 The second lecture of Julian Sonner on quantum filtering chaos Okay, so Sakura is loud enough All right, so we stopped last time we stopped this kind of structure and I want to go one step further and analyzing the structure and So in particular so the idea is that we have rewritten the The the spectral We have rewritten a gadget which allows you to extract spectral information about a given quantum system And in particular I emphasize that even though I put some averaging on this is generating gadget The averaging doesn't mean that we fundamentally are dealing with a system that we draw from an ensemble of random systems, but that in fact we're mostly interested at chaotic systems Individual chaotic systems and what we want to extract is the fact that they behave like random metric systems And in order to establish that nevertheless you have to perform some sort of averaging so the the averaging that maybe would be conceptually the cleanest would would be this kind of and Slight energy averaging that I described when we talked about spectral correlations already often individual system but other auxiliary Averaging procedures could be imagined like you introduce some Artificial parameters that that you average over or you average over a small amount of coupling constants things like that But what I want to emphasize is that is the next point is that the physics that follows for such chaotic systems Is actually universally determined Not by the microscopic form that this potential takes which depends on what particular quantum chaotic system you start off with And of course just for completely this x I remind you was a graded vector of sources where you put the energies in such a way that you That you account energies that correspond to the denominator as As fermionic and energies that correspond to the numerator as as Bosonic But anyway, so we're not so much going to insist on particular microscopic properties here We're going to just say that Further now in a chaotic system that is useful to analyze this By a symmetry breaking principle, so let's say There is Or this exhibit symmetry breaking and in fact eventually also symmetry restoration and we're thinking about performing an analysis at large D and D I remind you is the size of the Hilbert space. So there's a large number of Energy states that we're interested in Okay, and so this symmetry breaking so I'm Quite generally I have argued that we could write this as a Sort of n slash n graded system, which means that we look at a chain of ratios of determinants that is of length n but for the spectral two-point function, which is sort of the first non-trivial and also in some sense and most interesting target to look at this is two slash two This symmetry breaking pattern At least in the cases that I will return to later is of the type that you have Some graded unitary symmetry su two slash two which is broken by saddle point to Su one slash one times Su one Flash one And then what you do is you actually write a non-linear sigma model And so you write flavor space non-linear sigma model With You know the coset in this case su two slash two over su one Flash one times su one slash one As target space now that there is a Classification of the possible target spaces that could be of of interest which I will return to in this classification though we would call this type a three and So then you will write a theory in which you integrate over the the saddle point manifold and you Parameterize it with a field that I would call q Integrated over this target space manifold So this might be M of q for this particular symmetry case and It's an action which is basically Well, I'm just going to call it right now signal model action and Which depends of q on on? You know this the the takes the shape appropriate to the skill set and of course you would retain The insertions so for example, I'm going to write this in inverted commas because we need to Also write these things in sigma model language, but you could have you know and spectral densities in here I don't know. Let's okay. Let's just write two spectral densities, but you could of course have more and now We could at this point go into as I said a classification of of What possible cosets are allowed and how this relates to symmetry classes? And I could also write down of course More specifically what this action is but as I said I've decided we will derive such a sigma model for the case of gravity And in this case it will be of this structure and we will go into some more detail So rather than giving you now You know one more lecture on sort of basics of quantum chaos I want to leave it at this somewhat formal level But with the promise to return to a specific example and one that is of particular interest for Gravitational systems this afternoon actually, but I do want to say That this structure here, so this flavor space non-linear sigma model That is sort of a key sort of a key tool and In particular I'm written here It it gives exactly what wine right here So this this gives exactly the same spectral correlations as random matrix theory so it gives exactly the same spectral correlations as RMT so it's a it's in some sense In in one one way it's a rewriting of RMT So where you where you average over random matrices that have the same dimension as the Hilbert space But in terms of objects that are much smaller So as I said in the case of the two-point spectral two-point function This is we said yesterday is a four by four matrix and you nevertheless Reproduce the physical content of random matrix theory But what is even more powerful in some senses that this thing can be established for individual Hamiltonians, which which exhibit quantum chaos? So this is a way of deriving Random matrix theory for chaotic systems now, and I should still clarify that If we could show That this is true for a generic quantum chaotic system so that I can rewrite every Individual Hamiltonian of a generic quantum chaotic system in this form that would be a proof of this bgs conjecture for example that that I said Was an important out outstanding conjecture so to actually Give a generic proof of this kind That's very difficult and there are some approaches. I invite you to ask me about them in the question time But the statement is that there isn't such a generic proof, but to be concrete for individual Quantum chaotic systems that have been of particular interest to us for example We have been able to establish this exactly in those cases so when I say we by the way in this case, I should say that That is work with Alexander outland And like Sakura also I will put references on the website so with Alexander outland and also work with well Alex outland and with with Boris post Jeremy van der Hayden and Derek valenda and Finally also I should put so there's there's a series of papers which involve us too, but also importantly Pranjal Nayak who has contributed to many of these as well and is also a co-author and So so however the original idea of the supersymmetric Sigma model is associated with the two names I would say Principally Vegner and Yefetov I'm but this sort of symmetry breaking presentation and applying it to strongly correlated many body Systems there is for example a paper which I will put on the website by myself and Alex outland in which we describe this And okay, so so much for references. Of course, I'm also not really exhaustive about references Maybe if I produce some type typed up notes, they will be more exhaustive also of course many important contributions of other people You want to ask about references, okay? I want to ask Are you assuming the symmetry breaking pattern? Yeah, it depends on the form of the potential I guess no Only a little bit so my next point gives a partial answer to this which is In fact, you can classify the Possible target spaces then you can have and there exists something called the outland Cernbauer classification Outland Cernbauer classification RMT Symmetry classes which actually generalizes the Vegner Dyson default way So Vegner and Dyson I think Basically Dyson taking up the idea of Vegner They showed there were three of these that they called at that point the Gaussian Orthogonal the Gaussian unitary and the Gaussian symplectic ensemble, but in fact you can show that there is ten of these ten classes and interestingly by the way, this is mathematically exactly the same structure as the periodic table of topological insulators and as well as Well, it's a classification that comes up a lot and has to do essentially with Riemannian Super coset spaces so there's also a carton classification of those objects and those are exactly the ten same in physics This has to do with the presence or absence of Anti-unitary symmetries in their combinations, but I don't as I said I don't want to spend we could spend hours and hours talking about this is a very beautiful story about Essentially quantum chaos, but I want to go and talk about more modern developments so perhaps what I do want to say is that Whether by assumption or not and one way of saying what's going on here is that this flavor non-linear sigma model is Maybe epitomizes quantum ergodic physics So once you have actually found This symmetry breaking pattern you have found an action of the symmetry on the degrees of freedom of your system in some sense And you have been able to establish this and the symmetry breaking and you can write this Then you've really shown that your system has those level correlations and has this sort of hard quantum chaos and falls into this category So by the end of today the afternoon I want to establish this We will establish this or Well, let's say ADS to gravity and tomorrow I Will talk about What can be said in higher dimensions? So in ADS to gravity we can actually literally derive this kind of thing Okay, so so that's that The other thing that I want to mention which I just want to because it fits well into this context of Basics on quantum chaos, I will only take it up again tomorrow But that's I'm to think about not just the energy spectrum, but to think about operators and there it is Useful to introduce this idea of the eigenstate thermalization hypothesis So I'm just to say so okay So far only spectrum. So this this tells us about chaotic properties operators, okay, and this is often associated with the names of Deutsch and shred Nikki and Basically, it is a different approach Although it turns out to be very closely related as I shall explain But it is a priori a different approach that attempts to answer the same question Namely, how is it that unitary quantum systems? Thermalize and So what they propose is the following they essentially say that if you take a non extensive operator oh So I'm Non extensive meaning that it that it only involves a small number of degrees of freedom So if you have like some spin system, you know You could you could for example call this a local operator if it's a local theory But basically it refers to only a small subset of the spins now I prefer writing it non extensive because in this business and also because we've already Invoked ADS to we might talk about systems like syk, which don't really have locality properties But you can still talk about non extensive operators think of them as small operators then if you look at Their expectation value Sandwich between two energy eigenstates, so there is an energy eigenstate With energy E I and an eigenstate with energy E J then Statistically speaking those Matrix elements of non extensive operators in individual energy eigenstates Should take the form Okay, so for the time being should take the form because people talk about the ansatz So in some sense it is proposed that if that were the case then a civilization would follow Okay, though, so suppose that it takes the form that there is some smooth function that I call oh Oh bar of E along the diagonal where E is the average energy and it has a Function e to the minus s which I think of the micro canonical entropy at the average energy times some f of e Omega times r ij. So let me explain the remaining symbols. So s I already said is the micro canonical entropy So these are at least on the face of it highly suppressed f is So oh bar and f are both smooth functions of their arguments that I've already defined e Omega is as before the difference of energies e i minus e j and r ij is a In the usual way that people talk about eth is a Gaussian random matrix So here you see random matrix physics comes in as well, but in a slightly different guys and So What else did I want to say? Oh, yes One important thing is that i and j are supposed to be high energy eigenstates in the sense that if you take the thermodynamic limit The system is at finite energy density. So there's like sort of high energy part of the spectrum Small operators and these small operators sort of carry the information about thermal physics here But they also have fluctuations which which you see are these corrections here Okay, so I as I said, I just wanted to state it It's true that if you assume this answer it implies thermalization and of Small endpoint functions of the algebra of observables which falls into this Let's say a given spatial region and you take all the non extensive operators in there So this implies thermalization But I do want to already make one comment Which is that what I'm going to talk about tomorrow is a different set of developments which is actually in collaboration with David Colchmire Bauer Muhammad Sanoff Sorry just getting small and then Neil Jeffress in which We actually Provide a synthesis of this eth idea and this let's call it the The random matrix spectral approach and in particular what we first point out is that you should think of actually This Rij as Instead of a Gaussian a strongly non-Gaussian matrix. So in fact this Gaussian will no longer be correct And we're going to use this tomorrow in particular to think about Chaos as Pertains to high-dimensional ADS examples or even some it is two examples, but including matter coupled to gravity Okay, so but that will be what I will talk talk about tomorrow okay, so that's it for the chaos bootcamp and I will now go and Go back and talk about gravity So maybe this is also a good time to pause briefly for questions. So so we have basically seen Hopefully what is a viable definition of quantum chaos? I have actually said I did say that in some sense it's It's it's notoriously hard to really define quantum chaos. Well, it's not so hard to do it in the classical case and then I have Introduced in some detail, although I also appreciate that still technically speaking. We've been going a little bit quick to really major Major paradigms to think about this one is this non-linear flavor sigma model Or by the way flavor because you know, we work in this flavor space Which is the space of these matrices a which is much smaller than the color space Which in this in this business the Hilbert space sometimes gets called the color space and you can actually think of this very often Really in parallel to flavor color and color flavor duality that you find in gauge theories and the second paradigm is this of the eigenstate thermalization hypothesis which at first at least talks about operators not so much about the spectrum, but we will see that the two are deeply related and Well, one of the obvious connections that I've already kind of pointed out will be via this sort of random matrix And it's statistics that that appears here, but those two approaches So now we are talking about gravity and actually we're even going to talk about string theory Of course, that's what we like But I do want to say that they are major developments in statistical physics and there are big Interesting bodies of work by the statistical physics community and ongoing really active work on on both of these aspects So it's a fascinating. It's a fascinating subject in its own right So now let's go and talk about Gravity as a chaotic system so So one thing that's actually going to be very useful in this context is We have discussed all these things in energy space I have Omeleted things directly about the spectrum and so therefore in energy space But it's actually very useful in particular when we compare or when we develop these things in gravity to talk about it Also in real time. So I do want to make at least that one I'm overview one more time We were talking I had this sort of motivational figure of black all collapse and so on Let's be a little bit more abstract, but let's just say what are the sort of Specific timescales that we should be looking out for that follow or that that are distinct imprints that leave distinct imprints in these in these approaches, so So the earliest timescale is really a t-beta Which is which is what people just call local? Thermalization and that should be you should think of that as sort of the classical black hole for forming Then the next timescale Is this scrambling time or erinfest time? Which is yeah, this is scrambling Yeah, very good scrambling Scrambling time and actually I made a pig zero of that in the sense that Remember I wrote that things go like one over h bar squared times e to the two lambda t I was thinking of n. So one over n is like each part It is it absolutely has to be h bar squared here. Okay, so correction So the scrambling time basically has to do with one simple way of seeing it is when this small correction Because we're thinking of h bar being small in the semi-classic limit when this becomes of order one so This scrambling timescale turns out to be t s usually go Parametrically as log of the entropy And that has to do with breakdown of the semi-classical approximation And it is also the timescale that at which we discuss this Butterfly effect and so this has to do with otoks So this this thing here is your local termalization. This is otoks Okay, and if you want a gravitational Phenomenon that happens at this kind of timescale and this has to do with shock wave scattering Then the next time scale. So now we're getting into the timescales in which we start exploring What word that big that are sort of more Indicative of these kind of things here that the t h is called the TT is called the tautless time Okay, and this is the timescale Which is related to the range of energies where random matrix theory is a good approximation to the system So this has to do with RMT This has to do with RMT and you see sort of the imprints of RMT spectral statistics at and after the tautless time Now in the kind of systems that I'm interested in this tautless time will actually scale Polynomally like s to the one half log s with entropy But that's not necessarily universally so for all systems And the final timescale that is the one which we call T h which is the Heisenberg time The Heisenberg time is the one that is to do with a Timescale that scales as the inverse of the average level spacing So if you if you want to have a resolution of your microscope that is good enough to Resolve individual energy eigen levels. You better look at observables that are Inverse related in time So so basically at the Heisenberg timescale You must see imprints of the discreteness of the spectrum of the individual black hole microstates So this is maybe for us is sort of the black hole microstate time Okay, and so what what we will do is I Will show you basically That there is a non-linear flavor signal model way of thinking about adiast particular adiast to gravity I will tell you what this tells us about tautless time physics In gravity, and I will also tell you what it tells you about Heisenberg time and this Heisenberg time physics is Actually related to non-perturbative effects in gravity which in the context of string theory I will make a side remark are very closely related to what is being discussed in the lectures by a sin Okay, so by the way, why is this always non-perturbative? That's maybe still interesting to know I mean, you know if there are e to the s many levels then one over the level spacing is e to the Then a level spacing is e to the minus s. So one of the level spacing is e to the s So this is a time scale. There's this non-perturbatively large in entropy And actually you will find that okay We can reinterpret this as the coupling constant dependence which will be closely related to what Ashok has been telling us about So let's keep this in mind There is as I was alluding to Quite interesting physics also at these relatively speaking early-time scales. I'm not focusing my lectures on those But if you want Well, not if you want I can put some references anyway And they have to do with sort of as I said black hole formation And then this classical shockwave scattering and there's there are beautiful developments But we only have finite time of course, so I Want to talk about this because well, I think in some sense. It's more quantum in that sense perhaps more exciting Okay, so Very good in this section at first you said real time. So this time is euclidean or Please speak up I can't hear what you say is time time is euclidean or Larenzian you said it's Larenzian time. So, you know, essentially you ask about how do operators or observables depend on time evolution Larenzian time evolution and the Properties of the spectrum because you know if you do time evolution, you you know, you have phases energy differences and so on those Those features they translate into features also in real time. So this is actual Larenzian real time Okay, so I have a relatively extensive section here on two things which I will skip except for saying that Reminder black holes of an entropy, which is basically the area of the horizon Divided by 4g and this is of course the Bickenstein Hawking formula And actually if you want to if you want more details on this, there is of course the Answer question answer session, but this is proportional to the area and for compact horizons Which is the case that one is usually interested in? This is a finite quantity. So we're going to be interested in at least Restriction of the Hilbert space of our system, which has a finite number of states and a finite energy width and therefore a discrete spectrum And of course There is also Beta which I actually mentioned before is actually one of the Hawking temperature and for with respect to this beta We would should think of the the black hole as implying a thermal average, which which are already mentioned Now there is one thing I do want to mention Because I think it's it's sort of useful, but we discussed it already I think you all appreciate it But the point is that we can actually establish in this case even at the level of mathematical theorems in Classical gr or a semi classical gr that there exists a generic Set of boundary conditions where you time evolution leads to black hole formation Okay, so so indeed this This is a system which results in thermal physics by time evolution Okay, and that's maybe the strongest argument intuitively why gravity is a chaotic system But what as I said, I want to really establish and hammer down is that it actually has this Quantum ergodic phase at least in cases where we have more control over it Which is at the moment in lower dimensions and in this context of ADS CFT But of course tomorrow at the end I will I will also Give some some thoughts perhaps speculations on higher dimensions. Okay So right so But what I want to do now is I want to show you how the naive way of thinking about black holes Well naive is of course a Modern way of saying it so if we use the kind of semi classical calculations that Hawking did to revolutionize the whole subject. We will find in particular tensions with What needs to be true both from a unitary perspective and from a quantum chaos perspective around these timescales? And it's therefore these timescales where you really start probing interesting features of gravity So this is something I do want to explain in some detail and For this we will need to have this real-time picture in mind Okay, so So let's talk a little bit about unitary to you. Okay, and so the ideas go back All the way to Hawking But the way that I'm talking about them Stems from work of Maldesena so Recall this will be This will be a very important thing to keep in mind so Because the dimension of the Hilbert space for the black hole is finite We're going to get some sharp signatures Unitarity or loss of unitarity and at late times and There are several ways of doing this one would be to talk for example about endpoint functions of operators e.g. Two-point functions actually so just o t o In the black hole background would be a good probe already By the way there if I wanted to talk about this I would have to invoke some notions from ETH or it would be Let's say convenient to do so and but There is actually a sort of simpler way Which is to talk about Things that don't involve at the moment operator matrix Elements, and this is what people like to call the spectral form factor keeping without tradition often I will call it SFF spectral form factor now This is a probe that has popped up in in gravity literature a few years ago, but it's actually been very present in the chaos literature for for decades and What it is is basically well, let me write the formula so f beta of t is A sum oh Is a sum over The Hilbert space twice ij of e to the minus beta E i plus e j minus i t times e i minus e j so this has both a Euclidean time component and a Lorentzian time component in the Euclidean time component I have the average energy and in the Lorentzian time component I have the difference of energies and by the way It is really no coincidence that these kind of properties also appear all over the place But so what what what does it actually mean? Well intuitively you see the later you wait the longer you let this thing evolve The more it becomes sensitive to smaller and smaller differences of the spectrum until At the Talis time you start probing RMT type physics and at the Heisenberg time You start probing actually the physics of microstates And however, it is convenient to think of this or you can construct this as The analytic analytically continued partition function Namely you can convince yourself that if you take the partition function z now really the canonical partition function of Beta plus it times z Evaluated at beta minus it people like to call this z star Then if you write this as a double sum You get exactly well, yes, of course, it's equal to itself. So you get exactly the same double sum that I have here So that's a useful way of thinking about it Let me tell you another useful way of thinking about it, but then for actual calculations At least in the context of gravity. This is a very convenient way to think about it because we know how to calculate partition functions so So actually another way of doing this is that if you take Row of e1 Row of e2 so our spectral two-point correlation e to the minus i t e1 minus e2 And you integrate over d e1 d e2 You can convince yourself that that gives you f beta of t for beta is equal to zero So at the infinite temperature limit this is like a Laplace transform of two spectral densities Okay, and this is of course not not a big surprise because I already said sort of morally that this is starting to probe Differences of energy levels and as a function of time finer and finer differences of energy levels So that's why okay, that is sort of probably the reason why People called it the spectral form factor. So we can think of this as like a Spectral probe a very good spectral probe. Of course, it has a sort of drawback. I don't know I mean we're theorists. So maybe that's not a drawback, but it's not directly and experimentally Measurable quantity unlike two-point functions or endpoint functions, which are but in fact It doesn't take too much power of imagination to come up with experiments where if you have control of making Two versions of the system and entangle them then you can actually define this in terms of fidelities and decay probabilities So it's it's it's not completely Exotic But it's it's so it's the time domain version of the spectral correlator, right? That's what I want to say So now let's think about some late-time constraints on the spectral form factor and As I said or as I will probably repeatedly say similar things can be said about these operators But they are in some sense polluted by matrix elements of these operators So we need to input some more knowledge before we can make such statements. So it's a little bit more inconvenient So and let's talk about late-time unitarity Okay, so What we can do is? We can talk about something like f of beta and Now time averaged by this. I mean I take 1 over t Sorry, I take 1 over t integral 0 to t of f beta of t Integrated along some time interval and I want to consider the limit that this time interval gets very large Okay, so I'm just choosing to look at this because it will give us an interesting constraint There are of course other things you can say about late times that will also be in tension with unitarity But this is a particularly convenient one Okay, and so then what you get is you basically get the limit that t goes to infinity of the sum I J e to the minus beta e I plus e J Times this integral 1 over t 0 to t e to the minus i t e i minus e j d t Okay, and essentially because you know of defacing This thing projects exactly on to that the diagonal. This thing is non-zero only if e i is equal to e j Okay in this limit that he goes to infinity and so what I end up is I end up just retaining the diagonal ones e to the minus 2 beta e i Where I go from i is equal to 1 to d and now? Simple counting just means that this is You know of order e To the entropy because that's how many number of terms are in that sum Okay, but in particular We want to say that it is not zero Okay, this is a sum of positive numbers it can be small But it cannot be zero so that's the fact that this cannot be zero After having taken this time average is really all I want to state Because this is already enough to be in tension with Semiclassical predictions, but actually what people typically do so I said that it is small But it's actually e to the s so it's huge, but what people usually do is they normalize this So let me write this here so Normalized what people usually do is they actually look at Z Z star so this this guy here and this guy here could have also called it f beta averaged Divided by Z of beta squared Okay, and this is of course still order e to the s every partition function is of order e to the s So this is like order e to the minus s So this is very small, but non-zero. Okay, and actually that there is a there is sort of a Intuitive way of saying what's happening. What's happening is that this thing decays and it wants to negate a zero But actually because we are in a quantum system There are always going to be fluctuations Okay, and what we're looking at here is the variance of the fluctuations z times z star And so even though it fluctuates around zero wildly The variance is not zero because you have sort of some intrinsic quantum mechanical fluctuations that have to be there And okay, we could do the chain of arguments that shows you that you've really used unitarity of quantum mechanics to establish this but so this is basically That that you cannot kill off So cannot have kill off completely Quantum fluctuations if you are effectively in a system with a finite Hilbert space I mean if you take z and z star and also replaced it by minus it doesn't it This formula, yeah, you have replaced z Taken z z star and also replaced it by minus it right in the last in just here last line. Oh So you think you should have put plus it here the whole time. No on the next line next line Here yeah Yes This looks like you get back Z of beta plus it, right? who do Complex conjugation and replace t by minus no, I mean sorry what you do is you I said this in words, but It's not a good notation So what you do is you take z of beta plus it and then you multiply by z of beta minus it But notationally for some reason people want to call this is easy star this quantity I see so this is what you so when I when I write easy star without the arguments. It's what I mean here Okay, so thanks for clarifying it's bad notation, but I don't think I'm gonna take the blame for this because you see this often But you're right, of course Right, so I Think I can draw this diagram which basically diagrammatically says this Visualizes what I said in words, but it will be useful and it's a diagram that I think many of you will have already seen but And Since we want to reproduce it this afternoon in gravity. Why not draw it once? So this is this thing where you know you look at Maybe log a rhythmic function of time and the log of the spectral form factor and What you find is that initially it decays and it may look like it decays all the way to zero Just keeps decaying But in fact as we pointed out that would not be compatible with a unitary quantum system And in fact what happens is that this decay at some point gives way to a rise That may seemingly go on forever, but in fact again The estimation here Is correct so that the fact is that it's it does terminate and order e to the minus s Basically the contribution of an individual quantum level So what you find is that it does terminate at some point at where the fluctuation will be okay, so Fluctuation will be of order e to the minus s Okay, so there's some initial decay, okay, this is cured by some Some actually turns out to be exactly linear rise We in our field we like to call this sometimes the ramp and is finally Results in in this plateau behavior, which again our field we call the plateau Okay, and What I'm going to do now is I'm going to do a semi classical gravity calculation And we're going to see what actually you find for this curve Fine for this curve here and okay Can I just say that first and then maybe that will already be the question because What actually happens of course is that there is going to be the actual signal is going to be very Erratic here, so it oscillates a lot I don't even want to draw all the continuous oscillations what I'm showing is the average behavior Okay, and these fluctuations here is the sort of so this is the thing that comes from as I said the quantum noise so now That was a question Yes, that's very naive question. So the the plateau is very The height of the plot is very small e to the minus s. Yeah, so in comparison. How what is the order of the minimum? Good So I don't know how to say this sort of universally. I don't think this is a universal Quantity, but you can always in systems where you have more control you can estimate it because you will ask What is the decay so it might be something like 1 over t cubed? Okay, and then this thing might be proportional to t and then you ask, you know, what is the value here and Roughly speaking, where is the intercept? So where do these two meet? But it is actually a Measure the the size of this hole here is a measure of the amount of spectral correlations that you have in your system so people also sometimes call it the correlation hole and In in different quantum systems, it can be more or less deep But of course as I said we have well we and others have calculated this for example for syk and other systems that are of interest in gravity, but I'm actually Very good. Sorry. This also motivates me to write that this guy here is that is the Heisenberg time by definition and This guy here Approximately, okay, because here tastes differ, but approximately where this differs and where the linearized starts This is what people like to call the tallest time So answering the kind of question that you asked is also in the same class of questions That to say what actually is the tallest time and as I already mentioned before This is for strongly correlated many body systems is not yet understood to be universal in any way for weekly coupled Single-body quantum chaos. There is a universal formula which was given by Paulis, but yeah, so that's why I can't give you a sort of Good answer exactly. What is this? And I guess a similar question is about the proportionality in the in the ramp. What is the proportionality constant? That is that is fully universal that is determined by which of the Symmetry classes you're in and so if you choose the right units, which is actually units of one over Heisenberg time The coefficient in the unitary symmetry class is one in the orthogonal. It's two Okay, that there's okay, there's a lot of things that we can say I now Hesitated a little bit about the author not author orthogonal class because the fact that we like to draw this kink here is actually Also, not true for all symmetry classes So the orthogonal one will start sort of linear and then it sort of curves and really only asymptotically reaches a plateau so this sort of stark Distinction between what is ramp and what is plateau is not present in all symmetry classes and in the orthogonal one It starts with I didn't draw it like this now But it starts with twice the slope and then has corrections in the unitary class it has slope equal to one with no corrections and And Okay, so this is the last comment that I will permit myself the fact that there are no correction Is actually very nice because it's related to a super symmetric non renormalization theorem for this super symmetric non-linear flavor sigma model And that only is at work in the unitary symmetry class Whereas the others have corrections because such a sort of localization doesn't happen. Okay And So I have 15 more minutes, right? So good. So what I want to say now is I want to give you Yeah, very good. I want to give you a treatment Okay, so first of all there is a very similar story For correlation functions, so these also have sort of this Initial decay of no t3 by the way, sorry, I can't even write my exponents t cubed They have some initial decay Which may be initially something like a Gaussian then gives rise to a power law Then that's cured by some rise and some plateau in the end But as I said, we would need to also talk about the operator matrix elements And in fact This was clarified a lot in a paper where Prandtl played a leading role okay, so Right, so what does gravity predict for now? So what we're going to do is we're going to do the semi classical calculation of the partition function We're going to analytically continue it in this sense And we're going to see what happens at late times so So indeed so we calculate Z of beta Then we Analytically continue Well two versions of it. Okay to get ZZ star and then we see what happens so Actually Later later today. Well, okay. No, let's just let's just do it. So and one convenient Arena where this can be done so it can be done quite generally not not clear how explicitly in all dimensions but one example is ADS-3 and I will also comment on ADS-2 ADS-2 is the one that will give us this one of our TQ decay and so if the idea to calculate Z of beta is basically to calculate the Well one loop partition function ADS-3 gravity Around the thermal saddle. Okay, because what we already said was that the thermal state of the the field theory Is mapped to a particular geometry, which in this case is a black hole and the three-dimensional black hole is sort of particularly nice Is this BTZ black hole and things can be done Particularly explicitly I Don't have all the references. I think the main reference that I followed was by Sheen and collaborators, but I think Ashok. I think you also calculated some of these determinants But just to give you a sketch therefore so in ADS-3 what you get is Z of beta It's basically going to be e To the minus the Euclidean action of the BTZ black hole times Some one-loop determinant Delta of beta so this is the this is the on-shell action On-shell Euclidean action the BTZ black hole and this is the one-loop determinant Gravitons around it or you know what you call gravitons in Three-dimensions boundary gravitons. They are in inverted commas gravitons So It's it's technically not an easy calculation Although it follows standard techniques in the sense that You know gravity is a is a theory that has a local As a diffeomorphism symmetry So you need to fuddy of pop of it and you need to decompose into the fluctuation modes and of course because it's also a tensor theory there is you know the the structure of the Quadratic fluctuation operator around the saddle is a little bit complicated But it's it's nevertheless a standard analysis in actually this sort of yeah Euclidean approach to gravity People sometimes even call this thing Euclidean quantum gravity Which came from the Cambridge school like Hawking Gibbons Perry and so on Okay, and so and what we find is okay one finds And by one Okay, now I don't remember maybe I'll give the references later But as I said paper that I looked at was by she and collaborators So one finds that z of beta Takes the form Well first of all it's it's the pattern goal over these metrics DG e to the minus K s of G where K is LADS in three dimensions over 16 G Newton Okay, and you integrate over metrics that asymptote to a torus With thermal circle Circle beta, okay, so and once the dust settles Actually, what you find is that? Z BTZ Okay, sorry, and of course what you do is you evaluate it on the saddle point Which is the BTZ black hole you expanded in quadratic fluctuations around the saddle point the saddle point action comes out with this coefficient That's what gives you this part and then the integral over the Gaussian fluctuations around the saddle is what gives you this determinant The determinant is very important because that's actually the thing that gives you the decay Interestingly, so if you look at z of beta t Let me write it in form that we should Beta minus it what you find is the expression Well, it goes like one over t to the six times e to the 16 Pi squared beta is it was a beta beta, excuse me over Beta squared plus t squared Okay, times this constant K and as you can see This the case as t goes to infinity. This the case to zero like exactly to zero and Not Order e to the minus s So what this? Shows is that okay? If the actual function the case to zero then the long-time average you it's easy to convince yourself that the long-time average Also will be exactly zero if you take take time t to zero. So the gravitational answer Has actually produced something here for ad s3 now, which okay? Let me write it not log log, but just in real time. So t and this zz star for BTZ Which just decays to zero One over t to the six So semi-classical gravity Let's say misses The ergodic phase Okay, this is now in our language of quantum chaos But the way that it is usually phrased because it's also just in in flagrant violation of this very simple Unitarity constraint so even if we didn't know about quantum chaos and so on and we associated this to Time scales that have some meaning to us We would we would say that this can't be right for a quantum system That has a finite Hilbert space and that lives on a finite manifold Yeah, as you know in Euclidean quantum gravity, so he fixed the boundary torus and you have Even in number of many like a thermal ad s Besides a bit easy and so on so if you think into account this sum is not gonna change the result or as far as I know But I'm so I thought about this at some point Other people have If I remember correctly, it does not it does not just solve this problem So you actually the the kind of configurations that do solve this problem are other saddles Like not saddles that he would have included in this analysis We'll actually exist a we will explicitly show what these configurations are in this afternoon in ideas to and maybe tomorrow Make some comments on ideas street The Okay, yeah, let me write a couple more things So so as we shall see Shortly and we will be a bit more explicit about the calculation if you actually take the Z of the JT so this this is ideas to basically Okay, in ideas to we will work with the so-called JT gravity. So if we take Z Z star JT, we will find that it has this decay one of a T cubed But we will talk more about this and and as I said be if time permits more explicit about the computation But maybe the last thing I will say is that Okay, so that there is one There's one generic point that can be made which is quite useful which is maybe the most important one to to Extract from this so I've basically argued by by example I've shown you two examples Which maybe if we add one more for physicists that becomes a proof But it's not but what I want to say is there is actually a general lesson here, which is that this decay Comes from the factor corresponds from the fact comes from the fact that Super gravity or semi classical gravity in this approximation Produces a continuous spectrum That's sort of maybe we can say a continuous approximation of a row of E. Okay, which in the actual Theory we know to be a sum of delta functions because we are As we have said so many times in the case of a finite Hilbert space in a finite quantum system So so in some sense there is this sort of let's say the word information loss here because the system has lost The imprint of the discreteness of its of its level spectrum. So this is why sometimes people Also Say that this is a manifestation of information loss and the second second remark in the same vein is that the the goal therefore is to produce calculation that sees The individual microstates and so this is actually something which Of course is sort of a long-standing problem in quantum gravity. It's in in in some ways even maybe the program of quantum gravity and There are many ways to think about it and what as I said the many times What I'm going to describe or what I'm describing in these lectures is that I'm hitting this kind of problem in this specific setup with the machinery of Quantum chaos actually produces some very interesting results about precisely the Non-perturbative effects in quantum gravity which are sensitive to individual microstates and in the context of two-dimensional JT two-dimensional gravity in general This really is if I'm not entirely mistaken the same kind of physics that Is being described in Ashok sense lecture, but as you appreciate from a very different kind of physical perspective In higher dimensions, of course, I think there is less control over these non-perturbative effects Because we might not have sort of the nice string world sheet technology, but as I said I will tell you how This quantum chaos machinery can actually make some progress also in those cases. So I think I'm Even leaving four minutes for questions officially. Thank you. Yes. So that's thank you Okay, so now before lunch we have some time for questions Can you maybe explain a bit more how do you compute the one loop determinant? And No, because like let me just say like this. I will I will explain it in the ideas to case as part of the next lecture And I think that's maybe more useful than trying to go through the ideas three case Okay. Thank you. Okay. I would like to ask Are there possible phenomenological observer observables? Which may be Somehow signature of the of individual microstates in order to be clear I'm thinking about something like title of number was in normal modes something similar Yeah, that's that's a very good question, which every now and then I ask myself but So I haven't I haven't sort of thought about this too hard Maybe it's not sort of the the mindset that I have but what I can say is that there have been speculations Indeed of people that want to look for sort of This kind of physics here, maybe these fluctuations by looking at By looking at imprints. So they were sort of arguing about Gravitational wave echoes and that they might be visible in those things. I'm really far from being an expert on this I'm not Maybe qualified to judge on the possibility of this but what I'm one thing I suppose is that one thing I did not mention here Which is more closely related to what you're asking is that in quantum chaos. So there was this Well in chaos theory there were these Lyapunov divergences. There is also another thing Which is named after I think last year is a ICTP Dirac medalist at Israel resonances And those just correspond to quasi-normal modes But from our gravitation perspective, I mean gravity gravitational quasi-normal modes is something that we understand very well So it's not maybe You know the hottest iron to push right now But there is certainly some some chaos imprint that that comes in the quasi-normal modes And that's of course will be in the ringdown effect of gravitational waves and so on but Maybe even if I am personally not doing it actively now This is is a very important question to be very interesting to know if there are such imprints But one might say perhaps It's quite ambitious Because you know what experiment do you want to do which is sensitive to these small energy differences and also which has Sort of a reach to the kind of time scales that we're talking about But then let me not discourage you. I think it's a very important question to ask in general We have more questions Thank you Maybe I can ask a sort of vague question about the goal you've written at the end Is the idea to produce a calculation that In a statistical sense sees the individual microstates, but perhaps doesn't recover exactly this set of delta functions Well, so the story that I'm going to tell this afternoon will achieve that When we set out our goal was to actually find the individual microstates So the overall goal would still be to to try and find actually the individual microstates rather than a statistical imprint of individual microstates and So that's actually also a class of questions, which is very important Which I hope to be able to comment upon as we go on but the technology that we have That gets the plateau And that the gets the flavor nonlinear sigma model it gets the flavor nonlinear sigma model sort of after this averaging bar Whereas my personal hope was that actually this technology would allow us to go even one step further back And yeah, but that's that's one of the key questions. I would say our questions So this in the special form factor this ramp and plateau behavior is this universal for all the 10 RMT classes Yes and no, I mean in the sense that I already said it So the general feature that there is some rise and that it caps off that's universal But the actual way that it does it is not That depends from one ensemble to the next and in particular except for the unitary case there isn't this sort of Linear ramp and then suddenly plateau there is more like a continuous curve Which has an interesting and non-trivial diagrammatic expansion Perhaps you also have some questions for Sakura since we didn't have much much time for extra questions After her lecture, I guess we are all very hungry All right, so if not we can go to lunch and we will resume