the whole Hilbert space it's over you're not going to get that it takes whatever you get the highest complexity took you to reach a state after you covered the whole Hilbert space with your time you're not going to get anything more than that but you also have the issue of Poincare recurrences so that will follow you will go to states which have lower complexity as you as you move on but the existence of the bound is clear because there isn't there is the most complicated state given a certain epsilon is that helpful you will find you know so you will you will do it with the operate so you will ask how many this is much more much better from mathematicians you would have some matrix and let's say some unitary matrix and you will ask by sub by small unitary or emission matrices how many times do I have to multiply to get the matrix which Heisenberg gave me so both pictures should give them in you can ask the question in both pictures it's a question of the if you want to act on the state I want to act on the operator okay so as I said in the bulk it's conjectured to be the maximum value and it comes actually with one more one more statement and the statement is okay which you could see here and that's why it became more more interesting the state was like this that and it was born by calculations done when you add perturbations to the just this development described here artificially if you wish but well defined in the in the presence of the black hole you can also a chalk waves and see what happens when you are chalk waves to the system and the claim was like this if you have a system which is sought to be the black hole where complexity increases then there would be no problem at the horizon the horizon should be non-singular if on the other hand you have complexity decreasing then the horizon would be singular and very proudly they show that in this case there is no problem because the complexity all the time increases and therefore there is no singularity on the horizon and they did examples by adding a shock wave so they could see where you could have a singularity but then with a small thermal photon you would get rid of the singularity and they because it would shift enough the horizon so the system could develop without any singularity involved so they gave a lot of examples in the context of black hole physics that it makes sense to think that indeed when complexity increases there is no singularity and the wish here was to say there is no firewall because in this situation which has some generic properties you actually have complexity increasing and as I said the things which are unclear what exactly corresponds to the complexity on the conformal field theory side in the bulk what is this mechanism which stops the complexity increasing because it must reach a maximum so is there a semi-classical picture maybe there isn't but is there a semi-classical picture which we say this stop the advantages that you introduced here an extra time scale an extras objects which happens as extra mechanism which happens the increase in time of the Einstein-Rosen bridge its lengths of volume which is not a termalization scale all this happens way after termalization has been reached so if you made a perturbation you termalized all this happens way afterwards because you have time scales of the order e to the s please this is time symmetric so if I start at the bottom complexity or whatever we decrease to a minimum and then increase well what you would say here is that you reach a singularity so this no but here what they would say by the time reversed thing here that's for them an example of why in the presence of a singularity why complexity decreases when a singularity there yeah that's for that one but there is a white hole some somewhere here but remember we didn't even reach here the horizon forget the singularity slices that do like this they will always have the same Einstein-Rosen bridge no you know no you know again what you need is you can do and they did it take a general TB and the TA and analyze how things behave as a function of if you use boost invariance okay but that's a very particular case but I don't see why why the boost case in the boost case is the one where you would see the invariance but you want not to see the in there the whole point is not to see the invariance so it's right you can arrange you will get a function of TB minus TA and you can arrange use the boost you can arrange whatever you want but the generic case take TA and TB and just let them vary each one independently you will see the complexity increase keep one fixed keep the other fixed again the issue is not finding things which won't do that the issue is to find something which will measure it and they suggest they want to measure the quantity they want to measure it and they suggest that this is the quantity which will measure it you are suggesting quantities which will not measure it okay but they they know that complexity changes in time on the boundary I think this is clear independent of your universality issues what is not clear to me is that if the state on the boundary is not I mean if it is a thermal state about a thermal state and everything is eternal black hole inside and okay nothing depends on time I don't need to ask this question yeah so now I need to start making some action indeed in the CFT to create yeah to start with a simple state and then it develops in something complicated that thermalizers but then it won't be at all the black hole like this in the middle no it will be something else that will be totally no why that's what I said they do perturbations they can do perturbations around this yeah but this is a bad thing to protect around because if I have a big thermal state and I had a little bit of things let's discuss afterwards I don't maybe I don't understand exactly what you what you say but let's discuss no let's say I have five minutes so let's discuss afterwards it won't we won't now converge to that okay so so well with five minutes it could be also a moment okay so they give the motivation on why they want this object and as I told you anyhow it got changed but I think the motivation is not empty there it's very it's interesting enough and they're interesting enough questions to pursue which are not clear and the problem looks very similar as one over the problem the S dual problem of the of the long time correlation functions which have a lower bound here you have an upper bound very similar type of behaviors they calculate it according to their definition what happened for black holes and what we calculate is what interests us for this I will discuss the next time what happens in the presence of sing of singularities and we will do the same calculations but we will do them for very singular various singular backgrounds including the Kassner case including other cases and I will describe what types of singularities one is interested in and maybe I will so and and of course we can discuss next time if you want more of the motivation of the group why they think it's an appropriate thing for complexity what I want to end now is actually something which I will use the blackboard just to amuse ourselves because I'm going to use this for next time but there is nothing yet to release but there will be okay so we all know that in quantum mechanics with a potential x to the n there is no perturbation that you can do you are familiar with the fact that if you have this system you can always this is the issue of a ground state but for the problem I'm going to mention could be anything you can always turn this into a problem where you have the energy scale sitting here multiplying this by two operators which come with an equal weight so that means that perturbation theory means nothing for this problem you know the harmonic oscillator and here you would get h bar omega is a function of the hook coupling and the mass of the particle and this is general for any n almost so that such problems in quantum mechanics don't have any perturbation you cannot make perturbation all the relative couplings due to you is give you the relevant energy scale now you know there are many arguments you can say where perturbations these are not bounded neither operator is bounded so if you take m to infinity doesn't mean that expectation values of p won't grow to infinity on the so maybe it's not just a classical problem and if lambda goes to zero you don't know that if expectation value of x to the n will not contra the the simple lambda dependence and maybe but without all this you just do a simple rescaling and you get that this is the case now there are two cases where this is wrong so let's say one case where my statement was wrong and one where you can do a trick so the case when my statement is wrong I want to I want to discuss that is everything I told you if you do the exercise of rescaling you will find it works along as n is not minus 2 if n equals minus 2 you cannot do the trick namely the coupling which would appear here for n equal minus 2 is a real coupling in the problem this can also if you want to be an example of an anomalous dimensions in quantum mechanics because things will depend behaviors which you do by scaling arguments would suddenly be depend on the coupling and this particular problem which is of the form p squared over 2m plus let me take in general a if if g is positive the system is I will describe its properties but it's well defined if g is zero it's a free particle if g is negative I don't remember the normalization but g in absolute value when g is negative in absolute value it must be I think it's one quarter but maybe it's one half I don't I don't remember it's one of these numbers one point and beyond that this is not a self-adjoint operator anymore and you really introduce by hand a singularity at the origin but as long as not as this is not happening in none of these cases will you have a bound state and the system will behave the way the in the following way so let's take g positive in the case where g is positive you have a potential actually you if if it's one dimensional then the space broken to if it's more dimensions than it's just the radial part of it but let's take one dimension so this system you can solve exactly of course like in a free particle case you won't have anything in l2 but you will have for any energy e which is larger than zero you would have states which are plane wave normalizable but for e equals zero you will not have a plane wave normalizable state the state will be its normalization will be worse than plane wave and therefore you have a very interesting system it is the spectrum is continuous bounded from below but does not have a ground state if you wish trans time translation invariance is broken in this quantum mechanical case because you don't have the lowest energy state of the system and it's interesting what properties the system has and I'm going to use it in the next in the next lecture so I wanted to introduce you to it I will also mention mentioned again there is now something more fashionable this is something which was discussed from my point of view by Sergio Fubini the alfarlow in four land many years ago I did the supersymmetric case with Fubini but the main structure is already there before you do the supersymmetry and this model is called conformal quantum mechanics because it has a symmetry which contains you have the operators h d and k I will discuss them this the operator h translations in time d it measures the scale and k is a special conformal transformations and this is just xp plus px and this is x squared and they close on an s o 2 comma 1 symmetry which is exactly the conformal invariance so it's a very interesting system now recently in recent days months there people are looking for another type of system which is called related to the name Kittayev again from a different point of view and they are trying instead of having this model having here a Schwarzschild derivative here you have just x dot squared over 2m they are trying to use a Schwarzschild derivative and get another type of conformal model this I will not discuss but that's now part of ADS CFT it's a hot topic of research I will use this model in order to discuss possible how singularities behave in in string theory using ADS CFT so I think by this I exhausted this lecture and we remain I will maybe prepare for Tibor for the next lecture I can review in more detail the motivations of Saskin et al if you want that's it