 So, for the first lecture of the second day, we are happy to have Laura Doné, and she will continue to talk about celestial amplitudes. Thank you very much. Good morning, everybody. Thanks for coming. Recording in progress. We will keep exploring this topic of the infrastructure of gravity in flat space times. Let me start by summarizing the main point that we saw yesterday in the first lecture. So, yesterday I tried to present the rich fact that the symmetries of asymptotically flat space time are way bigger than the ones of Minkowski, asymptotically flat space times. And we saw a precise ansatz for the metric which is asymptotically flat as one approaches the boundary, the null boundary, a future null infinity. So, the symmetry of exactly flat space time are given by the Poincare group, which consists of four translations and six Lorentz transformations. Now, the symmetry group of asymptotically flat space time is an infinite dimensional enhancement of Poincare, given by an infinite amount of what people have been calling super translations, which are spanned by this arbitrary function of the angles t, which comes in these new components of the infinitesimal vector field. And similarly, there is an sense in which we can enhance the Lorentz part of the Poincare group to an infinite amount of super rotations spanned by conformal killing vectors. So, we have two copies of them in this complex coordinates, y, which depends only on z and y-bar, which depends only on z-bar. So, these are conformal local conformal killing vector on the sphere. Okay. So, this is what is known as the extended BMS group, extended because in the first place, Bondi, Messner and Sachs only found these super translations. But now, people thought it was a good idea to extend the Lorentz part to also an infinite amount of symmetries because in conformal theory, we are very used to having this VeroZero type of symmetries. So, this is what we saw yesterday. And today, I want to tell you basically on the importance of these symmetries for scattering problem in flat spacetime by showing you that these symmetries provide an infinite amount of conservation loss for the S-matrix. And so, this is the relation we will start looking at the BMS and the S-matrix. So, this is a result due to Strominger, which is now already a bit less than 10 years ago. So, the result can be stated as follows. BMS symmetries imply an infinite amount of conservation loss, which constrain strongly scattering amplitudes of massless particles in flat spacetime. So, the original references for these papers. And to show that, of course, I will not have time to go into the details of all the steps required for establishing such a correspondence, which is a neat mathematical statement. But I want to tell you what are the two main ingredients you need in order to achieve this. The first ingredient is given by BMS charges. So far, I've been telling you about symmetries, but as you know from Nutter's theorem, when you have a symmetry, there is a conserved quantity associated to it. And the BMS charges are nothing but the Nutter charges associated to these asymptotic symmetries. To the BMS, of course, asymptotic symmetries. So, I mean, actually, this topic of building charges associated to asymptotic symmetries is a whole topic on its own in GR. So, it could be a lecture just about that. There are many techniques involved, which trace back to the work of Wald and Zuppas and other people in the 90s. It's known as the covariant phase-space formalism. If you are interested in knowing how building these charges, I would refer you to this PhD thesis, which I think contains the most accurate and state-of-the-art prescription for BMS charges. But let me just give you one result, what this charge is for super translation. So, I will denote by qt, the charge associated to super translation symmetry. And this is just taking this simple form. So, it's basically a pairing between this function of super translations and the mass. You remember this M was the bandimath aspect that enters into this one over our expansion around flat space. And this integral is over the sphere located at the past of the future of null infinity. So, I will have to define this because I don't think I did it last time. You remember we had this null boundary, future null infinity, square plus, which was parametrized by retarded time u and coordinates. Now, if I take u goes to minus infinity, here at the past of future null infinity. Similarly, we had, I mean, I didn't present the things in advanced coordinates, but as Francesco was asking, there is all these stories also valid for incoming particles. And in this case, everything is labeled in terms of this advanced time v, which is t plus r. And taking v goes to plus infinity, you land on this location which people denote by square minus plus. That is the future of the past boundary. v goes to plus infinity. Okay. So, yesterday I've been writing everything in terms of just one components of this boundary in terms of square plus. And so strictly speaking, I have defined for you one copy of the BMS group living on the future. So, I will denote by, with a plus, yes, the coming, the charge living on, meaning on future null infinity. Hi. One question. What is the qualitative difference between the past of the future null infinity or future of the past null infinity and the special infinity i0? Yes. So, we'll come to that. But so, i0 is here. And on this diagram, it looks like they are the same thing. But actually, it's just an artifact of this Penrose diagram. Actually, these locations are definitely far away. So, and actually, more precisely, special infinity is not included into the conformal compactified spacetime. So, this is actually a very important point. And I will come back to it right now about how do you basically, what happens when you cross is special infinity. But yes, special infinity is not neither square plus minus neither square minus. Not at the limit point of this. Yeah, it's really speaking. I could not even draw it here in this diagram. Thanks a lot. Thank you. So, I have told you about BMS here on the future. But there is another copy of BMS symmetries on the past. And this charge will denote the BMS charge living on the future. But there is also, there exists also a charge living now on square minus plus. So, that's the first ingredient is actually to be able to construct these charges. And now you probably see where I'm getting at. What I'm getting at is that in order to define the scattering problem in flat spacetime, we need to know how the quantities that are defined at the past are related to the one living in the future. So, in order to make this statement true and actually to uncover that BMS symmetries provide you a symmetry of this matrix, a fundamental ingredient. And this is, I think the lack of observation of this statement was because nobody really realized what we needed to do to match these two disconnected pieces. And Storminger proposed that there is a precise way to map the quantity of the spacetime from the past to the future through so-called antipodal matching at I0. It's just called antipodal matching. So, this antipodal matching will give us a junction condition or gravitational scattering problem. Again, I'm focusing on massless scattering, which is needed to relate the incoming states to the outgoing states. So, without such a matching, you cannot talk about scattering, basically. So, what is this antipodal condition? Well, antipodal in just in the sense that we will relate quantities on the celestial sphere as cry minus plus with the one at the antipodal point of the sphere as cry plus minus. So, to do that, you can consider, you remember I used this Z, Z bar coordinates. So, a convenient choice for these coordinates is to say that a point Z and Z bar, the coordinates Z, Z bar will represent a point theta phi on cry plus, or the antipodal point phi minus theta phi plus pi on cry minus. So, this is just a way to choose my coordinates here in such a way that when I will give you a Z, Z bar, it will either represent a point on cry plus or the antipodal related point on the past. This is just a choice for these coordinates that will make this antipodal matching way easier to write down, because now, basically, this coordinate system implements naturally this antipodal relationship. So, what it means is that the condition that we will impose is that the bandy mass aspect, so it also depends on you, evaluated at cry plus minus will be equated to, well, okay, u goes to minus infinity, so I will not write it down, because it's not a function of u anymore, strictly speaking, on this location, will be equated to the sphere to the corner at cry minus plus. So, that's the antipodal matching condition we will impose, and it's really crucial, because not only without that you cannot make this statement about this metric, you cannot make any sense out of that. And second, we will see that this condition is actually the one needed, and very natural to a certain amount of extent. For instance, you can see that this CPT and Lorentz invariant, and they have been a long literature and general relativity about this sort of matching condition, so it's a very difficult problem to see in general how the field will behave when you take u goes to minus infinity. It's a delicate problem in GR and how they evolve from the past to the future. So, yes, yes. So, in general, m is different before the limit, so you have m u zz bar on a square plus, and some m tilde v zz bar, and then on the limit you do this, or the function as a function of u and v is supposed to be somehow related. So, why you use the same symbol, so to say? Yes, very good. So, indeed, in principle, the m here is this 1 over r term in the GVV components of an expansion of BMS type at cry minus. So, in principle, these are two unrelated quantities, but now, indeed, I'm making the claim that if you want, yes, this, in principle, would be a different function, but I'm identifying this function at these corners. So, is there a physical intuition between why you want this condition? Like, I understand maybe the math gets better, but... Yes, I mean, you can see that if you have, if you are a non-interactive particle and you enter the space time from the past infinity, you will cross the sphere at scribe plus at the antipodal relation. That's somehow natural in this sense. But it's a really non-trivial statement about the structure of the space time, which is... So, I want to emphasize that this thing has not been fully proven, it's proven at posteriori when we will see that it's equivalent to the Weinberg's theorem, so it gives a proof at posteriori that this matching should hold. Sorry, maybe you also mentioned that this is a... It seems that it is an artifact of the Benro's diagram that these two places are the same, but they are not the same. No, they are not. Okay, so for other quantities, this matching is not true. Other quantities, what you have in mind, like other... The other terms in the expansion, for example. Yes, yes. So, I will actually, I will also impose this matching on the other function. Here I'm doing everything for super-translation, so this will be enough. But it also matches... Remember this angular momentum aspect we saw yesterday? This will also be matched. And also the shear. This gravitational shear will also be required to obey a similar antipodal condition. Okay, but so if you match all the quantities there, it is not like if you are imposing that these two places are the same. You can analytically continue all the function in i0. So it seems like you are identifying these two. So these quantities are not strictly speaking defined at i0. There was an expansion over around the null hyper surface. And if you want to give me a data on, like, what people do on some kind of time, like because she slides, you will have to do a total... And you want to describe an asymptotic expansion around special infinity. You will have to resort to a totally different gauge and stuff. So these are a lot of work by Friedrich and other people where you have to resolve a space like infinity by either you can put a system of coordinate and foliate it by hyperbola, or you can use another gauge where you foliate it by... You can describe i0 as a cylinder. There are many ways to do that, but there are really different expansion around i0. And since I would be interested in massless scattering anyway, no massless particle goes there, so we will not need to do that for the purpose of this lecture. But it's an interesting story, and people have been looking at BMS symmetries around i0. Okay. So now that we have this matching on the phase space, so I'm just... I will not enter into much details, but I'm actually restricting myself to a certain class of spacetimes, which are the so-called crystal-dooloo-cliner-mine kind of spacetimes, which obey certain conditions. I don't want to go into too much technicalities about that, but if you're interested about the assumptions behind this, just you can ask me at the break or during the discussion. So this antipodal condition will break the combined BMS plus... So in principle, we have two copies of the symmetry, one leaving the future, one at the past, the BMS plus and BMS minus action, which act in principle independently on the future and the past. It will break it down to a diagonal subgroup that preserves this antipodal condition. So basically what he will amount to do is he will amount now to identify the super translation parameter of the past of the future with the one of the future of the past. So again, in principle, as Marco was making a point, this t function is in principle a very different function, but now if I want to preserve this matching, it will induce this choice, which basically fixes one frame in terms of one other. So we have this symmetry matching now, as we are going through around I0. And now we can put the two ingredients together to show immediately that what this matching will amount to do is it will amount to equate the BMS charge at the future and at the past, which is the statement that the energy is conserved, this we know, but now the super translation enhancement is telling us that the energy is conserved at every angle. And the quantum version of this equality is a wide identity. So the word identity associated to super translation symmetry is the statement. So if it's a symmetry, it should commute with the S matrix. So this will be the form of the word identity that is implied by a super translation symmetry. And I will tell you briefly how you can see that this word identity is actually nothing but a very well-known theorem in quantum field theory, which is Weinberg's graviton, where a self-graviton is a graviton whose energy is taken to zero. So I will present the main things we need in order to show this sort of statement by getting at a more defined, more affinated definition of massless scattering in flat space where I will introduce the things that we are more used to now in quantum field theory with free fields, creation, annihilation operators and most of the work to show these identities actually translate these two languages, the language of general relativity and asymptomatic symmetry in charges with the one of the quantum field theory that we are more used to. Is there any question on this conservation? The reason that we don't need this condition for simple flat space scattering, is this that M is zero or is there something else? Because I think there's a discussion in Weinberg's book about matching the generators of Lorentz group for the in-state and out-states. Is this related to this? So the first part of your question, no, don't keep the microphone because I'm not sure I understood the question. You asked about what is zero, if this thing doesn't hold, what? The quantity M, the bond mass. Yes, so what's the question, is whether this holds, what happens? This condition is trivial for simple flat space scattering. I mean, why we don't bother to talk about this? Oh, yeah, yeah, yeah, no, because we are interested into this asymptomatic flat space, and how this bond mass, I mean, Weinberg doesn't care, I guess, in the bond mass aspect and the angular momentum. This is just a different point of view. It's just really two communities talking different languages and working in a different setup that never came into the realization that they were actually describing the same physics in different ways. So let's go to massless scattering in flat space. If there is another question, just interrupt me. So I will sometimes, okay, I will write down very fast. Now I will be using, I mean, usually we are using not bondy coordinates, but Cartesian coordinates, and I just want to write for you how these two things are related to each other. Just a simple change of coordinates, but again, most of the work has to do with translating things into one other. So this is just a change of coordinates from Cartesian to this bondy coordinate I used yesterday. And this maps to the line element we had yesterday with gamma z bar, the round-sphere metric. So a massless particle of energy omega will cross the celestial sphere, the S, at the point. So now I will use a different notation. It's not a different notation. I will use different letters to distinguish the coordinate z and z bar, and the one that coordinates the moment of the particle, but this w, w bar, not to be confused omega with the w. So I hope this will not lead to confusion. But you can parametrize the foreign null moment of the particle, the mu, the on-shell massless particle, like so, where omega times q mu, where q mu is a null vector, which can be parametrized like so in terms of this angle w and w bar. So this parametrization is not unique, of course, but this is very convenient when we want to match things with Bondi because you can see, when you see how you go from Cartesian to Bondi coordinates, this parametrization looks not so crazy. And the particles, so I will be talking about spin 1 and also spin 2 particles, mostly about gravitons, but they can have a polarization vector, epsilon mu, with a plus denoting positive helicity, which in terms of this null vector parametrization for q can be taken to take this form. And similarly, for negative helicity, epsilon minus can write it like so. So you can check that these guys satisfy these relations. So they are suitable polarization vectors, polarized like this. Okay, very good. So this is just mostly introducing notation. So what is important is that we have these particles, they have an angel momenta, and I can write, instead of writing p mu, I can write everything in terms of three numbers, omega d energy and a point w and w bar, at which the particle will cross the celestial sphere. So this is the most very natural thing to do. And now we will want to talk about, let's take mass scattering of gravitational field. So at late times, the gravitational, the graviton will become free and can be approximated by the mode expansion that I guess you're all familiar with. So I'm considering an outgoing graviton, which at very, very late time is free. And this is the usual textbook formula that you can find. But I have to write down just to introduce my conventions and notations, where you have the usual creation and annihilation operators, exponential of each i to the p dot x, where x are the Cartesian coordinates. And you have a sum over the two helicities, plus and minus, labeled by this alpha, out dagger e to the minus i p dot. And now we have a polarization tensor. It can be plus or minus helicity, which is we're choosing a gauge so that it can be written as epsilon mu times epsilon nu. And a out and a out dagger satisfy the usual commutation relations. Out dagger is delta alpha eb times 2 omega. Omega is p0, 2 pi cubed delta 3 of p minus p prime. Okay? So I'm guessing you have seen this expression before. But now the main thing we have to do is we have to express this expression, well, express this tautological. You have to write down this expression in boundary coordinates, u r z z bar. Take a large radius expansion. If we want to match it with the BMS asymptotic expansion that I wrote yesterday, and then what you can show is that it will take the following form. So remember that what I introduced yesterday, this shear function, so that this encodes the two polarization modes of the graviton, was the piece in the z z bar components of the metric. So to extract it, I need to take the perturbation h divide by r, and take the limit as r goes to infinity of this quantity. So this is just a definition. Now it's an operator, you see. I have operator representation for this object in terms of creation and annihilation of gravitons fields, gravitons. So this is a definition if you want, but it will be identified with the shear in the BMS expansion. And now you need to do this business. Write down h in boundary and take the large r limit. And there is a little computation to do that I will not review here. I'm just writing the result minus a minus out. This depends on omega z bar. You have a dagger and you have exponential e omega u. So to obtain this, there is, you see you need to write this in boundary coordinates and take the large r, a rapidly oscillating phase in this exponential. You need to take a stationary phase-phase approximation that will localize this exponential. And it's done in detail in the exercise in the book of Andy Strominger that I've written the reference yesterday, this Strominger lecture notes. You can see the detail in exercises. But the important thing I want to emphasize on is that doing this larger expansion will localize the point, the direction in momentum on the celestial sphere. So you see before I had, I made a distinction between the z bar coordinates of the point in spacetime and the point w w bar with the particle is pointing towards the celestial sphere. Now I'm doing this large r expansion. These two angles will be identified. So I have only now something that will depend. So omega and omega bar have been identified to z and z bar. So this is this little computation that does this. It's just saying that we are identifying the point on the celestial sphere with the particle courses. Scribe plus and scribe well here. I'm talking about outgoing states of scribe plus. So I can pause here. If there is some question on the notations or on what we are doing. Sorry Laura, a very elementary confusion. So typically when we have particle states we take say a graviton with a definite momentum. So here we are integrating over this energy. So what is the relation with the usual particle states? Can you repeat again? So here I'm integrating over, yes if you want this is the expressing the state in a direct or position space. Which is related to the momentum space by a Fourier transform. Because you see we are, so basically this is basically a Fourier transform which inverts the energy with the time because I need to map precisely I need to express this in this basis to map it with the boundary story. But the state is going to be say against state of boost or what? The associated states will be an against state of boost or? So the associated states, so usually they would be, if I were in the usual momentum space there would be energy against states. Now if you want there are the Fourier transform of that. So they will, the transformation will be a bit, the thing that they will diagonalize will be a little bit different. So then eventually I will go to another basis, I will not stay very long in this, you basis if you want. I will go into the celestial basis where I think will transform nicely under boost and Lorentz transformations. So yeah it might be a bit not familiar to write the graviton mode like so, but precisely we, most of the things to establish the connection between these two topics is precisely to sometimes go to some basis which are not so conventional from one point or the other. And at the end I will go to a different, a totally different basis. Hope it clarifies the things. So this is for graviton and you have an analogous expression for a spin one particle. Let me, yes I mean it's very similar but in case you're, you prefer that. So well okay I will, I think I'm running out of time so I will not, I will not write that basically you have a similar expression for gauge field a mu. So the z components of that and I'm selecting the leading piece in the large r expansion which now normal falloff for gauge field is to start with r to the power zero and in them except that here you will have epsilon z instead of zz. Yes maybe I should say what this, this hat here is one of r squared times this epsilon I have defined here. And here the, well okay. So what is this expression means? Again to reiterate. So let me call this expression one. So here you will really have the same expression. You can find it into and this book epsilon hat is epsilon z divided by r. The powers are a bit different than from gravity but okay so let me write some words. So these are, this will define for a boundary operator leaving on scry. So these two operators one and two. What they do is they annihilate a positive electricity graviton or photon graviton for expression one photon for expression two and they create a negative outgoing graviton or photon at a retarded time u and at the point zz bar on the celestial sphere. Okay so here you have c zz so we describe one electricity but you have a similar expression for c z bar z bar with some details change about the polarization and so on and so forth. So does this make sense? I hope. So this will have a sort of a boundary operator that lives in this null hypersurface and now what I will tell you is that the zero mode of the field strength associated to these boundary operators once inserted into the s matrix actually leads to this universal formula known as Weinberg's theorem. I will not have time to review in details some theorems I will just sketch very schematic way of what these are. So this trace back to the 60s also so it's funny to notice that it was at the same time that Bondy, Mester and Zack was studying the infrared structure of scattering elements and other people like low Burnett, Gelman and others. So a soft theorem is the statement about s matrix which involves a soft particle. So let me just sketch what these are and you can go into the Weinberg's literature if you want to know more on that. So it's basically saying that you have a scattering process involving a certain amount of particles which all carry a momenta and you add to this process a soft graviton or soft photon but let me focus on gravity which with momenta will be written like this I'm calling this K mu. Now in the limit where the energy of this soft guy is taking to zero this amplitude will factorize and will be given the same amplitude but now without the soft one times a number and this number is called a soft factor. So probably speaking you have an amplitude with n particles so these particles which are not soft or sometimes called hard this amplitude will factorize as omega goes to zero like so where s0 is the soft factor let me write down what this factor is for gravity s0 so it's just just a number which only involves the momenta of the hard particle and a pole Weinberg's pole so importantly this pole goes like omega to the minus one and this eta here is just plus or minus just plus one if the particle is outgoing and minus one incoming so this theorem is called universal because the form of this amplitude factorization doesn't depend on the nature of the other hard particle remarkably there is a sort of universality into this formula it holds for gravity but also for scaring of photons and also of gluons there is a soft photon theorem a soft gluon theorem the form of the factors are of course different but very similar and now I can in the last five minutes I can basically tell you the main conclusion that I wanted to get at is that the BMS symmetries this white identity that I've written before once we do this dictionary between Bondi and Weinberg they are exactly the same the same statement so I will just this will serve as a sort of a review and it will amount it will help me to extend what I've been presenting to other cases just a summary table we have this white identity if you remember this was the statement that the S matrix commutes with the super translation charge what am I trying to say we have a graviton we take this the z bar component we read off the shear of that and then yesterday we saw that the asymptotic symmetry of a flat space includes super translations psi t psi du there are super rotations psi y yz dz plus yz bar dz bar and the thing I'm claiming and that Strominger and others proved in a rigorous way is that this white identity associated to symmetries gives us Weinberg's theorem so the leading is the one I have written here which contains a pole in the energy I will come back to the sub-leading story in a moment but before that let me just say that if you have a photon you make a similar expansion near a scry you have now the gauge version analog of these symmetries which what people call a large gauge symmetry which basically changes the field like so where the epsilon is an arbitrary function of the angle and the white identity associated to that is now not surprisingly the leading soft photon which also goes like omega to the minus one and how you do that well is actually the realization that this is implementing implemented by inserting in this matrix some currents for super translation there is this so-called super translation current which gives the following I'm just rewriting here the word identity so if you insert this current into this matrix it will take the following form there is an allegory story here let me just write everything and then I will explain a little bit more I'm just trying to collect a lot of statements and then we will see later that in the celestial holography program these currents will be implemented in a very natural way let me just mention this sub leading story what I remarkably what people found is that they knew that there was this super rotation symmetry and then they thought ok it seems that there is a identity between where there is a relation between symmetries and sub theorems so why don't we look for the extra term into this expansion which is an expansion in omega where now this sub leading sub factor doesn't go like one over omega but now like omega to the zero and Kachasso and Storminger found that indeed there is a theorem a sub theorem associated to super rotations which now is sub leading compared to the one with the Weinberg pole but that exists and remarkably these statements can be implemented through an insertion of an object which looks like a stress tensor the realization that there was so basically that this white identity was equivalent to inserting a current on the celestial sphere which has the exactly the right dimension to be a stress tensor in a conformal field theory was what I think really kicked kicked off this celestial holography program when people realized that there was something transforming like a stress tensor and they thought ok maybe we can dig better into the conformal field theory structure of flat space and we will find some constraints on the holographic nature of flat space times by identifying more of these currents so I know I was quite fast on that so I will stop here and take any question you might have thanks