 There are some seats here and in the front if anybody's Very good. Okay. Let's start This is a you can say the academic highlight of the year So we want to always Celebrate Salam's birthday by the way that he will have liked to celebrate by having physics and having physics presented by the best scientists in the world so we started this tradition in 2012 so we have been having already several Lectures so then lectures we have been very very good at a very high level and they're all Accessible by By in internet in our website you can follow them Let me say see if I can remind you all of them in 2012 we started with Neymar Kani Hamid and He gave a wonderful set of lectures about the past present and future of physics something so something like As big as he always is but then we have we had Bill Bialek giving us a beautiful set of lectures about the biology and physics Which was I mean very very very good and then we have super such dev who gave a Beautiful set of lectures also in contest matter physics and face transitions and so on By the way super is one of the our diesel the direct medal for this year Then in that order we had that don's a year top mathematician who is actually local for ICDP because now he bought an apartment and lives in Trieste for three months a year So he gave a beautiful set of lectures on the number theory modular forms and so on and that was followed by two sirs Sir Brian Hoskin on climate change and Sir Michael Berry on mathematical physics all of them excellent lectures and Last year we had the person who introduced the inflationary universe Alan good gave general lectures about inflation which is the leading idea about the early universe and The idea having multiverse and so on so giving all that high level of Speakers is hard to come up with something competitive but we always managed to to Improve ourselves and we found Juan he kindly agreed to give these lectures. I will say a few words about one you don't mind He is considered by Essentially the whole community to be the leading figure in theoretical physics for the last 20 years and So he has made one important contribution, which was in 1997, which is one of the Probably the most important result that in theoretical physics is what is called the ADS CFT correspondence his paper has Probably is the most cited paper in the history of high energy physics or so and We actually Remember this by having two t-shirts in our ICTP Shop one is about the standard model and the other one is about ADS CFT, which is in honor of Juan and So this is very important because it relates to the gravity that we do not understand in Higher dimensions to a theory without gravity. We are we are supposed to understand in a lower dimension so the mysteries of quantum gravity are being uncovered by this Result and it has been the main subject for many many years in theoretical physics, but He's not a one paper person He has been the leading figure every single year providing opening new ways to theoretical physics in many areas from cosmology to black holes and so on This the the Rating to cosmology for instance, which is totally Outside a string theory. He did the calculation for the non-Gal scientists for for the cosmic microwave background everybody follows essentially and and he has then Made a lot of other contributions in string theory and gravity in general It was a pleasure to have Juan here before I give him the word I would like to say I always Show of how about one for at least two reasons One is that his wife is from Guatemala. So since I'm from Guatemala, I'm very proud of that we always claim that By by being married to Juan his wife is the person that has contributed more to science from all the watermelon scientists and And then he's not only that he's very generous whenever he comes to Guatemala. He comes and gives lectures and Meet students and so on we open a few years ago Finally in the National University of Guatemala with hundreds of thousands of students was the first time we had a faculty or school in physics and mathematics and Juan was kind enough to give the opening lecture for the for that school and Yes, the second thing that you show up about Juan is that he told me once that he has done he He moved to string theory or he did a string theory partly motivated by ICTP Which is I think is you know, I tried to emphasize that as much as possible one of the reasons is that He supervised on our good friend herald on the salad Who was supposed to here as being a regular visitor here and so in Superperson in Argentina and So I think he motivated one to participate in activities by GDP and to follow up so that that I think is it's very good and again by Being attached to a GDP. He has been very supportive of our mission He's a member of our scientific council He has been lecturing in our schools several times. He comes regularly Probably more importantly, we have opened this center in Brazil in San Paolo and He's a member of one of the three or four members of the four members of the steering committee of the governing body of our center So he goes every year we meet in San Paolo and he Contributions to the discussions of the running of this Institute are crucial to keep the high standards that this Institute has So in that sense, we are very thankful to him for being always supportive to ICP and its mission And so let me just finish also by saying that he of course being the leading figure in theoretical physics He has won so many awards that are hard to to to list essentially every single award You can imagine he has won it and he fully deserves them I think I see people as the for probably the first one or not to recognize his Contributions beginning in the Iraq medal more than 10 years ago But after that he has got the the breakthrough prize the Feynman Prize the Einstein medal the Lawrence medal I think I saw another one in condensed matter, which I I need some explanation from one why It was the Richard Prince Prize and lecture shipping condensed matter theory, which is also very impressive So in that sense he has won the many awards that he deserves so So it's a pleasure for me to have one here and I would like to ask you to join me to welcome one Okay, can you hear me? Can you hear me? Yeah, it's on Okay, thank you from Mando for that very kind introduction. It's a pleasure for me to be here at the ICTP So I'll be talking about the relationship between quantum mechanics and the geometry of space-time Now In the ancient times the people thought that the geometry of space was Euclidean and this that's what Euclid thought Then people discover non Euclidean geometries So the geometry for example of hyperbolic space that was the first example of a non Euclidean geometry And he has fascinated artists for example Escher who wrote who drew that nice picture of hyperbolic space as an alternative possible geometry Of course when we consider Euclidean space plus time We through the theory of special relativity We have a new geometry which is that of so-called Minkowski space So that's the geometry that appears when we consider special relativity and then Einstein put together these two pieces and In that way describes the geometry of our space-time Through the theory of general relativity or sometimes called geometric dynamics. So the idea that Gravity is due to the dynamics of the space-time geometry itself Now this theory had two stunning predictions One is black holes and the other one is to expand in universe and these predictions were so surprising that Einstein himself didn't quite believe them. I shall have this phrase That apparently Einstein told the metric the metric was one of the people who proposed the expanding universe He said your math is great, but your physics is dismal And sometimes other physicists tell us string theory is the same thing. So that's why I like this phrase So I would be telling you later about some interesting mathematical relationships, which I think will have some bearing to fit in physics Now the reason they were surprising is that they both involved drastic stretching of space and time and Now then of course those were predictions of the classical theory and those two predictions Have been largely confirmed and observed in nature But when we consider quantum mechanics and space-time We we have some other interesting predictions Now general relativity is a classical field theory So it's similar as a field theory to let's say classical electromagnetism There are fields at every point in space, which are the the case of geometry The fields are the shape of space time itself or the metric of space time and As any other classical theory in principle in a world where H bar is finite We should quantize it and there is some there is some procedure that allows you to quantize it Now because it's very difficult to change the shape of space time In fact, for example here the whole earth is barely changing the shape of space time. In fact it Hadn't been noticed that the shape of space time had been changed So for that reason It implies that when you have matter it changes the space shape of space times, but very very little and so in most situations when you think about Quantum mechanics you can think about quantum mechanics the quantization of all the fields including the gravitational field In an approximation where you first start with a fixed background geometry with a classical background geometry And then you consider small fluctuations around that geometry and that's a well-defined procedure that you can follow and it gives reasonable results for many situations and Even the simple perturbative quantization has some very interesting consequences and will will go through them in a second But it's a method that cannot be applied in the beginning of the big bang so for the beginning of the big bang we need a full theory of quantum gravity and The main reason for trying to understand the theory of quantum gravity is to understand the beginning of the big bang Now this is the We don't yet have a full theory of quantum gravity So we don't understand this most interesting question, but we've understood a few other Interesting questions, and I'll be talking about those But I mentioned this here because that's the main motivation for Trying to understand the quantum mechanics of space time Now I I mentioned that there were a couple of interesting predictions of this perturbative quantization and The two surprising predictions are related to the two classical surprises The first one is that black holes have a temperature and the second prediction is that an accelerating universe and expanding universe also has a temperature and The formula for the temperature here was derived by Hawking is what made Hawking famous The saying that the temperature is inversely proportional to the radius of the black hole so a small black hole Has a higher temperature than a bigger black hole the radius of the horizon and the formula for the accelerating universe is very similar and It says that the temperature is proportional to again one over the Observable size of the universe a universe that is expanding with a constant rate of expansion has a so-called cosmological Horizon and then the size of this horizon sets the scale of the temperature, so you you should understand these formulas are saying that the When you have a temperature you have an associated thermal wavelength of let's say radiation with that Characteristic temperature and that wavelength is equal to the size of the black hole in the first case or to the size basically of the observable universe in the second case Now the similarity of these two formulas is telling you that they are both based they both are Due to the presence of the horizon and they conceptually are basically the same thing, so they're the conceptually the same effect and the first Black holes are predicted to have a temperature, but black holes that exist in nature That form naturally through Natural processes tend to have masses which are bigger than a solar mass and have sizes of a few kilometers And that makes this radiation very hard to detect actually basically impossible to detect and It hasn't been observed yet on the other hand this other effect Which is very closely related to the first one as I was trying to explain It's actually very relevant for us and let me try to explain why and so the reason is the theory of inflation and in fact Fortunately, I've been told you had a nice set of lectures by Alan both on inflation So inflation is the idea that there is a period of expansion with almost constant acceleration That produces a very large classically produces a very large homogeneous universe So if you start from any type of initial conditions and you have classical inflation for a long time You'll end up with a perfectly homogeneous universe. Okay, so according to this theory the classical version of this theory We would end up with a universe that is filled with gas a uniform a Universe which with our average density would have roughly one One atom per cubic meter roughly speaking. So that would be the state of the universe if it weren't for quantum mechanics so the effect that we discussed before that is analogous to Hawking radiation is supposed to produce small quantum fluctuations and and The small quantum fluctuations are quantum fluctuations in the shape of the universe which and Produce a fluctuation in the density of the universe and then after the universe evolves These fluctuations get amplified and they form galaxies and so on and in this picture Which is a picture of the cosmic microwave background that I'm sure you've seen many times We see those fluctuations imprinted in the cosmic microwave radiation Now I should clarify that the temperature I'm talking about was the temperature of the desicciter universe very early desicciter universe and the thermal effects are related to those fluctuations But we are that temperature is not the same as the temperature of the cosmic microwave background, which is a different thing So the fluctuations and quantum fluctuations. I'm emphasizing are what create those Regions of different colors which correspond to regions Let's say regions of over densities or under densities in the universe So you should view this picture as a picture of the shape of the universe at very long distances and So what you see is that quantum mechanics is crucial for understanding the large-scale structure of the universe now sometimes you probably heard the statement that quantum mechanics is very important at very short distances and that's indeed true, but the Universe acts as a kind of amplifier. It's a kind of microscope that When we look at the universe at very long distances We are looking at how it was when it was very small and when it was very small quantum effects were important And indeed we see those quantum effects embedded in the long-distance scale structure of the universe okay, so this is a beautiful story that Usually well that we should all know it's a beautiful piece of physics that It's very interesting and in particular quantum mechanics is crucial for understanding the large-scale geometry of the universe. So instead of yeah It's kind of surprising we thought that quantum mechanics was important at very short distances But not the longest but actually in fact it's important for understanding and our best description of The large-scale structure of the universe is through this quantum theory So we cannot predict the precise shape of the universe But we can predict the statistics of the possible shapes and and it matches beautifully with experiments Okay, so this is just the first instance of a connection between quantum mechanics and the geometry of space time As I said is a connection that is intimately related to Hawking radiation and it's been it explains the leading explanation for the experiments We actually that have been performed now. We'll describe some other more theoretical developments that suggest a stronger connection So one one of the developments is this relationship between quantum systems and gravitational systems It goes under many names such as gauge gravity duality or gauge string duality or adseft or holography and what it is is it relates Theories of quantum interacting particles or for example quantum field theory, but it could be just ordinary quantum mechanical systems It relates them to Theory that describes a dynamical space time So it's a kind of bridge that relates these two theories The bridge was developed over many years by many people and it's been It's been understood better and better and so that's why you see all these people holding this bridge Now in general The this type of equality relates a concrete quantum mechanical theory to a concrete strength theory Unfortunately when you try to be concrete There are lots of details that you have to give and this sometimes clouds the understanding of what's going on but in many cases When you take the very strong coupling limit of any of these systems you get some very concrete Let's say very strongly coupled Quantum mechanical theory and that ends up being something simple than simpler than a strength theory It's just a nice time gravity theory plus perhaps some extra fields if you want to be very concrete there will be Lots of details you have to give but in general the spirit is that strongly coupled strongly interacting Quantum systems very strongly interacting quantum systems are going to be related to Space times governed by Einstein's equations to lead in order now I'll give you one example just to give you a flavor and I've chosen the example that is normally not not given but I've chosen this example to Show to you because of somehow the same somehow conceptual simplicity the example so is It's a system of coupled harmonic oscillators. Okay, so you have You have you start with a bunch of bosonic harmonic oscillator So that's an ordinary harmonic oscillator that you you learned in your quantum mechanics class And then this is just a Majorna fermion. So Majorna fermion will actually yeah, so Well a system of Majorna fermion, so you need at least two for this lagrangian to make sense But so this is just a fermionic harmonic oscillator. This is the same thing. This is a System, let's say with two levels two energy levels Now you take a bunch of this is a bunch of these systems So large number of them or for their n squared and you couple them to each other you will couple them by Putting some interactions. Let's say for example for the Unharmonic terms for the harmonic oscillator. So there are many harmonic oscillators. There are many possible and harmonic terms And also you will couple them to the fermions, okay? Now in principle you can couple the systems in many possible ways and the dynamics would be hard to solve and Would we have to understand but there is one particular coupling which Has some nice properties and is the coupling for which we know the gravity dual and in order to describe to you what the coupling is is a coupling where you take these harmonic oscillators and you organize them in a big matrix so an n by n matrix and So there will be an index a that runs over all the components of the matrix can also be viewed as The adjoint index of UN if you know what that is if you don't just think of them as a matrix and the various components of the matrix and Then there are nine of those matrices and so Then the full aggression is fixed by this UN Symmetry and also some funny symmetries called supersymmetry So that is a symmetry that relates the bosons and the fermions and that fixes all these couplings Okay, so the reason for right now the only motivation for introducing this is that is that this is Nice symmetry and with this nice symmetry we can understand this model a little better Okay, so it's kind of the spherical cow approximation. You know the story about the spherical cow so then Then this this theory has an effective coupling constant Which is the coupling appearing in front of the these harmonic oscillators and its unit in units of the energy So as when you have an unharmonic oscillator for very high energies that system is weakly coupled But at low energies, it will be strongly coupled and effective coupling is of this form and The N appears here because there are many possible Interact for any given any given harmonic oscillator interacts with N others through the interactions We have defined and that enhances the effects of the interactions So we can consider the system at finite temperature and then the characteristic energy scale that appears is the temperature and if that that temperature could be such that the coupling is very strong then we can We can ask what the system does. Okay, so it will be something And the statement is that in that regime the system is described by a ten-dimensional black hole So it's a it's a solution of some ten-dimensional Einstein gravity equations With a gauge fields with a charged black hole and the black hole has Is in ten dimensions has an eight sphere with an s or nine symmetry And that's related to the s or nine symmetry that the harmonic oscillators had and then it has a time direction that's related to the time direction of the in which the Quantum mechanics evolve and then it has a radial direction which arises from some mysterious way out of the interactions of this harmonic oscillators And then there will be a horizon so this is the geometry outside the horizon So it's just a sketch of what the geometry looks like and Then the the curvature of this geometry is such that The curvature has is proportional to some power of the effective interaction strength So that when the effective interaction strength is very large the curvature Radius is very very large. That means the curvature is very low. Okay, and we can apply Einstein's equations Now with this geometry, we can use the Hawking-Beckinstein black hole formula and it gives us some entropy So it gives us the entropy of this black hole. So that's whenever we have a theory of gravity. We can apply the Hawking-Beckinstein formula to compute the entropy of the corresponding black holes of the black holes in that theory So we can calculate it and then it will give us the entropy as a function And the entropy and also the free energy as a function of the temperature So we can do this computation or as a function of the energy. We can really know what the thermodynamics of this system is okay, so In other words, we have a strongly coupled system of harmonic oscillators And if you accept the idea that is related to gravity, I didn't explain why it is related to gravity I'll try to explain that later. I just first made a statement just to for you to get an idea of what the statement means So if it is related to this gravity theory, then you can calculate the thermodynamics of that system without having to solve the Schrodinger equation and so on for that quantum mechanical system. You just apply this, you just solve the Einstein's equations for the solution and get the answer And then you can compare That answer that you get from gravity with the answer you would get from the direct quantum mechanical model Okay, and This is this is the result. So this graph is a little busy. Let me just try to explain What these people did so they they took that quantum mechanical system of harmonic oscillators and they they solved for the thermodynamics by calculating the partition function Using numerical methods just using the methods of so-called lattice gauge theory because I'm well well understood methods and they found this square dots for the for the energy As a function of the temperature of course from that you can read off the free energy and also the entropy and So that's this is the result of the numerical computation and the result of gravity is this one. Okay So naively they don't agree. Okay but It's okay because what is plotted here in the bottom is Is the temperature and this is the reason where the theory is weakly coupled and this is the reason where it's strongly coupled And what this graph is showing is that as you go to stronger and stronger coupling the two results Start matching more closely as was predicted and in fact you can fit you can expect so from the full string theory answer you can Think you can take into account the stringy corrections to Einstein gravity and that Those corrections are expected to have the form of some power series in terms of the temperature with some particular exponents And so you can fit to a power series in the temperature of this kind and extract the leading piece That would be the gravity answer and by extracting this piece. They find that the piece is in agreement With the theory. Okay, so it's in agreement with gravity So the numerical calculation is in agreement with gravity within the error bars of this calculation So the the error bars of the calculation are of about 7% So that's if you wish a direct test of this relationship, so I've shown you this in detail just to Give you an idea that is a concrete statement for a concrete quantum mechanical system that you can go and check in the computer It's a mathematical statement and you can go and check whether it's true or not Now I'll now try to explain why we have this type of relationships So I'm not trying to get some intuition for how one concluded that this type of relationships exist so Let me give you a very brief History of why we expect a relationship like this and it's all based on an experimental observation So normally it is something said that string theory has nothing to do with experiment In fact, the string theory was motivated by experiment It was motivated by the observation that strings are producing hadron colliders So next next time someone tells you strings don't exist to say, you know what they actually exist in nature They are produced in hadron colliders. They're producing the LHC. They are used to match some of the data produced by the LHC and these strings are actually Now you might say but wait Aren't the hadrons made out of gluons and so on and yes, yes, they are made out of gluons and in fact the idea is that these strings are made out of gluons and The emergence of these strings is most clear when you think about the theory where that has many colors So instead of taking three colors that QCD has quantum chromodynamics has we take n colors That's a simpler limit because in that case you can argue somewhat abstractly that something like a string should exist And this this string is basically like a necklace of gluons Where the we have a gluon has a color and an anti-color So the gluon is a particle that has a color and an anti-color And if you form a chain of gluons so that the color of one is Correlated to the anti-color of the other and so on then you find that Neighboring objects in this chain interact strongly but objects that are far away interact quickly because their colors are not correlated Okay, the color for example of this one is not correlated to this one. So these ones do not interact strongly so then this object behaves like a String in the sense that a free string in the sense that it can go through the string and cross and not break and so on But it's a string in some kind of color space in the space of colors It's then what what the neighboring gluon is is a gluon whose color is correlated where it is in space It's not clear at least from this argument But the idea is that in in a theory Even in the theory of chromodynamics, we will have The strings that are observed in experiment will be made by gluons in this way Of course, since n is not very large these strings The strings can break and indeed they do break the strings do break in in the experiments But they if you take a limit where n is very large and the string cap and the Gluon coupling is very small when this combination is fixed then we The strings become almost free and they were expected to speck to behave like continuous strings Okay, this is what tough the argued in 1974 and The idea is that these closed strings are the objects that in QCD would call glue balls things that could exist even in a theory without quarks, okay? then If you have and the string coupling would be a proportional to 1 over n And as I said, let me emphasize again that we have experimental evidence for these strings And it was initially thought that the strings when you produce them this way should always be thought as propagating in four dimensions And But it was understood that in some sense the the string should propagate in at least one more dimension And this was realized by polio code And we don't know the precise geometry of the extra dimension but the idea is that the extra dimension is curved and for the case of QCD will have a This curvature would be such that the string would like to lie at the particular value of the extra dimension so that it Effectively behaves as four-dimensional, but that the food string really moves in in five dimensions And but unfortunately, we don't know the precise string theory background for the large end version of chromodynamics Which is the theory relevant for describing the strengths in nature But we do know it for the case of the maximally supersymmetric casting So there is a version supersymmetric version of chromo quantum chromodynamics with super symmetry You can view it from the purposes of this talk as simply something that allows you to solve the theory And so if you have supersymmetry, you can know the string theory and the way you derive this string theory is the following so You Go to the ten-dimensional super string theory that people had studied So when when people realize that strings are producing hadron colliders they developed the theory of string and that was It was started by Veneziano and many other people contribute to it first They developed they realized that the strings make sense only in special number of dimensions or are simpler in special number of dimensions Let's say 26 dimensions or ten dimensions and then with supersymmetry. It's ten-dimensional And in that ten-dimensional space there are objects that are not only strings. They are brains That are some objects that obey very precise rules that were derived by Polchinski And when you consider those brains, they also Curve the space-time geometry and produce a kind of black hole or black brain as was found by Horowitz and Strominger and In one picture you have explicit objects that With excitations we understand precisely and in the other picture we have brains that curve the space-time geometry And at low furnaces we find that the precise description that Polchinski had found reduces to that supersymmetric version of quantum chromodynamics And the the geometry near the brain near the horizon becomes ads 5 times s 5 so some particular Curve space-time geometry ads 5 is the same as hyperbolic space with some time direction So it's a negatively curved space with one of the directions is time-like And then the idea is that the probabilities in the quantum particle theory so in the theory of let's say young mill theory or Interacting particle theory are computed in terms of the propagation of particles in the curved space-time geometry So for example the propagation of a particle in the bulk space These can be used for for example the propagation of gravity waves is related to the correlation functions of the stress tensor in the boundary theory And well there are many many examples and there are many checks that are very impressive for large-hand specially But and I won't discuss any of them. They're very precise mathematical formulas that you can match So I describe the numerical test that there are many analytic tests that have been derived It's something that this has not been proven and so by not being and Not only that but we given any given quantum field theory. We don't know how to derive the gravity theory There is no general method for deriving the gravity theory So there are some examples like in that case where We had the brains in string theory then we could derive what the relationship was between the two But we are not always so lucky that the theory we're interested in is the theory that arises on some brains in string theory It might might be more complicated to find the gravity the world so this you should view this as a kind of tip of the iceberg of a phenomenon where you can find an example but perhaps it's a more general phenomenon and people have thought about this phenomenon in generality and They have found relationships between this phenomenon This this relationship implies some connections between various areas of physics and so here we've discussed this holography and This is related to black holes and there is related to the quark luon plasma this phase of Quantum chromodynamics It's also has some aspects in common with nontrivial phases of condensed matter physics and There are also some connections with quantum information theory and quantum error correction and This these are all different areas of physics and they share some Some similarities they are not necessarily identical or but We understand these arrows perhaps only some aspects of these arrows But the central point is by that by realizing that there are some connections You might learn something about your system. So for example, let's say you are only Interested in non-trivial phases of condensed matter physics, right? then by realizing that this might have some connection in some cases to black holes and You and knowing about some properties of black holes You might learn something non-trivial about these phases of condensed matter physics And I'm going to give you one example of that So I'll discuss only one example and This is a universal bound on quantum chaos that was inspired by black holes Okay, and this work mainly done by Schenker and staff and Stanford and Kitaya But let me just remind you first about Classical chaos So in classical chaos one of the signatures of classical chaos is that if you start with a dynamical system Let's say the plane here is the phase space of the system and as you start with some point and as you evolve in time The system changes and if you start from a nearby point the system also changes But the distance between where the system was without the perturbation and where it's after the perturbation starts growing in time, okay, and It grows in time exponentially with a coefficient with an exponent that is called the Lyapunov exponent Okay, so that's classical physics and you can view this as So let's say that at time equal to zero you change the momentum of one of the particles Then the position of the particle at time t will have changed let's say with this Poisson bracket between The position at time zero and the moment and the position of time t So this this change of the position of particle t time t is the same as this Poisson bracket You could all the Poisson bracket could be either positive or negative But if you take the square then it will be always be positive now if you And well both the Poisson bracket or its squared would grow exponentially with time now in the quantum theory You can diagnose something similar by taking a system and considering the position of a system at time t the operator time equal to zero and Let's say a similar operator at some later time and then you can compute this commutator and You could start with things that initially commute and then as time progresses They will commit they will cease to commute and this commutator will start growing exponentially with time Now in this if we consider a large end system and Let's say V you and V is a V and W are two Operators that act on different parts of the system. So let's say two different particles or two different degrees of freedom Then initially they will commute because they are doing two different degrees of freedom and then as time evolves the evolution will start mixing all the degrees of freedom and The operator at time t will not commute with the original operator at time zero Okay, so you can view this as taking one operator and the operator starts growing and eventually will Spread over all the cubits of the system So there is something you can define which is the quantum version of the Lyapunov exponent Which is basically the thermal expect Expectation value of this commutator. So you you don't consider just the commutator for well You could just take the trace of that but you could also Consider it at some finite temperature and then that would project them to that commutator onto the states that have an energy characteristic of the energies that the system has at that temperature Um So the idea is so we have these two initially almost commuting operators And this is the definition of just the ex quantum Lyapunov exponent and so this is a quantity that So initially the two operators are simple operators Let's say could be like poly matrices acting on two different degrees of freedom two different spins for example Then in a large and system we expect the commutator to be very small of order one over n But rising exponentially and then it will saturate at the value which is a further one Which is the type of commutator which we could have for two. Let's say poly matrices. That's the largest value and so the the Lyapunov exponent is Defined by this period of initial increase. So this can be defined for any system Okay, this is for any system any quantum system. You can define this You can define it even for a free system and if you define it for a free system or a weakly couple system You will find that it does not increase it just initially was small it stays small Okay, that's for a free system for a weakly couple system. It will grow, but it will grow very very slowly Okay So the the more strongly coupled the system is the bigger this Lyapunov exponent will be the bigger it will grow So now we'll we'll we'll apply this adseft and the connection to the black holes and see what happens so if we have a black hole in antideciter space or in Then that will be a fluid in the corresponding quantum system Which in the case of antideciter space it will be a conformal field theory. I didn't explain this, but that's the relationship in this case So it's a fluid of very strongly interacting particles. So we can apply the formula we discussed before And then we are going to compute that commutator using the gravity theory Okay So before we said in the previously we saw an example where we computed the free energy using the gravity theory now We are going to compute this commutator using the gravity theory. So briefly we'll outline here how the commutator is computed So a commutator will involve an operator at time zero and an operator at time t Okay So one will act at time zero and the other one in this this is the diagram of a black hole So the vertical direction is time the radial direction is essentially the the radial position this horizontal direction and this This black lines or in general 45 degree lines are directions of null null rays Okay, so we have an operator at time zero when at time t these operators when they act on on the state Well, they will create a perturbation that will propagate into the interior I mean we can evolve the system So for example this operator at time t prop creates a perturbation that goes into the interior and if we evolve the system backwards in time then Then this perturbation that falls in should have come from a perturbation that comes out of the black hole And then hits the boundary here and then goes back into the interior And it turns out that the commutator is related to the scatter in amplitude between these two these two computations this these two perturbations, okay And Now so in order to compute the commutator, so we We have to compute the scatter in amplitude So if the theory in the bulk is very weakly interacting in the bulk then the scatter in amplitude would be zero Okay, we'll commentator with vanish and the But in general there could be various interactions between the two particles and and the commutator will be non-vanishing Now something important here is that when the time is when this time is large the This this time translate that this whole diagram has a time translation symmetry Okay, far far has a time far away. It has a time translation symmetry, which is the the time translation symmetry of the quantum system That time translation symmetry near a black hole horizon at acts as a boost So that's a feature of black hole horizons And it's a feature which underlies which is the reason for Hawking radiation and all these interesting properties of black holes Is that the what looks like a time ordinary translation far away looks like a boost near the horizon? And for that reason if we let the long time pass here the the boost angle or the relative velocity between these two particles will be very high, okay, and if it is very high then the leading interaction is the interaction that Is mediated by the particles of highest speed highest spin and in a gravity theory that is the graviton And so that's an interaction that grows with energy So as you give the particles more and more relative energy The interaction will be growing and it will grow by a growth that is determined by the spin of the graviton Which is to it's just the fact that the energy the interaction is gravitational interaction is proportional to the product of the two energies and so that implies once you translate it to the two time that the well the energy goes like the boost angle and that implies that the The interaction grows like e to the lambda t with a lambda which is related to the black hole temperature so this conversion factor between the locally defined boost angle and the time at infinity is the same factor that Appears in Hawkins calculation for the temperature and for that reason it appears here, too And we find that then the Lyapunov exponent is given by the temperature, okay? So that's what we get for black holes Now We said before that in a general quantum system this Lyapunov exponent can be anything right But when we did the calculation in gravity, we just found one value. We just got the temperature, okay? So we didn't get anything The one question we can ask is whether it can be different than that and indeed it can be different if Instead of exchanging gravitons we were exchanging strings for example in the bulk then it could be different and it could be modified But one important feature of this modification when you do the calculation in string theory is that it will become smaller Okay, so I'm not giving you the details. Let's ignore the details you do the calculation using some string theory approximations and you find that the When you include the stringy corrections the this Lyapunov exponent becomes smaller So then this leads to a conjecture, right? So it can't well first of all it can be less we show that some corrections can make it less and If you have this category in amplitude in flat space then The power of the energy has always to be less than one to have a causal theory So that implies that this Lyapunov exponent would be always less than something Okay, less than the gravity result, but cannot be more Okay, so this inspires you to think that there might be a bound that there might be a bound based on general principles on For this Lyapunov exponent, okay, because in gravity there is a bound, okay? And in fact you can prove from From general principles that any system very general system very general large-end systems Just use in general properties like Analyticity in Euclidean time unitarity and so on in particular. You don't use ADSF T You don't use gravity. You don't use any of that. That was just inspiration. You just now use purely general properties of Quantum systems, then you prove that this Lyapunov exponent is less than this bound, okay? And this is supposed to be true for any quantum system regardless of whether it has a gravity Duel or not. So it's an example where Knowing that there are these relationships you can get results that are valid for any quantum system, right? so if you have your condensed matter system or your let's say high temperature superconductor and you did this experiment of trying to compute the the The the commutator for for two operators at different times, then you should obey this bound, okay? so So it's an example where you can go the whole way so from inspiration to final proof Removing the inspiration somehow, but there are many many examples where people are sort of part part of the way through that The maybe they have a conjecture They don't know whether a conjecture is true or not and there are other examples where people also have found Have gone the whole way and they have some properties of general system another example is Anomalous transport I won't explain it, but you So it's the correction of hydrodynamics equations when you have anomalies in field theories Okay, so now let's go back to quantum mechanics and their relationship between quantum mechanics and geometry What time did I start? It seems there was a time dilation effect here Okay, yeah, okay. Well, I'll have to go but So there is some so one of the We were discussing the relationship between quantum mechanics and geometry and one of the recent developments is the Connection between entanglement, which is a fundamental quantum mechanical property and the geometry of space time So it was found that the entanglement pattern pattern present in the state of the boundary theory Translates into geometrical features of the interior and that The shape of space time and the geometry of space time is closely connected to the entanglement properties of the fundamental degrees of freedom of the Quantum system now this is based on a formula called the Ruta Kanagi formula for the Entropy of sub regions so in theories that have a gravity Do also if you have the boundary and you have a sub region of the boundary you can compute The entanglement entropy for when the quantum system is in the ground state so the full grounds the full system is in Quantum mechanical ground state a single state But if you consider only a sub region it will be in a mixed state and you can compute the entropy And it turns out that in theories that have a gravity dual that entropy is given by the area of a minimal surface Defining the bulk that ends at the boundaries of this region so here what you should get is the idea that this entanglement entropy, which is a quantum Mechanical notion is connected to the geometrical notion like the geometry of space time the area of some minimal surface So this is a formula similar to the Hawking-Beckonstein formula. In fact, it's a generalization of the Hawking-Beckonstein formula and This formula led to many other many other developments that I won't discuss I wouldn't I will discuss only one one example of this Which is the two-sided black hole so it turns out that the simplest black hole solution So the black hole solution with no matter is actually a solution that describes two black holes First one to realize these were Einstein and Rosen They didn't get the full geometry. They didn't get the full interpretation of the geometry right It was finally got the one that will go finally right was Kruskal So it took about 50 years to get this geometry right from the Schwarzschild solution So Schwarzschild solution actually describes two black holes So what we call it a right black hole and a left black hole the two are identical outside the horizon And they are connected through the interior. So they're connected through a kind of wormhole. It's a time-dependent solution. So These regions here which we call the future interior and the past interior look like a cosmology So they look roughly like a cosmology in the sense that The universe contracts here and we have a big crunch singularity here in the future We have a kind of big bank singularity in the past. It's not isotropic, but somewhat similar conceptually and So that's the geometry. So that's a geometry that was found a hundred years ago Okay, but more than a hundred years ago and the idea is that In the case of and I see their space where we have a dual quantum system. That geometry can be interpreted as As an entangled state of The two quantum systems a particular entangled state which is called the thermo field double Where you take all the energy eigen states of one side and you correlate that you take a copy of that Energy eigen state in the left side and then you sum with this Botsman like factors Okay, so that creates a state. It's called thermo field double because if you only look at the right system You trace out the left system then you get the thermal density matrix on the right system But this is a pure state of the combined system And the idea is that this pure state of the combined system is described by a geometry Which is connected. So there is no dynamical connection between the left and the right system It's just that we chosen a an entangled state So the geometrical connection here in the trash that solution is due to entanglement It's not due to a dynamical connection between the two systems So that's the the interesting aspect of this and this has been studied in various ways So if now Because it's related to an entangled state you would expect that if We have two people here in one in the left system Let's call him Romeo and Juliet on the right system if Romeo wants to send the signal to Juliet The signal will go in and will fall into a singularity, but will not come out on the other side Okay, and indeed that's reasonable because you cannot send signals using entanglement Okay, so that matches that property of entanglement And indeed you you can check that this is even if you perturb the system You will cannot send the signal if as long as it is given by an entangled state And there is some interesting reasons using the null energy condition and so on that for why that is true So there are no this is a wormhole, but it's not the traversable wormhole. So So you could have two black holes that are very far away They will be connected through the interior, but you cannot send signals Using that purely using that connection So if you had Sorry, I usually like to take the story of Romeo and Juliet. I Get permission to tell the story of Romeo and Juliet So we have Romeo and Juliet. So this is sometime far in the future They like each other, but their families don't like them to like each other So they confine Romeo to this galaxy and Juliet to the Andromeda galaxy. Okay But Romeo and Juliet are very very smart. They exchange pairs of qubits and they Produce a pair of entangled black holes. Okay, they have lots of technology and lots of time to do this and Then at some prescribed time they jump into the respective black holes So let's look at the bottom picture first. So their families see them jumping into their respective black holes And they think they committed suicide And the narrator of the story also thought that and wrote that in his book now on The other hand We now think that if they did that properly what the what would have happened is what we see here in the top So when Romeo was standing outside the black hole and Juliet in outside her black hole one in this galaxy The other in the Andromeda galaxy They were separated by a short distance through the wormhole because they had created these black holes In this precise entangled state and then when they jump in they would meet in the interior. Okay Fortunately, then they died of the singularity so So this script wasn't allowed for a Hollywood movie Okay, and so so then but they they read this other paper. Okay, they they read this paper About quantum teleportation. So they realized that it's true. You cannot send signals using Using quantum you cannot send signals using entanglement But you can use entanglement plus some classical communication some unrelated class some classical communication to send the qubits and the way this works is that Then we have Romeo and Juliet here and let's say They share some entangled qubit and then let's say Charlie gives Qbit to Romeo. So then Romeo can mix these two qubits thoroughly Make a classical measurement send the classical information and Juliet can do an operation here and extract the qubit Okay, so that's quantum teleportation. It doesn't contradict anything because some information was sent Okay, so it's fine but the interesting aspect of this is that you only need to send classical information and then Somehow a qubit gets sent by using the pre-existing entanglement This should surprise you because to specify the state of a qubit You need an infinite number of classical variables, right? You need to specify the direction in which the qubit is pointing but by sending By the classical information you need to send the just two classical bits and so by previously Having an entangled qubit and just sending two classical bits. You can send the full state of the qubit from left to right so the resources you need is You in doing this you will spend one entangled pair. Okay, so you have to spend one anyway, so Now so then they then then Juliet was telling Romeo. Well, why why don't we do this and You can be the the guy that gets sent the qubit that is sent, right? and then so We sorry we go back to here, so we'll have some very complicated system, so Romeo will be sent here will play the role of this qubit and and then Alice will recover him on the other side, okay, but Romeo didn't like this idea because This teleportation protocol involves mixing thoroughly Romeo with these other entangled pairs Okay, so Romeo didn't want to get thrown into a blender where he would be cut into little pieces and mix the very thoroughly Okay Juliet said well, it doesn't matter. We'll get you resurrected here on the other side But Romeo said no, I won't like my the feeling of being cut up into pieces. Even if I forget, okay, so Didn't like that But then they said no, but let's do something else. So let's do that but with these black holes, okay? So now we can do this teleportation protocol, but the entangled state that we share will be This black holes in the thermo field double, okay So the thing that we'll do is the following Well Romeo will jump in Will fall into the black hole the dynamics of the black hole mixes Romeo very thoroughly with the degrees of freedom at least from the point Of view of the left exterior then we measure the Hawking radiation here and send that information to the right side and When because the Hawking radiation here and here are correlated It is possible for Alice here to use the information about the Hawking radiation To extract some energy from the black hole or to send some negative energy show wave into the black hole Now what happens is that from the black hole picture? So Romeo is been sent in he doesn't feel anything special When he crosses the horizon From the point of view of the left exterior he's suffering strong interactions But from the point of view of Romeo nothing special is happening when he crosses the horizon And then he crosses this negative energy shockwave and in suitable situations he that also doesn't feel some very bad when he that happened when that happens and And when that happens he can suffer a time advance as opposed to a time delay. So normally when you When light passes, let's say near the Sun it suffers a small time delay But if the Sun had negative mass they would suffer a time advance or would arrive earlier than before and arriving earlier Means that it can go outside the horizon So Romeo will survive and get here to the right side and would feel perfectly fine throughout the whole process. Okay? so in this particular class of entangled states that has this gravity dual this Process of teleportation takes place through the wormhole. Okay, so the qubit when it travels from one side to the other Travels through this wormhole and doesn't feel particularly bad or doesn't feel particularly special okay, so I think I'm already enough over time So what's interesting about this is how the qubit is being moved from one side to the other and this is So we are analyzing this and trying to understand better what this implies for the behavior of black holes actually thinking about this has some interesting implications for For the black hole information paradox and we'll discuss some of those things tomorrow So In conclusion the we've seen that quantum mechanics. So first point Is that quantum mechanics determines the large-scale structure of the universe? That's according to our current theories We also discussed that certainly certain strongly interacting quantum systems have concrete gravity duals We discussed that the near-horizon dynamics of black holes is related to the chaotic dynamics of the quantum system and The Lyapunov exponent is related to the near-horizon properties or the scattering of excitations near the horizon of a black hole We discussed that the structure of space-time is intimately connected to patterns of entanglement And we also emphasize the fact that entanglement is related to wormholes and that quantum teleportation is related to travel Traveling through wormholes Well, there are future challenges for the future Maybe I'll just mention only one of them Which is to understand better what happens at black hole singularities? Or for example the big banks singularities So they are both very similar and we hope that by understanding the black hole singularity will learn about the black the big Bank singularity and that's for me one of the main motivations for studying black holes for to try to understand this Particular quantum aspects of black holes is to get some insight about the black big bank singularity And also a related question is how do we describe the black hole interior so people have Different proposals for doing this, but we don't have a complete understanding that we all Agree with and an understanding that can teach us what a complete understanding would be one that tells us what happens at the singularity And what that's not to extract from there. So there are many problems that are open and hopefully will You will solve them. Thank you Thank you very much everyone. This is very exciting lecture There are questions something about Shakespeare No in the expression for the commutator with the exponent Into the lambda t. What about the pre-factor? What does determine the pre-factor in this? Yeah, the pre-factors depend on the operators in question So in general the end dependence on if you have a system with many class many pieces Will go like one over n For the for simple operators but the detail of the pre-factor depends on the System and the operators. There's no general statement of this questions Do you know a real life systems that get close to saturating the bound the chaos bound No Well depends on what you call real so I'll describe in the next lecture the so-called s yk model Which is a model of involving interacting Majorana fermions and It's believed that this model might be you might be able to simulate it in a I mean you might be able to build it in a lab and there have been There are proposals for building it building it in a lab It hasn't been built so we I guess by the definition of a real I mean if you the definition is something you can build in a lab We don't know Hasn't been built yet, but it looks like a type of system that you could build in a lab It's a relatively simple system question myself This bound first of all you had to have the minus sign from the gravity side that it was crucial to motion That it was is it straightforward to think that from gravity you had you you have a one minus something Yeah, so the the the methods you used to prove the bound show that in the case of gravity you should have that minus sign This is okay, but it would be a general thing for any ADS EFT or any gravity dual you you know that from the gravity side you always get a minus sign Yeah, the force is attractive. It has to do with the fact that The this this this effect is related to a time delay And so for causality, you need a time delay and not a time advance. So that's what And if I understood correctly then Without gravity people who have thought about Finding this bound I realized I answered the previous question sliling correctly So you had me asked me about the pre-factor the sign of the pre-factor is fixed The sign of the pre-factor is fixed. So I said I said I couldn't say anything about the pre-factor Without gravity people could have thought about looking for a bound and then forget about gravity just do so they could have done it Many years before it's the correct. Yeah It's only that gravity the gravity dual motivated Yeah, it inspire you to think about bound. Yeah, very good possibility of a bound Yeah, I mean there are other examples where people first thought about the bound from other points of view So there are for example a band on viscosity which people have proposed before Before the connection to gravity and in gravity also is the viscosity some particular value and It was believed for a while that there was a bound but actually there isn't the bound so you can be violated slightly It's not clear where there is a real bound in that case And going back to the beginning of your talk Just more elementary things and when you mentioned that Paul Jakov had You concluded that the strings move in an extra dimension. Yeah, do you be more explicit on that? Yeah, so so so Paul Jakov realized that the First of all that the reason that strings make sense in certain number of dimensions is It's due to a particular anomaly or problem that you get when you quantize the strings so if you assume that the strings moving four dimensions you get a kind of anomaly and You have to somehow fix it and the simple way to fix it is to add one extra dimension or be so you can add one extra dimension But then the extra dimension will be curved or if it sees to be in in flat flat space Then you have to have a specific number of dimensions So what that what Paul Jakov realized is that you could add just one more dimension and it could be curved it could be That space could be a curved space Okay, and in your model your harmonic harmonica selectors model Yeah, you pick nine as a particular number of the number of measures At the end it ends up related to the string theory in ten dimensions But could you have thought about another reason to put nine? No, no, I certainly couldn't have thought of any reason for picking this model The reason the history was different that first there were the deeper first where the difference And it turned out that a low energy The low energy dynamics on particular so called the particles or particles which are like the solitons The theory that describes them is this particular theory So in some sense we got this as some example of this duality when we derive them from string theory, but no, I First was the string theory and then was this model I Chose to present it this way just to show well here is a model particular model as I said There is no generic. There's no general method for picking Dual pairs like picking a particular quantum mechanics model and picking its dual Gravity theory, so there are some we got a window into this through a string theory And maybe there are some extra techniques some other techniques for finding General gravity duals that we'll learn in the future Okay, if people don't ask more questions, I have more to ask, but we have more questions Yes Basically, so in mathematics the non zero Lyapunov exponents exist for hyperbolic dynamical systems So this is a very special feature that two trajectories actually Dalit more and more and if you consider dynamical systems on hyperbolic manifolds, there are bounds on Lyapunov exponents So in this case number one, why are Lyapunov exponents non-zero? Is this a feature of ADS geometry or? I think this question probably will have part of this question would be answering the next talk But for now, let me just say that the I the Lyapunov exponent here is arising From the dynamics of the particle theory and in principle, it's not related to hyperbolic space Okay, so it's You can have You can have a complicated dynamical system with a complicated phase space and the trajectories will diverse Even though it's not hyperbolic space hyperbolic space is an example of a simple dynamical system But a particle in hyperbolic space is an example of a particular system that has a Lyapunov exponent and It's particularly simple to understand But it's something more general There was a confusion between so the dynamical system is called hyperbolic as well if the Sorry, sorry. Yes Yes, so in that system there in that sense they are hyperbolic, but it's not a classical dynamics So it's been this defined for quantum systems So even in systems that don't have a classical analog like a system of fermions, for example, you could define this Commitators, I will we see that next time and of an example of that system and And then And find this Lyapunov exponent, etc. So That's not. Yeah, I don't know if you're definition of hyperbolic includes anything that has a Lyapunov exponent. Yes They're all hyperbolic in that sense Just wait for the microphone wait for the microphone fix so we can yes, I Understood it so that if you have s by k4. Yeah, so you know what kind of Gravity corresponds to it. Yeah, so it's some black hole. I don't know details with something like this But also, we know that if we add to s by k4 we add s by k2 The changes drastically the situation so what happens in terms of the holes in this situation where you add quadratic term to Well We don't have a precise correspondence between s by k and a particular black hole What we have is a correspondence between some feature of s by k and some feature of black holes are explaining more detail later But at the big level what should happen is the following that when you add the s by k2 The Infrared region of the geometry should should develop Some extra symmetry some more symmetries so called higher spin symmetries So beyond beyond having purely pure gravity. We should also have other higher spin fields And that's so it should be a version of what's called Vasilev gravity. So Vasilev gravity is a theory Where you have higher spin fields? Beyond the but we don't know that the precise relationship this this in general what should happen when you go from when you have some theory That is free. So the s by k2 in the infrared Is this a free theory? So if you've only had s by k2? so if I k2 means Hamiltonian with a with the quadratic with with a Hamiltonian, which is quadratic in the fermions so that's an exactly soluble system and It in the bulk theory should have this higher spin theory symmetry. So that's the general principle Let me lower the level of the questions so that to encourage people to ask more questions and like I said you mentioned That you study you want to do so the backshots but holes in light is partly Because you want to understand the big man singularity. What do you think that black holes are simpler than the big man? It's roughly I'm going to explain it with an analogy So if you want to understand how ocean waves behave right you can be standing in a boat and try to figure out Trying to figure out what the waves do. Okay, so it's a little complicated because your boat is moving and the wave is moving and it's kind of On the other hand if you start on the shore and you look at the waves That's a little simpler you can set up all your instruments on the shore and look at what the waves are doing Okay Now the the black hole is a bit like the system where you have a shore You have some region far away from the black hole where the gravitational Where there are no big fluctuations in the geometry of space right the geometry of space becomes very cold and Very rigid far away and then you can set up experiments and ask precise questions about the black hole in cosmology It's more similar to being in the middle of the ocean where you You have these fluctuations of the geometry and you have to take them into account So our existence as observers and so on depend on our cosmological evolution and There are no very very sharply precise Mathematically completely mathematically precise questions we can ask. It's a more complicated situation. So that's why one looks simpler and When I was young people used to say that black holes were kind of science fiction things But now our real objects and so what people say that wormholes are science fiction and now you are working on more holes What do you think about that? Well, I hope the Metra's phrase would be true apply to this I think Mathematically, I think it's correct and How they will be realizing physics. I don't know but I hope that there will be Well, I think for example when people realize this is why cave model in the lab And so on you will be able to to see all this teleportation Discussions and it will be very similar to This two-dimensional black holes. So that's some similarity, I guess but I think Yeah, and you can find by thinking about these things you can find the black hole solutions for black holes even in four dimensions that look like wormholes or traversable wormholes and I Don't know if at some point you would be able to make them. But so for example The idea is that if you manage to make Magnetically charged black holes in nature. Okay, so black holes Then if you bring them close in if you have small enough black holes and you bring them close enough You should be able to create a traversable wormhole So now there are lots of details which are not understand How long it would take to do that and so on But that's an example of something that in principle could be done now It's very difficult because first you have to produce magnetic monopoles and then collapse them into black holes and so on So it's not some experiment that will be done in the near future I think the the the idea is that we we analyze this these systems because of their general nature and the principles they expose and And also because of the connections to other physical systems. We might learn more about other physical systems we might Discover new laws of physics for other physical systems and so on You say for some when you start talking about the Romeo and Juliet story that you couldn't The the teleport you couldn't entanglement couldn't send the information But then in the last example, you say in principle, you could do it. So in that sense, you say it could be good Yeah, yeah, so the the difference between the two cases is that in the last part of the talk We allowed some classical information to go between the two sides and So if you wanted if you want to do if Romeo and Juliet wanted to do this They would Romeo would have to fall into the black hole a machine would have to be left out measure the Hawking radiation and send information to The Juliet's black hole and then after when that information gets there It will be processed and Romeo will be extracted from the other black hole. Perfect plus removing. Yeah. Yeah, that is better for a movie Very good. So before we finish Let me remind you that this is the first of three lectures tomorrow. We'll have a same time the continuation of this lecture on when is they will be a bit more more More technical, but that's that will be that the end these lectures have been Generously founded by the KFAS, which is the Kuwait Foundation for the advancement of science, which we are very thankful to them and So as usual when we have lectures of this type or colloquia we have the tradition that there are some refreshments outside for everybody, but We asked everybody to leave the room except for the students for the diploma students who will stay here with one to ask more personal questions or that they can be too shy to ask In front of everybody. So and we promise also full for them. So let's send one again for