 So let's go live. Awesome, I think we are live. So welcome back everyone. Thank you for joining us. Let's go live. Okay, now I have an issue with this. Awesome. Okay, sorry about that. My name is Alejandro and I'm going to be your host today. Today we're presenting from Quantum Theory to Black Holes by Alfredo Gueva. We're very happy to have you. Alfredo earned his bachelor in physics from University of Conception in Chile and then he moved to Waterloo to do his master's and PhD at Perimeter Institute. Alfredo is currently a junior fellow at the Society of Fellows at Harvard. So remember as usual, you can ask questions over email, through our YouTube channel or Twitter and then the questions will be read at the end of the talk. So without further ado, we'll turn in time over to Alfredo and thanks for joining us. All right, awesome. Yes, let me know if you can see my screen. Perfect. Yes. All right, great. Yeah, so I really, really appreciate this invitation. I think it's one of the first times, hopefully not the last one that I get directly in touch with how physics are evolving in Latin America. And I of course look forward to many more opportunities like this. I think what you guys are doing is really, really amazing. So in this opportunity, I would like to sort of present a sort of review talk of some stuff I've been working on the past few years, but also many other people together in wonderful teams. Some of them are my collaborators and some of them are actually in different groups, but there's a whole new sort of ideas that are emerging to connect these two topics that I described here, amplitudes in quantum field theory and general relativity in particular black holes. So without further ado, let me just try to give you a motivation of why these two topics are intrinsically connected. So we know that general relativity is one of the prime examples of effective field theory. So that means that it describes a low energy regime of a putative quantum gravity theory that of course is yet to be discovered, as we all know. As such an effective field theory, it has had many beautiful successes and in particular, it's making really strong predictions at the scales we can measure right now at this very moment with our current technology. Okay, so we're talking about gravitational detectors, but we know as technology progresses, the ability to measure the precision that we can reach also progresses. So while quantum measurements are still kind of out of reach, so we don't really understand how to test the putative quantum gravity theory, we can certainly know what is test ER to then present the precision. And that's of course expected to increase over the coming years. Okay, so the fact that we can actually test to more and more precision brings up two major problems. So if you are familiar with general relativity or maybe if you are just sort of acquainted with general relativity, you will realize that Einstein equations are really, really hard to solve. And most of the textbook solutions that we are taught in grad school or sometimes in undergrad is that are very, very idealized situations in which you have like complete spherical symmetry. And in the universe, of course, we respect more realistic processes, more dynamical processes, non-stationary process to take place, such as collision of two big neutron stars to massive compact objects that don't really have much symmetry. And so that's a problem because we don't really know, I mean, if you open the textbook, you don't really know how to address the sort of more realistic situations. And the other problem that has not been so emphasized at least in the last decades is that observables, like actual measurements, actual numbers that we can get from the theory, are not clearly well defined because as you are put it, there is different morphism in variance. So there is like gauge symmetries and redundancies. So you have to pick a gauge, pick a frame. Asymptotically, you have to fix your asymptotic observers and you have to sort of fix a sort of asymptotic structure of your space time. So all these sort of boundary conditions, if you want, or like gauge fixes, gauge choices are sort of hiding the actual gauge invariant or on-chill content of the theory, which is already something that we should be measuring. So these are different problems. Of course, here I just put some pictures just to catch your attention. The first one is, of course, recent observations of the event horizon telescope pertaining what we call the photon ring of a supermassive black hole, which is something that was predicted by the theory and then observed in 2017. And the second one is the waveform of two light-in-black holes, in particular the gravitational wave that emanates from those things. So this already makes you wonder what is the actual number that we want to compare? You can think of the waveform as some sort of intrinsic distance between two detectors, but we want to sort of make that intuition more precise, like more fundamental. Okay, so for the problem one, so we were talking about the textbook solutions for general relativity, right? So we know many stationary solutions. And in particular, this nice book by Eddard St. Kuhn is one of the oldest references on analytical solutions to general relativity. You just open it up and you find all these nicely very symmetric solutions. Most of them have some very nice geometrical meaning like you can draw them, you can play some Hendanken experiments with them, try to understand the causality, the topology and all that. But they all have some diverse degrees of symmetry. Some can, it can be static, it's cylindrical, spherical and so on. Some of them are GPS state of supersymmetric vacuum. So as I said, these solutions fall short in describing more realistic time-dependent scenarios. Okay, so the ones that are measuring, including the gravitational wave emission by black holes is one of those more realistic situations. And regarding the problem two, well, we know in other theories, not in general relativity that observables are very precisely given by scattering amplitudes or by cross-section, right? So what happens at particle accelerators is that every time there's a particle shooting in a particular solid angle, there is a detector that clicks and then it counts like the density of particles and every time there's an event, you can measure more or less the probability of that event occurring and that's a number that doesn't really depend on your gauge, right? It doesn't really depend on the frame or the Lorentz frame or the redundancies of the deer and so on. You just count this cross-section of particles. So those are the intrinsic observables of quantum field theories. But we know that in quantum gravity, in particular, in general relativity, these sort of scattering amplitudes are not so easily defined. In particular, there is some asymptotic structure of space time that needs to be taken into account to define what scattering event means. And in particular, we have UV divergences. And more importantly, we have infrared divergences that prevent us from actually defining observables in this theory. So what can we do in GR instead? Well, it seems that for the experiments that I was talking about, right? So this photon-bring process or gravitational wave detection, we can still define something that is independent of the coordinates, its gauge invariant. And it actually happens to be more or less related to the idea of scattering. So in particular, for instance, if you have the old Edington experiment that sort of tested GR in 19-something, you will find that there's a light bending of, you know, provoked by a massive object and the action of the gravity, right? So this light bending angle is something that you can define in sort of a particular intrinsic gauge invariant way. And it's measuring some asymptotic event because you just sort of propagate light very, very far away from your black hole. So there are other similar situations that you can measure polarizations. Asymptotically, you can measure gravitational fluxes. Asymptotically, and all these quantities, you know, are measured very, very far from the events. And so they have this relation to the cross section to the scattering situations. When we try to actually compute this, it's a little bit hard because as I mentioned, these Einstein equations, right, are very hard to solve. They are non-linear, very non-linear. So if you don't assume any symmetry for a realistic process, you have some very, very hard problem. And the way in physics we deal with those problems is, of course, by using perturbation theory. And perturbation theory we usually perturb away from something that's very symmetrical. Usually it's the vacuum, but it doesn't need to be. In the case that you perturb around the vacuum, which is some sort of Newtonian vacuum, the expansion is called post-Newtonian. We can also perturb around the Minkowski vacuum, you know, just completely flat space, and that's called post-Minkowski. And we can do all sorts of medical experiments using numerical relativity that was developed in the 2000s. But at the same time, all these sort of more realistic approaches are at the same time sort of have a bottleneck in terms of how much we can compute, right? So eventually, either the, you know, processing like computing power or like our ability to perform perturbative expansion has like, it doesn't really escalate as we go to more precise experiments. So, and all this in reality, even the numerical case is coming from the fact that these equations are very highly non-linear. So just to give an example, in this post-Newtonian expansion, what happens is that essentially you're just summing the same objects that you sum in quantum field theory, right? So in quantum field theory, again, going back to this area of the cross-section, we just have to sum over these different Feynman diagrams. And as we go to more and more precision, it doesn't really escalate, right? We need to add more and more diagrams and it becomes combinatorially hard and also algebraically hard because every diagram has more terms. And in the post-Newtonian expansion, it's kind of the same effect. Now you actually have to sum all these different interactions because you're expanding sort of away from the Newtonian potential. So you have to include corrections from the Newtonian potential. These are corrections in the Newton. And eventually, you just reach this bottleneck, so it's just too much diagrams, okay? So too many diagrams. And so you can think of these two situations as being reminiscent, right? So corrections to a Newtonian potential in GR to corrections of the, if you want the Coulomb potential in QED or QCD perturbatively are sort of the same degree of complexity, right? As you go increase precision. So what can we do? So I would be happy if by the end of this talk, you end up sort of convinced that a scattering amplitude in QFT actually have a more simple structure than what I just showed you. And in particular, we can sort of resum these amplitudes into simpler terms. We don't need to compute all these boring diagrams. And the relation actually to GR is exactly that. So in GR, we actually don't need to sum over all these different post-Newtonian diagrams. We can just sort of find a simpler structure, a simpler guiding principle that allows us to compute it. And in particular, the same way that we talk about Feynman diagrams related to observables in quantum field theory can be used in GR and we can sort of derive observables in GR inspired by the cross-section and scattering process. And these are more realistic dynamic scenarios because obviously we're talking about two-body or three-body problem in GR, which is not something that you'll find in a textbook. Now, what's the problem of GR not being a quantum field theory? Well, usually the problem is that there's UV divergences and you need to renormalize them. So you get this infinity of terms. But at the level that we want to work on, so we want to work at the scales which are low energy effectively way below the Planck mass. So we are thinking about GRs and effective field theory for these UV divergences, right? So actually, if you think about three-body problems when these two, where these three-body are very far from each other, there's no divergence, right? So as long as you separate your scales in a consistent manner, you won't face, you can treat GRs like quantum field theory and not deal with these UV divergences. They are just encoded with some coefficients at any given order. So one way to separate the scales is, of course, to consider the radius between two different compact objects in general relativity. And so you perform an expansion in GR where R is the separation. So it's either like with gravity, but also like long distances. And when you don't consider the velocity of the objects, it's actually called the Postman-Costian expansion because they can be actually relativistic, but you can also get the Postmutane if you consider non-relativistic velocities. Okay, so I'm telling you that there is some relation between quantum field theory amplitudes and observables in GR. Now, how does that happen? Well, the scattering amplitudes, we can compute not by summing them and items, we can actually compute by employing this word here called unitarity. So even if you think about the classical theory, that classical theory actually arises as a classical limit of a quantum theory. And in the quantum theory, the unitarity principle actually helps you compute things. It's not like a further constraint that you have to deal with is actually a tool. And once you take the classical limit, unitarity is still present, but it's not evident from the classical perspective. Now, this was of course enforced by Feynman using the Panintera approach. And we know how unitarity is respected in the Panintera because probabilities are up to one and all possible paths that you can follow actually follow a probability distribution. But in terms of Feynman diagrams, it is a very simple way of realizing unitarity, which is essentially you do these cuts. So these cuts, you just take these diagrams and then just cut them by putting the internal particles on shells. So they become real particles. And once you cut them, what has to happen is that there is some factorization and then the factorization is related to, again, other amplitudes. So these are usually called the cut-cost-q rules or the cutting rules or generalized unitarity and there are many fancy names that people have come up with for this idea. But in essence, it's just making manifest the fact that probabilities are up to one in the quantum theory. And that survives after you take the classical limit. So this provides a new analytical window into perturbation theory in GR because in GR, we don't usually think about probabilities. So this actually is the tool that we're gonna use. Okay, so let me just try to explain how amplitudes these diagrams need to observable. Well, let's just consider the case in which there's some gravitational radiation. Okay, so we will be interested in observables that can be measured very far from the events, right? Very, very far from the sources. And by far, we mean actually the asymptotic region. So we're considering some asymptotic space-time where there's some black holes in the bulk and so on, but very far from it, we have flat space-time. So what we usually do for this situation is that we know that the metric has an expansion in terms of flat metric plus a correction. And this correction actually need not to be linear in Newton. It actually can include all orders if you want more precision. In this talk, I'm expanding in Kappa, which is the square root of G, but that's just because the propagator in the quantum filter language has another power of Kappa. And if you just try to compute the metric, this H menu, right? The sort of deviation from flat space, you can actually compute the very, very, very far away by solving the wave equation, of course, sort of linearize if you want, but actually you can compute any order. And the wave equation is sourced by this sort of localized compact object, which here I call T menu. Now, what T menu actually encodes is all sort of non-local sort of back reaction, interactions of different gravitons. So here you can think of green and red as some two compact objects, maybe black holes, and they interact gravitationally, so they exchange internal gravitons, right? So each of these gravitons is like a power of the Newton potential if you want. And so you have many sort of back reaction effects. And at the end of the day, you just resum all this into this bluff, right? Which is summing all these Feynman diatoms and you just compute the effective source, which is this T menu, right? That these bluff sources, right? So you just plug that into the wave equation and at the end of the day, you can, very, very far from the source, you can use the subtle point approximation and essentially just relate the radiative metric to the source itself, right? This is some textbook argument that H menu actually is thinking on shell. So this includes an effective source, which has matter, of course, is red and green line and non-linear gravitational interaction. So all these sort of non-linearities of Einstein equations are in this bluff, okay? So these are like just more Feynman diatoms. But someone may say that this metric is not really absurd because you can just do some gauge transformation and it shifts, right? Well, today with that, we actually go back to sort of all construction of Penrose, Rindler, Kinersley, and some other people refer to as the null tetrad. So what happens is that we can extract, like from this metric at null infinity, we can actually extract the gauge invariant information by just projecting it in some tetrad. So just take your flat space and expand it in M, N, L, N as in this equation and then these are just null vectors and then you just project in some direction of null vectors your metric. In this example, actually I'm projecting the vial tensor. So I'm not taking the metric, but I'm taking the curvature and that's because of course the curvature is nicer because it's gauge covariant, right? So you take this gauge covariant object and then you project it on some null tetrad. So that's kind of extracting the two graviton modes if you want like the actual plus or minus circular polarizations of your gravitational wave. And in disguise, well, this is the simplest guy, it's called Psi4. It's just to project in the vial tensor into polarization. I'm not gonna explain how the null tetrad works, but it's just something which has LECD2. So it's usually what we call the plus mode of a gravitational wave. And you just eventually realize that by just writing the vial tensor in terms of the metric and then metric in terms of the energy, stress energy tensor, then this Psi4 quantity which is something spin two quantity actually becomes some polarization tensor, right? M mu and mu dotted with T mu nu which is a on-chill source. And this contraction actually is what we refer to scattering amplitude, right? But if you go to a textbook definition of the amplitude in quantum activity, you take a non-chill source and then contract it in some polarization direction and the LSE reduction tells you what that's actually a scattering probability. And so this sort of Psi4 scalar that Penrose and Newman defined many, many years ago actually matches the intuition for being a scattering observable, okay? So that's one of the main points of this part of the talk. Like this sort of gauge invariant quantity is a scattering amplitude in that sense. Now, I told you essentially how to compute it with this Feynman diagram. So that's not actually what we want. We want to find simpler ways. So obviously there are some cancellations when you sum over all these Feynman diagrams, right? There's some perturbative expansion and actually you see that all these terms start like canceling with each other. And that's has been observed for several years now. It happens in QED, it happens in QED, it happens in GR as well. So we want to say that essentially there are simpler ways of computing this amplitude. I'm not gonna sort of go through any of them because there's no time for explaining but let me just mention that, of course, this is the same effect that you would see in QED if you start computing all like two or three loop corrections to define a structure constant. There are certain situations in which for certain algebraic spatial space, the space times which are the most common ones such as, you know, strategy on care black holes. This sort of simplification is very evident already in writing down the metric. Like if you think about it, the fact that we can actually write down a metric for this like rotating black holes to any order in the spin is some sort of resumption, right? Is some sort of cancellation that had to occur. And this happens very explicitly if you start computing it in terms of Feynman diagrams. And it's what's usually called the Kershiel gauge. In technical terms, actually, what we need to do is to incorporate a coherent state. I'm not gonna explain how a coherent state appears, but it's roughly related to the asymptotic nature of these space times being asymptotic flat. Usually when you have some classical, of course, feel arising from a quantum calculation, you know that there is a coherent state. So that's exactly the mechanism here. Okay, instead of trying to prove that there's a coherent state and there is some sort of resumption, I'm just gonna give you a simple example actually after I plug in my computer. Yeah, so in this example, what's happening is that we're taking, for instance, three values, okay? So you have three compact objects in general relativity. And then what you do is you consider the source, okay? So the source associated to this matter distribution, right? Now, the simplest calculation you can possibly do with you when you have three values. So that's already very complicated. It's like the three-body problem. People write books about it. The simplest calculation you can do is actually expand in the frequency of the radiation of these bodies. Okay, so you are considering the lowest possible frequency order in the radiation. You don't really see, because of the wavelength being too long, you don't really see like the internal details of these objects, right? So the result for this lowest order is actually universal. It doesn't depend on the internal like multiples, like the internal interactions that they have, that if they are neutron stars or, you know, they are fermions or bosons, it doesn't really care. What you see at the leading order in the wavelength is this very simple term obtained by Bravinsky and Tern in the seventies or something that, you know, that only depends on the initial and final velocities of these three massive objects. So that's a very, very nice result because it's universal. Now, I guess, in a moment, I'll mention how this is related to amplitudes. Of course, it's what we call the sub theorem if you're familiar with it. But even the, without talking about soft theorems or quantified theory or anything like that, just to compute this source already shows you that there is a nice structure. And even the simplest case, instead of having three bodies, we have just one body, right? Suppose we can only have one particle that sort of emits a graviton. You can immediately see that this source, T mu nu contracted with the polarization vectors, this null tetra, it's a delta function. And it's, it contracts the polarization with this moment of, you know, one body. Now, this delta function is kind of mysterious because usually we don't, you know, radiation is not localized. But the reason this delta function is that if you have one body, there is no radiation, right? So if you have one body, there's no acceleration. Because there is no, like, driving force. So actually there's no radiation. Actually this delta function tells you that the amplitude, the radiation is zero, right? It's kind of a stationary. But still it's in the situation in which, you know, this P and K, so K, sorry, I should have mentioned K is the momentum of the radiation and P is the momentum of the body. There's a situation in which this P and K are complex. And in that case, actually you can have radiation in the complex kinematics. So this is very familiar for people who have explored these amplitudes because they are usually defining complex kinematics. But in the real world, of course, this is zero. Nevertheless, it makes an interesting point. Now, what happens when you have two, okay? So if you have just two bodies, well, what happens is that now there is no delta function, right? There's just a simple sum of two terms and there's a P ampere prime and there's P1, P2 and so on. And what you get by contracting this source with the polarization tensor, again, the seminal tetra, you get this amplitude again. It's a five point amplitude and it actually corresponds to what we call the soft theorem. This is sort of the well-known Wember soft theorem, okay? So this is something that Wember who was, you know, studying quantum field theory and gravitons and photons observed very long ago and obviously it's related to its general relativity result of Braggian's gain term. Now, there's a little bit of a puzzle because the soft theorem technically is a factor, is a factorization of, you know, the radiation part which is what we call the soft factor. And the sort of stationary part or the matter part which is essentially a four point amplitude without the gravity. And you see that here, it might not be as clear but essentially what is not a factorization what we obtain from the result of Braggian's gain term, right? So they just computed this stress and retensor. It's not really a factorization. It's actually just the soft factor, right? So it's just this S zero here. It doesn't really tell you what happens with this other hard part. And so that's a little bit of a puzzle because, you know, in quantum field theory we have the hard part and it's a factor. Essentially, if you actually follow the classical argument like trying to take this quantum amplitude of four massive particles and gravitation and take the classical limit and take H by H bar to zero. What happens is essentially this hard part doesn't really contribute. So you only get the soft factor, okay? That has to do again with the coherent the coherent exponentiation of the gravity. One way, I mean, we need to discuss the classical limit and what does it mean to start with something that's quantum that has Feynman diagrams and then take H bar to zero and get these classical results. It's a little bit of a conceptual or philosophical issue. I mean, what are we computing in Ether case, right? In one case, we're computing a probability, a transition probability. We have the initial states and we have the final states and then we ask the probability of that process to happen. That's what we do in quantum field theory. In GR, what we do is we start from an initial configuration. We don't get our probability. We know that there's something that's gonna happen, right? So we evolved that from an initial value problem, right? So these are different problems. And of course, as we take H bar to zero, these problems happen to be related, right? That's the whole, like, bad internal idea. Now, you can think of, you know, just replacing this sort of in-out problem by an in-in problem in which we start only from the initial value of the quantities. So I'm not gonna exactly describe how that happens because of time, but trust me, you have to come up with some wave function that sort of localizes in some position and then take H bar to zero. And if you do it very carefully, actually you end up realizing, as I said, that the whole hard part of this quantum field theory amplitude doesn't really contribute. It's just a soft factor that you need to include. And that's explained very nicely in a paper. Actually, I should have cited here, but I also cited my own paper, I guess, that's also explained. But the sort of key point is that there is some sort of Fourier transform to map these two problems, right? So we start with some in-out problem and then we map it to an in-in problem. There's a Fourier transform there. And that Fourier transform, it appears in many contexts. Actually, you can just see it in a textbook sample when you try to compute the deviation, like the deflection angle of some scattering process. So you just integrate the momentum, but the momentum is the force and the force is related to the potential and so on. So the potential is actually the amplitude by virtue of the Born approximation. And yeah, so essentially this relation that we learned in quantum field theory of computing the Coulomb potential from an amplitude, right? Like the Born approximation giving the Coulomb potential, that's exactly what's at play here. Right, so this is actually a paper that I was referring to for on four point amplitudes. These are parts of what's explained this nice paper, very beautiful paper by Kosovo, maybe Anna Connell that sort of links these two in-out and in-in approaches, right? I would like for the time being just to provide a small complementary perspective on that problem. And that perspective is gonna be suitable for higher orders in G. And moreover, it will help us sort of translate between the classical and the, sorry, the classical scattering problem and the classical bounded orbit problem. Now, so far, these are kind of unbounded problems because various thinking about massive sources sort of going from minus infinity to plus infinity, whatever, but for the experiments, we are sort of more interested in bounded situations in which you have like two black holes and then they merge and then they form this massive compact object. They don't escape to infinity, right? So let me sort of try to argue how these problems are related. The relation between bounded and unbounded problems. Well, let me first just say that the unbounded situation is the one that scattering amplitudes compute, right? Because you are considering a scattering situation in which you have initial conditions and final conditions, they're all at infinity. So you don't really have these bounded orbits. But there is a relation between bounded states, unbounded states, scattering states to bounded states that we're also learning quantum utility. Essentially, by looking at poles of the amplitudes, we can get information about bounded states. So when we solve like the hydrogen atom, you can compute scattering of an electron of this Coulomb potential, but then eventually in this scattering amplitude, there is this poles in the complex plane that tells you about the bounded situations, like the situation in which the electron is bounded to this potential, right? So it orbits this potential. And this is something that appears also in this context. Okay? So a very, very nice way to understand this problem happened to be obtained by Tuft in 87. It's called a sort of a conal approximation in quantum utility. So there's these follow-up papers by now very classic papers. So let me kind of explain how that approach works. Suppose we consider just a four-point situation in which you have essentially just two massive bodies and they interact in many, many ways. Of course, you just have these Feynman diagrams depicted by these blocks. Now you have to add all possible Feynman diagrams, right? But if you take this a conal limiting, which both bodies are sort of ultralativistic, what happens is that essentially, you have to resum all these small blocks which encode local information and at the end of the day, after you resum them in this limit, they exponentiate, okay? So they exponentiate because you're summing over permutations, right? You're doing quantum field theory and so you're summing over different permutations. And of course, you have to iterate over the internal momentum. But if you actually fully transform this convolution, you get simple exponential, okay? So this is what's called a conal phase. And I think this is the last point I want to make. The a conal phase is essentially the amplitude. It's the exponent of the amplitude. And it's an exponent because we know by unitarity that S dagger S is one. So that means that the amplitude in this particular basis has to be a phase, right? It has to be unitary or unimodular, right? So that's why we have this exponent. So the exponentiation is something that is a consequence of unitarity. So it doesn't really requires you to compute all these super complicated diagrams. Unitary details you immediately that there is an exponent there. And this exponent actually contains physical observable information that's sketching variant. And I'm not gonna explain exactly how but essentially it's the WKV approximation, right? So you know that once you have this phase which is the on-chill action. From the on-chill action you can derive the scattering angle you can derive the time delay and all these observables just from the on-chill action. So this is just Hamilton-Jakowic theory that's a textbook as well. Again, this is also related to the coherent as we're position and I will be happy to discuss that in more detail. So just to make that intuition precise let me just give you a simple example. So you have like one body is scattering of another body with some impact parameter and then there's some deflection angle. So you compute this amplitude and then eventually you find that it's exponentiates in the column phase and to exponentiate you actually have to go, as I said you have to the Fourier transform. So you realize immediately that this sort of phase is nothing but the on-chill action of a classical particle which has time translation invariance or you have energy, it has rotation invariance, it has so it has angular momentum and it has some radial component that's not fixed. But of course it can be computed from the amplitudes. Once you have this form, you effectively don't care anymore whether it's bounded or unbounded because you have the form of the on-chill action and that's of course the same action for unbounded, unbounded orbits. So this thing that you get from the amplitude actually tells you information about the bounded orbits not only the scattering orbits. Okay, so just to wrap up with some small correspondence in the bounded case, of course, we have this a column phase, essentially some scattering angle. In the bounded case, we just have, I mean the same, but it's just some radial action that's bounded by our plus and our minus which are the two points of closest and farthest approach. The scattering angle in the bounded case is just the derivative of the real action with respect to the angular momentum because it's just the conserved quantity associated to a rotational symmetry. In the bounded case, you don't get a scattering angle but you get some sort of peri-astron precession which is the complete analog of the scattering angle. So it's just a mercury perihelium precession is exactly this computation that you can do by taking the on-chill action and then vary with respect to angular momentum. You can vary with respect to the energy. In one case, you get some sort of sharp video time delay because time is the conjugate to energy and then in the bounded case, you get also a period precession. So all these sort of analogs between unbounded and bounded are actually a consequence of Hamiltonian-Jakowic theory but in terms of QFT is also something that we know again from the hydrogen atom, right? So you have the scattering amplitude and then the pulse of this amplitude tells you about bounded orbits. And this is most of what I wanted to talk about just the relation to amplitudes and the observables that we don't compute. And maybe just in the last minute, I'll just very briefly mention one last point that sort of follows from this correspondence and it sort of opens a new connection of black holes in particular to amplitudes. It's something that's currently being explored and I would be happy to discuss with any of you. Maybe it will lead to a different formulation of what black hole mechanics is. So just from the relation to amplitudes, we know that there is a particular classical limit. Usually this classical limit, if you think about all quantum mechanics is the thing that takes particles into waves, sorry, waves into particles, right? So we have this wavefront, we have this icon approximation that tells you that the wavefronts are like essentially just classical trajectories. And you can start thinking about what are the corresponding amplitudes for these waves instead of just massive particles, right? So instead of having like a particular massless geodesic, we have a wavefront of a massless wave and that massless wave is some gravitational wave, of course. And that gravitational wave has an amplitude that we can compute. So the idea is that to compute amplitudes of gravitational waves, we can learn about trajectories, classical trajectories for massless particles. And that's something that we have been playing with and you can just compute amplitudes for massless waves using the Taukowski equation. You might have heard of the Taukowski equation and the Ray-Wheeler equation for these situations because these are perturbations of care or Schwarzschild if you want. You can solve this Taukowski equation, compute perturbations in terms of the spherical harmonics and then eventually relate these spherical harmonics again to the same on-shell action to a converse phase and to the scattering amplitudes. So it's actually a very, very, very nice game to play that I don't have much time to explain but essentially it's all in this, in these papers that we have been working recently. Okay, so that's a wrap-up and the idea, of course, has many important consequences philosophically. We think of these classical black holes in terms of quantum field theory so we can sort of relate them to elementary particles. So I will spare you all the philosophical discussion but I'm happy to talk about it any other time. So I guess for now that that would be a wrap-up. Thank you, Alfredo. Thank you very much for this very nice talk. There's always a delay in the YouTube channel. So let me just see if I see any questions. Okay, so one is saying, thank you Alfredo for the talk, have a good day everyone. Okay, let me start with a question Alfredo. You describe all these analytical tools and then in principle like this translation to one to another. And I was wondering how far in time or in complexity are we like to be able to compute, I don't know, even a simple waveform using these techniques. So I don't know if you can, somehow if you are able to compute a PSY4 that is related to the second derivatives of something that I can measure in an astrophysical source in some sense. So through this scattering amplitude approach, will we able to get soon a way to compute a waveform? Just I mean, just because you were able to compute PSY4 for I don't know a merger or something similar. Right, right. Yeah, there are many, many sort of fronts of calculations happening at this very moment. There are people losing a sleep working on these problems while we talk. So I'm not gonna say much. I mean, there are so many, so many results. I would have liked to summarize it all but it's actually very, very complicated. If we look at the waveform and just to very briefly touch that point, the waveform has different stages and different stages have been addressed with different tools. And of course there is a stage which is mostly analytic computation of post-newton expansion and so on. All these sort of analytic computations can be immediately like re-sum or like re-derive if you want from the amplitudes. So it's not something that you really have to do much work. It's just sitting and using unitarity you eventually just reproduce all the post-newton expansion and way more than that. Essentially you get all the orders in the velocity. And that's something that's right there in the spiral phase. It's not just completed the flux and the binding energy and essentially just get the waveform. So it's not something purely theoretical. It's actually happening. And if I may, let me ask my question then in a different way. Yes, I understand like these earliest stages you can do in post-newtonian and people have been done that then you do some stitching to numerical relativity to compute like waveforms and then okay the perturbation or the post-normal modes who goes to the thing you were mentioning at the end of your talk. But then so the question is like in this other enterprise because in the way I see it and maybe I'm saying it differently it's another way to compute this type of stuff, right? Like people have been at least in the gravitational wave community have been computing this with post-newtonian in a post-newtonian way and in the way it's sort of described. Of course we can do this fully numerically but it's too expensive. So my question is more or less will this effect approach will be, I don't know easier, faster or just complementary just for the translation? In some sense, I don't know if I now want to compute something that is asymmetrical in mass that's going to be very challenging for numerical relativity. It's in principle doable but then it will last too many cycles and it will be very hard. So my question is even we can do this translation I'm asking more as an astrophysics person. No, of course, yeah. How can I get this? Are the advantages will be faster just complementary? Are there regimes where post-newtonian will be very hard to compute and here is easier or something like that? That was more the flavor of my question. Yeah, that's a very broad question but honestly, I mean there is a sort of bird's eye perspective to this question, which is essentially you see this unitarity approach, right? So after you go to the details you can solve these Feynman diagrams to precisions as are far exceeding the experiments that we currently have. Of course, with Lisa, you would need more but if you compute Feynman diagrams you are just sort of realizing unitarity in a way that is guaranteed to be realized. So at the end of the day, after you sum the whole set of Feynman diagrams you know that unitarity has to be there and that's something that you can start with that's something that you can sort of bootstrap from that's something that you cannot see in the classical theory by doing like disposing to an expansion and trying to match coefficients and trying to get sensitive answers, right? So what I mean is essentially there is something that the quantum theory has that allows you to sort of not only speed up the computations but actually have some principle as to which from which you can construct. Now unitarity is a sort of multifaceted tool because of course it applies to the poles of amplitude so it tells you about where the question on my mouse should be essentially using bootstrap approaches so you can see where the poles of the amplitudes have to be you can see the thresholds of the amplitudes. Yeah, I mean there are many sort of avatars of unitarity. I'm just saying that as a guiding tool as a guiding tool is something that was not considered before this connection to quantum field theory emerge. Awesome, thank you, thank you. I mean as someone here in the soon call have questions on the coordinator. Sorry, let me before maybe yes, let me briefly mention the point of the EHD image. It's also, it's also some, you know just computing just image in a projector screen you put very easily, you know just trace your ethics and do classical computations but if you think about the scattering process again there is a notion of unitarity and people have analyzed where are the sort of photon rings where are the poles of, you know this question on my mouse and on all of that from the perspective of the S matrix and it's something that, you know I feel in that case is kind of under explore. Thank you, thank you. Let me see, sometimes we have a meme Alfredo so sometimes I don't see if the questions are real or not but I'm gonna read it anyways. So, how does the novel interpretation of BOR correspondence principle suggests that classical black hole collisions governed by GR? No, I think it's, I don't think we have to address this one. That's a very nice question. I think that's part of the original abstract but you know, I didn't really go into like BORs correspondence but it's sort of there. I mean, it's just, you know setting H bar to zero doesn't really mean going to the big events. It actually means that effectively you're going to very far separation, right? So you're like essentially taking this very large objects which you may see in the classical but once they are separated far enough so you are considered an expansion in H over R, right? Separating them far enough they look like this point particles, right? So these are like, you know for quantum mechanics this would be like, you know just two electrons which are spinning and that's all you need to know. You need to know about the multiple moments of these particles but you don't really know about their microscopic interactions. Alfredo, and if I may like this way to take like H bar to zero, is that procedure sort of unique or that you can get different answers or like how is it typically performed just out of curiosity? Right, so if you write down an amplitude, you know we usually learn that the H bar powers count the number of loops. So you may say, okay for taking H bar to zero the classical term is only the first, you know like what's called the three level, right? There is no loops. And that's actually not true because after you consider waves and it has to do with this coherent exponentiation, right? After you consider this gravitational waves or this radiation the power of the momentum of the wave actually is because it's a wavelength has to be multiplied by H bar. And so there is a contribution which is leading in H bar even at the loop level once you consider this momentum is a wave number. And that was observed in 2000 something by John Donoghue in this paper that's called like classical physics from quantum looks. It's a very nice paper. That sort of challenges the textbook derivation. So yeah, I mean it's unique in the sense that you have to tell me what are the states, right? What is the scattering data? If you have like massive particles and waves then there's some extra H bar that it doesn't appear if you only have just particles. Awesome. Thank you. Let me see. I don't see more questions here. I need you to try. I have a question for Alfredo. Yeah. Thank you. Yeah. I was wondering because with all the connection with GR itself and I mean I was wondering how this expansion can change the interplay that we kind of tried to make with between gravity and particles, I mean the standard model of physics in the sense that when you have measured you have output in electromagnetic waves and neutrinos and so on and so forth. In this approach, could you have other type of signatures that may differ from the basic connection that we used to put to get this observable signature beyond the gravitational wave itself? Right, yeah. That's an interesting question. I don't really have an answer. I feel that's, for one, that's under-explored. At the recent standard explorers because we usually think about effective field theory separations. So say there's no neutrino production as long as it doesn't affect the multiple expansion it's not really something that's gonna enter into the computation. At far distances, of course. I mean if you have some neutron star you need to know the equation of state, right? But the question of state essentially informs some particular multiples in this large distance expansion and that's essentially what you measure, right? You don't really go into details of the TOB equation or whatever. You just care about the multiples which are this wisdom coefficients that appear when you have like fairly separated bodies. And that's sort of the question sort of gravitational wave answer to the question. There's also some sort of EHT answer to that question which is, you care about this accretion disk around the black hole and the accretion disk has all these magnetic effects. It has this very, very hot plasma or whatever. And at the end of the day, what the sort of quantum filter approach tells you is like what happens with the geodesics when they are far from this accretion disk, right? So not really what's happening inside whether there is like, you know, production of particles or not, but very, very far away tracing the geodesics using quantum filterity. It's something more universal than that. So this whole has to do with the separation of scales. I mean, it's something that lies at the very core of general relativity, right? The fact that we can talk about general relativity is because we can sort of separate the intrinsic components of the matter from the gravitational effects. Awesome. Thank you. Thank you. I don't see more questions and we are past the hour. Alfredo, thank you very much for this lovely webinar. I guess people can find your email online if they have more questions. And then as usual, Alfredo, the viewers will see your talk at a later stage. Awesome, awesome. Thank you so much. Thank you very much everyone for watching and then we see you in the next low physics webinar. Thank you. Bye. Good, we are, Diana, estamos live. Muchas gracias, Alfredo.