 We are very happy to have Julian Sonner from the University of Geneva, who will tell us about quantum field theory and course. Right, so first of all is this on? Yeah, okay, very good. So thank you very much for the organizers for the invitation to talk about some of our work. And so I'll start right now by writing an overview of what I've planned for those four lectures. So first of all, maybe an abbreviated title would have been recent developments in quantum chaos. In fact, I want to talk about also how this relates to quantum gravity. So first of all, the good question to ask is that also readable from the back? Is that large enough? Okay, and already also if I start writing too small, don't hesitate to tell me and also don't hesitate to interrupt for any other kind of questions while I'm going on. So this lecture, the first one, basically will be on background material related to quantum chaos as I assume that that's maybe material that not everyone is very familiar with. So this is something like the chaos boot camp and lecture four is when we talk about the most recent stuff, which will basically be in a technical sense the chaos bootstrap related also to the conformal bootstrap. In lecture two, I want to tell you how to think of gravity as a chaotic system. In lecture three, I'm going to talk about a specifically important notion which I think we can use to great profit, especially in the gravity context and that's to talk about ergodicity in gravity. But of course I will have to define even what I mean by this ergodicity story. So to start off with though, I'd like to give a very general motivation as to why I think it is very interesting and profitable to think about chaos. In field theory of course but in particular in view of these dualities that we have that Ashok already mentioned also in gravity. So of course let's start with an introduction and motivation. So the very first sort of slogan I want to say is that quantum chaos really is the right way to think about or maybe even a prerequisite or possibly even the same thing as the idea of quantum thermalization. So however, so this is something that I will of course explain and by the way this is also true maybe even more well understood in a classical context. But let's add to this statement the fact that in semi-classical gravity quite generally we can think of, we have been taught to think of a black hole as a thermal system. So for example a precise statement can be made for example in ADS-CFT. This is something that is more generally or that is appreciated to be true more generally but we can make the precise statement in ADS-CFT for example that you know if we consider the gravitational setup that is ruled to some field theory setup then typically the geometric background is mapped to the state and so that means for example that if I calculate the correlation function of some operator or some insertion of an operator where in the gravitational background I include the black hole which will have some mass for example or other parameters but let's just stick formally with the mass and the mass we should think of as determining the temperature, the inverse temperature beta then I should think of this kind of computation from the dual field theory point of view so here we are in the gravitational setup. I should think of this as the thermal trace of the dual system where I am inserting the canonical density matrix e to the minus beta h where this beta is determined by the mass of the black hole so that's why I was saying here the black hole just has its mass but the mass will determine what beta I put here of whatever these insertions are so in the field theory this is the thermal ensemble and so then the idea is that if these black holes in fact are thermal systems then they should have an explanation in terms of an underlying microscopic theory and this microscopic theory should be such that it allows us to understand for example how you establish ever such a thermal state or the black hole in terms of the notions related to quantum thermalization and therefore to quantum chaos so I will go into more detail regarding this but just as a motivating example this is rather important and so let me write still working on the conceptual level how this usually works so what we are saying is that black hole formation as well as the process of radiation which in the right asymptotics will lead to evaporation so let me say and radiation perhaps evaporation is in fact a process of thermalization and so for example what we should do is we should think about what happens before the black hole forms we have some initial configuration which might be a star that has exhausted its fuel and so on but let's just say this is some initial state and we might say if this is some t is equal to zero t is equal to or t zero more generally then we have some state psi of t zero that gives us the initial condition and from a gravitational perspective we know quite a lot what happens we certainly know that under the right conditions this state will collapse in upon itself and it will form more or less colored horizon if I find some trucks well let me actually reserve the color for the singularity it will form some sort of horizon which surrounds the singularity so black hole forms so we have something like as time progresses we eventually will find something like a thermal state so let's say yeah did I want to be yeah so we want to say that under time evolution psi of t eventually will be something like thermal in order to define what I mean by thermal in these inverted commas we will need to go a little bit more into the background details and that thermal state will be the end state that results from this black hole that has formed and that has radiated for a long time and so there are of course some initials and intermediate states of this thermalization process which one of the goals of this lectures are we want to understand but the point is that there is some time evolution u of t, t0, time evolution and the supposition is that all of this is described by chaotic dynamics okay and so the point is in some sense first of all that I want you to recognize this problem of black hole formation, radiation perhaps subsequent evaporation as an intrinsic non-equilibrium process of thermalization so this is non-equilibrium, non-equilibrium physics and secondly that the precise way of how an initial state can be transformed into some seemingly thermal state is one of the key questions that we've been asking for a long time about really the proper interpretation of these kind of formulae okay so now one more remark that I want to make here is just that this is a very general statement I appreciate this, the statement being that we have thermal states ergo the system should be chaotic however it is true that basically thermalization itself is associated to chaotic systems as opposed to and so whenever you make a statement you should also say in some sense what would be the other option so for example systems that don't thermalize well typically they will be integrable or in some sense perhaps a little bit more interestingly of course they can be effectively integrable but for example this will be something like a strongly correlated system that is many body localized, MBL or perhaps some other exceptions to thermalization but the generic statement is that since gravity fulfills all of the hallmarks of the system that thermalizes we strongly believe that it should be described by chaotic dynamics and as experience with generically thermalizing systems and the study of quantum chaos in such system shows the insights that come from formulating things specifically in terms of the quantum chaotic properties are very powerful and lead to pretty deep insights so this is basically the point and so the point that I want to make maybe to write one last sort of slogan before we go into actually some of the more contentful parts so well technically contentful I think that on a conceptual level this is hopefully quite useful so I want to argue that quantum chaos can be or is a key tool and the point is both conceptually and also technically let's say calculationally for understanding black holes and through the understanding of black holes also some interesting aspects of quantum gravity so that's kind of the guiding thought of these lectures so well as I was saying the way that I'm planning to proceed is to now essentially give you some of the necessary background on chaos and in particular quantum chaos to understand the subsequent points that I would like to make but of course maybe this is also a good place to pause for some questions okay very good so now let me start by defining defining some notions in chaos so actually notoriously so let's say Lyapunov exponents so Lyapunov random matrix theory, RMT actually maybe before I use abbreviations I should once at least write it so random matrix theory which I will always abbreviate as RMT and all that so there will be more things to say like including other three letter abbreviations and we'll have to foist upon you but before that let me say that the notion of quantum chaos like actually quantum chaos is notoriously hard to pin down so we will of course nevertheless do our best so quantum chaos is hard to define but classical chaos is actually rather beautifully understood and so let me start with that some of those notions also carry over in fact despite naive expectations they do so classical so well the first thing that comes to mind when people think about classical chaos is the famous butterfly effect that somehow if a butterfly bats its wings here it may change drastically the weather in China in some such analogy however that's of course true but I want to say the same thing in a slightly different way which I think is more clarifying in this context so in fact chaotic systems so somehow there is something about dependence on initial conditions so chaotic systems in fact have a dependence upon their initial conditions and that makes them tend to forget about these initial conditions so a chaotic system forgets about its initial conditions and the way that it does so actually in order to say that it's useful to adopt a phase space point of view so if we think about phase space so the space spanned by all the coordinates and all the momentum and then so say Q0 P0 at some initial time then classical mechanics says that the Hamiltonian equations of motion act in such a way that they generate a flow of phi T that gives me for a given initial condition Q0 P0 the state on phase space of the system at some time T so this gives basically just the notion of trajectory Q of T P of T and so this sensitive dependence on initial conditions that is at the heart of the butterfly effect and in fact at the heart of this idea that chaotic systems are very good at forgetting about the initial conditions that is quantified in the following sense for a chaotic system if I take and if I just say that X0 for example is this point in phase space Q0 P0 and then by analogy X of T will be this trajectory so then if I look at X of 0 here and I look at a nearby initial condition X0 plus delta X0 here then the chaotic dynamics are such that the flow I'm running actually out of space because there should be exponential dependence but the flow acts in such a way that the divergence of these trajectories delta X as a function of T should be exponential delta X0 times e to the lambda times T exponential divergence for these initial conditions maybe if we have many directions we should actually think about the length where lambda non-zero is the Lyapunov exponent this is called the Lyapunov exponent now right so but what does this mean so first of all okay I guess I'll continue on the other side maybe you want to say agent and microphones so everyone can hear said that chaotic systems forget its initial conditions but except and X0 and X0 plus DX0 make something like Lyapunov exponents so Lyapunov exponents is not for chaotic systems sorry we'll get to what I mean by forgetting about initial conditions now so plus DX0 makes something like Lyapunov exponents so Lyapunov exponents is nothing to do with chaotic systems which forget its initial conditions I'm not sure I understand actually whether you're asking a question or you're making a statement if you're asking the question so in a non-chaotic system there is not supposed to be this exponential sensitivity so there is no e to the lambda T and if you actually define the Lyapunov exponent as let's say the limit of T goes to infinity of the logarithm of the linearized Hamiltonian flow then for a system that is non-chaotic it is supposed to be exactly zero now the question about forgetting about initial conditions is going to be probably my next point but I mean feel free to ask I have a couple of questions but tell me if you're going to explain it in a moment so one is is this lambda somehow constant on the phase space or it really depends where we compute it so what you really should do again is what I was just saying here is you should actually linearize the Hamiltonian flow around some initial condition and then what you get is essentially a matrix that has eigenvalues which for chaotic systems typically are non-zero and positive and what you call the Lyapunov exponent will be the largest of those and that will be the direction in phase space where the exponential spread of course dominates ok and so perhaps this answer my second question that was is there a reason why lambda should be positive and cannot be negative but perhaps because it's the maximum well if it's not positive you don't have this divergence of trajectories right yeah but I mean can we have a system where you mean systems where you have some directions where some directions where it is at least zero or negative I mean is there some reason why lambda cannot be negative well I think the trajectories are not allowed to cross so I think you would probably get crossing of trajectories right well not if it is exponential but I don't know I can ask you more so right so initially close trajectories diverge exponentially as a function of time and what that means is that roughly speaking the phase space distribution evolves towards constant density given conserved quantities so for example on constant energy shells ok and that's basically the point that you get uniform let's say uniform and such uniform phase space distributions basically are the ones which give you the thermal properties of the system and so thermalization basically via well this mixing behavior and this Lyapunov behavior lead to uniform distributions is basically approached no matter what the precise initial condition was ok so the end state that this dynamics tends to is one that has let's say no imprint of the initial condition ok in that sense this butterfly effect also allows the system to forget about its initial conditions now let's note one thing which will be interesting in a minute that this sensitive dependence on initial conditions sometimes can be written by saying that if we look at the Poisson bracket of q of t with for example p of 0 this is precisely the derivative of the trajectory with respect to its initial condition and this thing itself should also basically encode the Lyapunov exponent or the largest Lyapunov exponent ok and the point of having written it as Poisson bracket is that this suggests actually that we use Dirac's idea about how to go from Hamiltonian dynamics to quantum mechanics and we think of something like i over h bar times the commutator of the operator q of t p of 0 ok and the question is whether this defines some kind of quantum version of the Lyapunov exponent and also whether that will be a useful notion and to take these classically chaotic systems into the quantum realm ok so we will have some more to say about this but the actual so is this something like a quantum version of a Lyapunov exponent and if so is that a particularly useful way of thinking about quantum chaotic systems so however with this semi-classical bridge let me now go and try and describe what actually is the good notion of quantum chaos that we want to employ so one obvious point is actually that quantum mechanics we shouldn't really think about trajectories I mean that is one of the first things we learn in quantum mechanics and moreover in thinking about quantum mechanics in a phase space way is maybe not the most convenient way of thinking about quantum mechanics so however so let's say no good notion of what is a trajectory no good notion of trajectory and there are many other reasons why you might not want to really use it however there is one way to make this notion a little bit more precise so before I go to what is a more sort of intrinsic let's say intrinsic quantum way of thinking about chaos let me nevertheless mention this so you can define something like the idea that a commutator defines so a commutator related to the Poisson bracket which the Poisson bracket encodes the sensitive dependence on initial conditions you can define a notion in quantum mechanics related to this and that's by asking what is the expectation value of this commutator however squared and with respect to a particular state so most of the time people will choose here the thermal state and this thing does have a notion of so it comes with a 1 over h bar squared which is again an indication that we're actually talking about semi-classics not really you know hard quantum regime e to the 2 quantum Lyapunov exponent times t and in fact the way that people think about this nowadays is in terms of the so-called out-of-time order correlation function or OTOC because there are two things that one realizes one can do or should perhaps even do one is that once you think about the commutator of these two operators actually there's no reason why you should think of q and p you can just think of two arbitrary operators a and b for example and moreover you can say that the interesting part the one that gives you the exponential divergence here actually comes from the part of this it's a square of a commutator of two operators it's a four point function it comes from a contribution that is of the kind so in this case it would be q of t p q of t p but you can write it as I said for more general operators b so more generally we talk about the out-of-time order correlator and out-of-time order because you know time doesn't as strictly increase as you read this expectation value from right to left okay now this is very nice it's a very nice object and in fact Maldesena, Schenker and Stanford showed that in semi-classical systems you can bound this you can actually show that this the up and off exponent is upper bounded by two pi over beta where beta is the inverse temperature that you use here in this expectation value but I do want to say that it is really a semi-classical notion and in that sense well, okay, as such the words that people also use is it has to do with you know the scrambling or more previously people will call something like this the Ehrenfest time and so it actually has to do very much with thinking about a quantum system semi-classically importing a classical idea of chaos into the semi-classical regime of a quantum system and I would say that it's still it is still really the jury is still out whether thinking about this kind of imprint of classical chaos in quantum mechanics is right correct or robust way of defining quantum chaos so instead what people usually like to do is they like to think about chaos in quantum systems in ways that are much more intrinsic to quantum systems themselves there's a question up there thank you my understanding is that in classical chaos there's no analogous bounds on the Lyapunov exponent is that right and if so that makes it a little confusing for me to understand the precise sense in which this is a semi-classical like statement because it's not like it then limits to the classical Lyapunov exponent and some limit is that right well the way that I want to say this is a semi-classical notion is that all the contexts that are well understood you have to have some semi-classical parameter which I'm calling 1 over h bar squared in front here so it can also be some effective semi-classical parameter like large n or more generally large central charge or maybe we can think about large entropy density or something like this but there has to be one of these semi-classical parameters so that's the reason for my statement that it's really a semi-classical notion however what you say of course is correct and this bound I think this bound is a very deep statement about quantum chaos so I would agree with that yeah but yeah anyway that's maybe a discussion to be had right so yeah Less than beta is there any h bar there that you have put to 1? Yeah I think there is yes presumably because we want to think of this the units of time to be in terms of let's say the iron first time yeah alright so now let me see how I'm doing for time, expectedly bad so let's say an alternative idea which is actually older than this is to think instead to work directly with the spectrum of the system so let's say the system is described by some Hamiltonian H of course there's nothing more intrinsic than all the eigenvalues of the Hamiltonian of the quantum system that you're looking at now all the development that leads to this point of view follows from an observation and this observation is due to Wigner so this observation is that a the what defines a chaotic Hamiltonian and I'm going to draw a figure here is that I can think of its its microstate spectrum as being in the appropriate sense well described by a random matrix theory. So for this we need to understand in what sense this is supposed to be true so but again to be clear what the statement here is that you take your favorite quantum system in that case of course it should be some gravitational system but of course many other interesting systems have been looked at under this microscope of quantum chaos so you have one Hamiltonian but in some sense its quantum spectrum is well described by drawing the Hamiltonian from an appropriate distribution of randomly chosen matrices and averaging so the observation by Wigner is about the following quantity so let's think about the let's think about the distribution of energy levels so for example you take something that people like to call p of s where s is actually the difference in energy levels so s will be omega and it's usually normalized in units of delta I think people like to put a pi here so delta we need to say is the average spacing between levels so in other words you ask about the differences between adjacent energy levels across the entire system so the average of that is what you call delta and then a nice way of asking about distribution of spacings is to ask about differences of energy so e1 minus e2 will be omega divided by delta and what you can do is you can go through the entire spectrum of your system and you can ask how many times do I find two adjacent levels that have for example s so if s is one then we're approximately one pi of the average spacing you can ask how many of them are there in which the subsequent level spacing is much less than one and you give yourself some kind of width of s and you bend them in here and in this way you start making some histogram which I'm just sort of inventing here but it will look like something like this and it will be peaked at one now the statement about this the statement by Wigner and that was of course elaborated upon by many other people, Dyson Meta in particular is that this histogram is well described by a probability distribution p of s which people like to call Wigner's termites because he essentially guessed it but that follows by choosing an h from a probability distribution p of h and calculate p of s so p of s one could say maybe one way of saying something like you know you ask e energy level minus e energy level plus one and you want this to be omega weigh it by the correct measure on all the energy eigenvalues and average it over the spectrum now this p of h so the probability distribution for your matrix is what defines this measure so the statement was by Wigner and subsequently as I said by many other people that an actual chaotic Hamiltonian one Hamiltonian will produce such a histogram this histogram will basically be given by instead of having an individual Hamiltonian drawing your Hamiltonian at random and calculating the corresponding quantity namely the level space and distribution of subsequent levels with respect to this probability distribution and the point is that a quantum chaotic system is supposed to be one whose level spacing statistics or level statistics is given by random matrix theory and this is sort of some intrinsic quantum mechanical definition of what a quantum mechanically chaotic system is or should be now of course I'm just giving you a glimpse so already here I've given you the level spacing statistics I'm saying level statistics because there are other statistical properties of the spectrum that one can define and calculate and compare and of course also many interesting things can and should be said about what this probability distribution is and in fact again time permitting we will discuss some of those things but I want to discuss them on the on the actual examples that I would like to actually treat rather than you know going into like some category of calculation right now in a very systematic way but ok basically the one thing I should still say is that the original distributions that Wigner and Dyson proposed were basically distributions where you sample from a Gaussian distribution for the matrix H which is only restricted by the fact that the matrix should be her mission because we are quantum mechanics and maybe should respect something like time reversal invariance or not as the case maybe Omega will be something like EI so the nearest neighbor level spacing but that's a good point let me just write it here nearest neighbor spacing and the nearest neighbor spacing well I've written it here in units of Delta but Omega is just the nearest neighbor spacing sorry are we considering only quantum system with discrete spectrum or we can consider also continuous spectrum yes so that's a very good question so in principle this really only makes sense for systems that have a discrete spectrum but what you would typically do is for example if you have so but here we we're going into territory where things are technically maybe not so well under control at this point but for example you might ask what happens in a quantum field theory so if you put this quantum field theory on a finite manifold and you restrict for example the energy to be some kind of micro-canonical energy E average then you still expect that in this micro-canonical Hilbert space you will have effectively a discrete spectrum and you can describe it by these notions but of course if the spectrum is continuous if you just think about the full spectrum and include all the continuous parts for the continuous part this makes no sense while your previous definition makes sense also for a for a continuum spectrum right there is nothing intrinsically that I would say here needs to be restricted to a continuous spectrum but the first definition seems to be a bit more general well but it has the drawback as I said before that it is a semi-classical notion so very good so wait there is another question the double-scaling limit of the random matrix model is related to a phase transition in the quantum counting model say again please yeah the double-scaling limit of the random matrix model is related to a gravity thing or is related to a phase transition of the quantum-cautic model we will talk about these notions for matrix models where we take the double-scaling limit so I think maybe we can postpone the discussion alright so good so maybe I can still say that in terms we should think of the quantum chaos in terms of looking at the level-spacing statistics of these systems and perhaps we should do this in a way that is like say micro-canonical Hilbert space or actually as I will argue in my last lecture also in such a way for example that takes account of local symmetry structures of more non-trivial systems that are not just quantum mechanics with a finite number of states but we will see that now there is also another notion here that will be very interesting for quantum chaos and this is probably what I will finish on today but let me just write it here because we have only talked about energy levels but there is of course more to life than energy levels and so for example if you want to talk about operators and if you want to talk about the expectation values and the time evolution of operators then we need to add a bit more structure and the thing that I want to introduce is this idea of a great thermalization hypothesis now before I go there though what I want to do is I want to say one more thing just one more sort of general background there is a very interesting conjecture there is a conjecture by Bohigas, Giannone and Schmidt which people also like to call of course the BGS conjecture we seem to like three letter acronyms and what they are saying is that the level statistics or the spectra of classically chaotic systems if quantized so if you quantize such a classically chaotic system once quantized well this is not a very good sentence because of course once you have a spectrum you are already quantized so classically chaotic systems once quantized their spectra follow RMT statistics so this is supposed to be true and of course even saying it like this I think invites comments about ultimately in those systems if you have quantum systems that have a good classical limit then there ought to be there ought to be some connection between these classical notions and maybe then semi-classical notions of chaos and these hard quantum notions of chaos that I was arguing for so I try to maybe make not such absolute statements because much of this is not known but it's true that maybe these notions are not so different after all maybe one actually implies the other but if that is the case then we are certainly talking about systems which have a good classical limit so quantum systems which have a good classical limit and in the context of such systems as I said this is the subject of a conjecture for which there is plenty of evidence and is known in the field as the BGS conjecture now right let me see how I'm going to I think what so to do is I wanted to give you an argument which certainly does not come to rise not to the level of proof of this but which gives you some insight as to how this could be true and in particular it will introduce a technique that I want to make use of in later lectures called the chaotic supersymmetric sigma model but because that is a somewhat more technical development instead of squeezing it in now I think I'm going to skip that part and start it at the beginning of the next lecture so I do want to directly make some comments about this EtH and so postpone this development to the beginning of the next lecture I can go on until three I know but even so maybe I'll start it and then we'll continue this discussion yes yes no the reason is that even so I'm worried that I'm going to squeeze it too much but let me actually start it so one thing that I want to say two things I want to say one a slightly more technical point is about the behavior of this distribution near the origin so what this is telling us is that two energy eigenvalues really don't like to be very close to each other so because they don't want to be very close to each other there is actually also a way of understanding this in terms of a force between two eigenvalues as you approach them they really want to resist that and so this tail end near the origin is a sign of what people like to call level repulsion so these chaotic systems show level repulsion okay let's say p of s or p of omega tends to zero as s tends to zero secondly though from this also again whenever you say you cannot read this particular part yeah yeah thanks or because it's too small or because it's too low down it's meant to say level repulsion but I will write bigger in future so well you know we want Sakura to know what I said so the other thing is that for a system that is actually integrable so that doesn't thermalize that is not chaotic this level spacing distribution near the origin behaves very different in fact this turns out to be so this is the integrable this is the integrable part but in particular it has a finite probability of having levels that are very close to each other including allowing for the generate levels whereas for this chaotic case once you have gotten rid of all of the symmetries which might force the generacies so in the absence of those symmetries the probability density that you find the coincident levels is strictly zero very good so and the third point was that there is a great deal of universality here enough space to write it big enough here okay there's a great deal of universality here in the sense that once we have specified symmetries of this system then the quantum chaotic systems universally show these kind of curves appropriate to their symmetry class of their level spacing statistics so there is a great deal of universality here and this universality is going to be sort of a key point about what I'm talking about next so what I want to give you is I want to give you an argument for this behavior from symmetry breaking so I'm only going to give you a sketch of this and maybe as time goes on we will be able to fill in more details but again as pertains to the kind of applications that I have in mind so let's focus on something that is very similar to this sort of thing that we talked about here but certainly not the same so we will actually talk about the correlation of two energy eigenvalues but not necessarily adjacent ones so let's focus on something like row of E1, row of E2 where I will already ominously put some bar over here that will imply some form of averaging and this is just the two level correlation and row of E is the spectral density which is the trace over the whole Hilbert space of delta of E minus H this is the spectral density so I'm focusing on the two level correlation because this is the first non-trivial example where the kind of argument that I'm about to make can be made because as I was saying already here statistics you could also think about higher moments much higher moments and perhaps other sort of let's say contentful probes of the spectrum of the system but this is one that gives rise to a nice story however instead of directly focusing on these things let's focus on something that is like a generating function for this so this generating function I will call it Z and of course I can write these quantities unaveraged first and when I need to average them in the appropriate sense we will discuss these bars so Z I will want to think of as a determinant of E1 minus H times a determinant of E2 minus H divided by two further determinants which I will just call E3 minus H and the determinant of E4 minus H I could continue adding ratios of determinants if I were interested in higher moments of these rows for the time being as I said let's focus on that and for regions that are related to convergence of certain integrals as well as certain more technical arguments about contours I put a positive imaginary value on one of these and a negative imaginary value on this energy so these are four energies which label my function Z those energies will eventually descend to the energy insertions in the spectra so those are the same objects but here I need to be careful about the analytic structure and moreover typically speaking in this generating function I will have twice as many energy arguments as I have in my spectral probe and well I will be set after certain manipulations to be equal pairwise and so I end up having the right number of energy insertions so now the whole story follows because of a very simple trick so just by exploiting the properties of Gaussian integrals so exploiting the properties of Gaussian integrals I can write these determinants as exponentials so there is the simple rewriting whenever I say something like simple in a paper election outside I kick myself because what's simple of course depends on on your taste but this one is really just using Gaussian integrals and I can write such determinants I can actually write an exponential for each determinant but I can unify that I can write something like psi bar and then let me see what notation I chose well first of all I put an I out front and then E I call this hat minus H and let me put a 4 here psi is a vector which has four times the dimension of the Hilbert space so it is actually a vector that has let me call the dimension of the Hilbert space d so it has two d directions which are bosonic because bosonic Gaussian integrals will give me one over the determinant of the operator sandwich between them and it has 2d permeonic directions for the determinants in the numerator so psi psi bar is a 2d slash 2d graded vector so there are Grassmann directions this is now too small for Sakura again sorry and there are not directions so just C numbers so now this simple rewriting basically so far is really just a rewriting so however then the idea is that okay so I am going to just refer to this before I completely rub it away so we are going to now consider C with some appropriate averaging so now this is a in practice very subtle point what I mean by this is that for example even to make this definition here for one single quantum system the fact that we have defined some finite bin size okay this is in effect some averaging because you average above over a number of nearby states there are also other notions of averaging that can be used so for example I could even think about writing down this same quantity when I directly sample H from a random matrix distribution then what the averaging is is very clear maybe so I am not going to be able to push this through technically now but just to say maybe if that is also useful for some people also in some of the questions we can do these developments more explicitly in cases where the steps are mathematically well not too involved and one of these examples is when you actually sample from a random matrix distribution but what I really have in mind is more like you take an individual system and you do some sort of averaging on the same spirit that you do here okay so now so what we want to consider now is we want to consider this generating functional of spectral correlations under this averaging so then eventually what we are going to do is one is able to rewrite z with this averaging procedure as a non-linear supersymmetric sigma model the supersymmetry is basically the one that we introduced here as an auxiliary technique in order to exponentiate a quotient of determinants the fact that it becomes a non-linear sigma model is related to the idea that I only mentioned so far in the title is that there will be some symmetry breaking and the symmetry breaking the right description of the physics around the symmetry breaking point is in terms of a non-linear sigma model so basically this non-linear sigma model first rewriting this thing as an integral over a matrix which I like to call a which has an action which is minus d remember this was the Hilbert space dimension times some gamma of a and plus a super trace of x times a so now what is a so a is basically you should think of a as the following field so a is a 4 by 4 in this case because I have 4 energy arguments a is a 4 by 4 graded matrix in fact you should think of a I call it free indices a b it's something like this psi psi bars a b where I'm contracting over the indices mu what does that mean you see as I said these size where I had twice the dimension of the Hilbert space bosonic directions and twice the dimension of the Hilbert space fermionic directions I am calling mu here and I'm summing over them the only indices that I'm leaving open are the indices that tell me if you want which determinant I'm referring to so a is equal to 1 will be this determinant b is equal to 1 a is equal to 2 will be this determinant so these are some sort of flavor indices if you want I have 4 flavors of fermions each flavor labels one of these determinants and then I have d colors which will wear on the Hilbert space I am and these are the indices mu and I'm contracting over them that leaves me a 4 by 4 graded matrix this is the matrix I call a and I introduce this matrix a by as a habits Trotanovic field for psi psi bar that will feature here after this averaging procedure so that's what a is then gamma of a is a potential which we need to write as a super trace of some potential function v of a so super trace because I have now a graded theory again this was a technical trick that I have here okay I'll just be able to write down the final answer so to speak and then I will take up as I said the more explanatory part at the beginning of next lecture so is a is a graded matrix and it's okay we can we can be explicit it's a sorry 2 slash 2 graded matrix and and x is a diagonal matrix in this flavor space so this 4 dimensional graded flavor space which just takes e sorry I should actually write it as e3 e4 because those were the bisonic directions and I had e1 e2 were the fermion directions this is a graded diagonal matrix and it encodes the dependence on energy that I had right from the beginning and they need to carry through and these energies appear here as sources for the field a and finally maybe okay no I should I should still say how to say that big enough for so this is averaging procedure okay and okay more about this soon and finally gamma of a okay an example for gamma of a for example it would be just a quadratic potential or it could be for example a cubic and other things can be imagined and actually other things will appear as we go on but before I finish let me just say that this is not yet the nonlinear sigma model the nonlinear sigma model follows very generically from this structure because one will argue that this system here shows a certain symmetry breaking pattern and once you have that symmetry breaking pattern it doesn't matter very much what the precise potential is because the theory that follows is dictated by the symmetry breaking pattern and not necessarily by the microscopic structures that I would have written down but I think I've gone already one minute over the extra time so let me pause here for questions and we'll take up the rest next time sorry I have not understood why you divided by these two other determinants which was the origin of the super symmetric that's a good question that is easy to answer by writing the equation so let us consider determinant of E minus H and let us already divide by in fact determinant E2 minus H okay now what I want is rho of E now what I know is that trace of 1 over E minus H has a real part plus or minus whatever minus plus i pi if I put a plus or minus here times delta of E so now if I put the trace over this then the part that is trace delta of E will give me my spectral density so if I get trace of 1 over E plus or minus minus H it's by taking a derivative D E2 of this ratio let me call this ratio let me call it D and not calling it R because people like to call this a resolvent they like to call this R so let me call this D even though it's a ratio so if I take D of E2 of E2 then what I get this I get determinant of E1 minus H divided by determinant of E2 minus H times trace of 1 of E2 minus H okay and what that means is that if I now actually put E1 is equal to E2 then I can forget about this determinant factor if I had not done that I would have not just ended up with the trace of the spectral resolvent so it's a normalization trick if you want so at the end you will put E1 equal to E3 and E2 equal to E4 or actually in the opposite way and the fact that you can do it in two different ways has some very nice technical consequences thanks I should have actually said that anyway okay unless there is an urgent question I propose that we postpone the questions to the discussion sessions which will be in half an hour is there any like very urgent question okay well let's see I don't know what very urgent but perhaps a very naive question so when you take the average of Z presumably it's an average over many Hamiltonians right and there would be presumably I mean for some large in matrix there would be many components to integrate over but in this expression you're integrating over a matrix which is a four by four matrix it seems a little counterintuitive that you can get some average of this partition function by integrating over just a four by four matrix well so there is something here which I think we're getting ahead of ourselves but let me answer the following question let me tell you the following point if the original system that what we put in here is actually a fully random matrix so this will be a D by D random matrix a huge random matrix my statement is still that I can write for you a theory where I only integrate over a four by four matrix and it will be mathematically exactly giving rise to the same spectral correlations so you can reduce a D by D matrix for sure to a four by four matrix now I think there is a different sense to your question but that maybe we'll discuss some other time let's thank Julian so we will resume in at 3.30 with a discussion session where you can ask informally questions to our lecturers from today and now we will have a coffee which I think will happen just