Stefano de Angelis - The hidden simplicity of the classical limit

Okay, good morning everybody. So, uh let me start by thanking the organizers to inviting me. It's very nice to walk down the hill and uh meet friends and new friends. I'm actually sorry that Rub is not here because I really want to be one of those that uh get reselected too [laughter] and I really wanted to impress him but uh uh but uh yeah I hope he will come he'll be impressed later. Good. Uh so today I'm going to talk about um well some project that uh as I start when I started my post in Sak uh with David he gave uh to me and Fabian and still didn't come out after a few years but uh at some point I didn't feel like putting some uh preliminary archive number at some point this will be out [snorts] and uh it's basically trying to understand resumation for observables from uh you know resumation of perturbation theory of various observables which are relevant for the gravitational to body system and then by the end if I manage to get there I will probably mention uh some other work that I've been doing recently on my own. [snorts] So let me start with a outline of the talk. Uh okay maybe I should do this. So we'll start with motivations for our interesting rotational waves. This will take onethird of the talk. Uh I was you know is a string theory workshop. I didn't give for granted that people are familiar with what we do and why we do it and since I'm the first speaker uh I felt uh I should do this part and uh and then I will talk about the gravitational dynamics through the eyes of an effective field theory. So the scales involves the separation of scales, the different regimes and I will briefly sketch on the perturbative computations that are usually performed uh in this the usual perturbative computations that uh the community the scattering amplitudes or gravitational wave community perform. uh and at some point I will get to the part where I explain some hidden simplicity of the classical limit that um some curious structure that comes from taking the classical limit of observables in quantum field theories and at the end I will mention this work which is bas is based on uh uh the gravit the kish integral to compute um waveforms and interactions. So the let me start with the very broad motivation. So in November 2005, the LVK has completed this fourth observ observing run and the fourth observing run has given us hundreds of new signals that uh we are ready to be studied. And among these 100 signals, some of them are really they really stand out in the sense that they are very loud compared to the average signal. And this feature and obviously the energy involved uh made it possible for for example to observe for the first time the ring down phase uh of the of of the of the the ring down phase of the merger of two black holes. So let me give just very you know the uh bachelor uh considerations Newtonian considerations that uh suggest like give an estimate for the amplitude of these gravitational wave signals. Uh here you have the frequency of the gravitational waves, the masses involved, the total mass, the reduced mass and uh so far the signal that we observe are very very uh small and uh you know since the observatories LVK and LISA are very different in size uh one is a few kilometers the other will be uh thousands of kilometers. They they will observe frequent very different frequencies and since they will observe very different frequencies uh they will also capture very different events uh given these masses. So yeah that that's the take-home message like uh the LVK observation that we have currently today are very different uh from the one we will have in Lisa and for Lisa one is very essential is essential that uh we uh provide very precise and very precise um predictions. Why is that? Well, you know 10 to the minus 21 at peak is extremely small which makes the signal buried by the noise. And uh to observe some uh some strain, you need some u theoretical waveform to match filter the the the signal and basically look for some statistical uh um overlap some significant significant overlap. Okay. So basically this is saying that we need the template uh a template bank for waveform signals. Uh and these uh waveforms depends continuously on various parameters. the masses, the spin, the spins of the black hole or the neutron stars, the their finite size, the sky location and uh you know we have no priority knowledge of this uh of this uh data. So to not miss any signal you must fill the parameter space densely densely enough that uh any you know physically significant um uh you know that you you don't miss any signal. So you have to basically feel fill the space of the parameter space uh these parameters here. Now this is very sketchy. I'm sorry to bo uh but uh uh you know the framework so far to get uh these waveforms is that we have some uh analytic computations where we have analytic control of the systems and uh we make an for what for the Hamiltonian of the system the flaxes and then we match these results where they both hold. So where the two bodies are very far or where the masses are the two masses are uh very different like the ratio of the masses is very small or large. Um and once you have done this matching this fixes basically an amonian that uh you can trust beyond you trust beyond the regime of validity of perturbation theory. Obviously this uh you know the the the this the waveform you get from this resounded waveform uh the this resounded strategy um like in the long run towards merger start defasing with respect to the real signal and uh so through some resumation some resumation and uh some calibration You match it with the numerical results to you you tune this difference like you make this uh difference disappear in uh when you match it with the numerical results and then basically you glue it with black hole perturbation theory to have a signal that uh goes all the way from where when the bodies are very far from each other to the single body in in the excited state that decays into a stable state. Now okay so what are the scales of this problem obviously uh there is so in this talk C equal one H bar is equal one and I'm only talking about length scales so there is the gravitational the scale associated to the gravitational charge which is basically the there is vi there is a typical separation of the two bodies which for example in a scattering setup would be the distance of minimal approach of the two bodies. This B. The larger this scale is, the weaker the gravitational interactions. And then there is the characteristic length in linear dimensions of these two bodies because they don't have to be black holes. They could be neutron stars. So these are an additional scale. I think we we will hear about finite size uh later um in Anna's talk and uh well the this is usually larger than the zwar shield ready and is typically for neutron stars is 10 times order 10 uh this length. So there is a separation of scale between this and then another typical scale is the wavelength of the emitted radiation. This is usually correlated to other uh kinematic parameter like the other kinematic parameters. Uh and this also plays uh quite an important role and the least interesting uh uh scale is the distance uh of the observer from the system which is usually the largest scale in this problem. Okay. So there are various setups like various scenarios where we can apply perturbation theory. As I was saying, if uh GM over B is very small, you have a weak field expansion. And uh if you combine this with the assumption that the two bodies move uh slowly compared to the compared to light then this is basically the the PN uh the PN expansion. And if you are looking at uh bound orbits basically the viral theorem tells you that these two scales uh sorry these two parameters dimensionless parameters are correlated and once you are in a postnewtonian setup is very the convenient the the description of this system is very convenient you know you have the single ball like this is distance this is energy so the shorter distance is this single body which is a almost spherical matter distribution with its dynamical moment because it doesn't doesn't have to be well if it's not a black hole but a neutron star this could be um some very nerial distribution of matter as you zoom out there is the dynamic of the dynamics of the system so you can model these two particles to be point particles that uh interact dynamically And uh and okay and the dynamics okay sorry yes and the dynamics of these two bodies is well is almost uncorrelated uh uh with the dynamics of the radiation outside the body which is now a multipolar source and um yeah because the w the typical wavelength is the typical distance between the two bodies over the typical velocity. And since we are assuming the postonian expansion, you know, the these two scales are uncorrelated. Well, they they are parametrically separated. [clears throat] Now there is another setup which is also interesting and is the one I will talk about in the real part of the talk is where we consider relativistic velocities. So we um yeah we give up the assumption of the small velocity and in that case uh in principle there is no separation between the dynamics of the system and the dynamics of the of the radiation and I mean obviously the these setups are very common for well they are very typical of scattering but uh one can think of um uh the this uh sorry the this approximation to be very relevant if we consider uh a very eccentric orbit where locally the velocity of the the respective velocity of the two bodies can be almost um can be uh can be order one. There is one last uh one last uh setup that uh like uh that we would like to consider and is the one where actually this separation the separation scales between the the dynamics you know the of the of the matter distribution or the single body and the dynamics of the two body uh is uh parametrically separated but only for one of the two objects. So we take B we this is basically the an expansion in the symmetric mass ratio in the small symmetric mass ratio um and uh is basically a resomeation in one of the we coupling uh so now we take uh GM over B or for for one of the M's order one and I mean this is very important for for Lisa Because you know uh this is actually the the setup where extreme mass ratio intermediate mass ratio systems um live. >> Okay. >> So >> yes >> in the ratio limit now there is again a separation between the radiation and the because the radiation is suppressed by which is why you can do like this. Oh okay. Yeah. Yeah. Yeah. Yeah, good. So I mean since the beginning I I mentioned uh uh amplitudes uh and how amplitudes can uh can be used in this setup and uh some hardcore physicists should complain that uh uh scattering amplitudes in in the classical limit are very uh they are almost meaningless in the sense that uh the probability of some event in the future to happen given some initial state is either one if it is the solution of the of the equation of motion or zero if he's not uh but you know why am I talking about amplitude scattering amplitudes at all well even at colliders we do not measure scattering amplitudes and when we compute scattering amplitudes we uh from there we compute uh cross-sections and energy energy correlators which are actually in inobservable so you prepare some initial state you make it evolve. You measure something in the future and you sum or ignore things [clears throat] the things that you are not interesting in measure interested in measure. Uh so I would claim that u in the gravitational physics this is exactly the same. So we have a binary system that evolves emit radiation. At some point in the future we collect this radiation without really observing the final uh the the you know the rest of the well we collect the radiation at one point without really observing the details of the system um after the merger. Okay. So usually these observable are quadratic in the S matrix. So uh this will be very useful because it gives some computational advantages uh in the computations because we can use some properties which are otherwise hidden in the classical dynamics such as uh unitarity. Okay. So let me give a very quick sketch of the perturbative computations. So what we do since uh you know modern ampute techniques were developed in the context of perturbative QCD what we can really do well is perturbation theory and in the case of the gravitational systems we compute observables in the econom limit. So the small angle scattering where we are mainly focused on uh uh trajectories which are slightly deflected by the like they're almost straight lines and slightly deflected by uh by the gravitational interactions. And uh the the way to do this is that you basically write some final rules for the deflection of the deflection of the of the trajectories. Some final final rules for for the gravit like the interactions between the word lines and the gravitons. And basically you compute uh your your diagrams use identities to reduce these very complicated integrals to a set of simple integrals which is a very nerous step and at the end you compute these integrals with f new fancy techniques. Um and okay and then obviously here in this talk I've only cited these two because I those are the people I I've stolen these uh these plots. Uh but uh there are like several equivalent uh approaches and uh well depending on the order in which you do the computations in particular depending on the order you take the classical limit uh the there have been several different approaches which at the end of the day bring to the to the to the same result. if they are computed correctly obviously. Now uh so let me mention the main problem of this of this approach. The main problem is that uh this is an assumption here that uh if we compute an observable in the in the scattering setup then we can use that information for the bound uh setup and for for actual um uh for you know to for the templates we need and [clears throat] well and if you go high enough uh in perturbation theory actually you find that uh the two body gravitational interactions are very different between uh unbound and bound orbits and basically in four dimension this originates because of uh from the nonlinearities of gravity and the long range nature of uh gravitational interactions. So the setup is the following. You have your system here which emits some radiation which uh goes on shell and far away from the system and then um the background source by the binary pushes back to the system. So the dynamics of the body at some point in uh space time depends on all the past history. So the inside of the of the lyon and so basically the interactions are an integration over the past history of the system and if you integrate a scattering or integrate a bound system this the result looks very different. So yeah, you have something like this this any observable uh well obviously this is this was not a compelling uh problem in the sense that uh the these interactions this kind of uh interactions you see they need a lot of gravitons and gravitational inter um gravitational vertices. So you can imagine that this uh is very high order in G. Let me say that uh a nice play toy model that uh in which one could study this setup could be just looking at uh systems uh gravit for example gravitational system in odd dimensions because in odd dimension this happens without uh nonlinearities and without uh long range natur uh the long without long range. So understanding there the the problem may be interesting and useful. >> Can you say why odd dimensions? >> Yeah, this is I think the greens fun the support of the green of the greens functions in odd dimension is uh inside the the litecoon while in in even dimension the greens functions the free green functions are only supported uh on the lite con. So the this is uh so in one case you know is in four dimension is the interactions that then uh pushes the the the support of the green functions inside the lyon but uh in dimension it comes a leading order you have these kind of interactions already a leading order >> sorry >> yes >> so in the previous slide you mentioned icon >> so what I know from it is just that you take the that I'm thinking slowly learning fast physics. Is it the same here? >> Well, icon in this setup was simply meaning that we assumed the two bodies to be very far from each other like a large angular momentum uh uh expansion. So they are very far from each other the interaction are very weak. So the result is just a slight deflection of of the trajectory that that's what I have in mind when I say icon just large angular momentum. [clears throat] Okay. So one thing okay now we go to the part of the talk that is not review. So [snorts] one way of uh trying to attempt like the understand like one way to uh understand this problem or solve this problem of uh of um uh the the map between unbound to bound is actually to get some resumation get some more some some results which are uh resummed in the angular momentum. And uh so beyond the small beyond the small angle uh scattering and once you have an an analytic function of the angular momentum you can uh try to really analytically continue this result. Uh so when when I talk about resomesh I mentioned several scales. There are several there are obviously there are several ways you can uh um you can res well as I was saying before post minkoskin is a resumation of postonian for the scattering orbits. uh just for the scattering orbits. Uh so this kind of resumation is understood but what I have in mind is actually trying to understand the self force resumation. uh what we had in mind uh was to uh trying to understand the self force resumation in one of these um two variables and I'm sorry now I will switch to blackboards because the this becomes very very quickly very technical and uh the formula are simple but long uh so if I put you know like if I when I wrote them on a slide. I thought okay I'm going to lose everybody here. So I will just uh shift to a blackboard. So to understand to try to understand this resumation we actually start as usual with simpler setups not from the gravitational case. We we started from um uh something which is exactly solvable from from a classical point of view that has no no problem which is the relativistic scattering. some relativistic scattering of central potential and this is something well understood since uh forever. >> Can you write a little bit bigger? >> Yes, this is just ratric scattering of central potential. uh and um so the the setups we we study I we we study this uh relativistic scattering in several different theories which have something in common well except that they are simpler than gravity. So first of all what one of these theorem uh one of these theory is electronamics. So the colum scattering and then there is uh some extreme black scattering in an equalite sugra and uh some very simple theory some scalar theory scalar mediated theory and the scal mediated theory it's something like um this so let me write the action for this you have some massless mediator and then you have two objects two heavy object just one much heavier than the I there. [snorts] And uh this is uh cubic interactions if you want. And then you have which is coupled to the mass. And then you have also some cortic interaction. Okay. [snorts] So if I tell you at this point that these theories are very closely related, you will uh doubt my words. uh but I can show you the computation and uh so basically you compute the scattering angle in this central potential and you find that for the theory like for the scalar theory with this cubic and quarting coupling the scattering uh angle is this. So is some square root some aran is super simple. uh in n equal 8 when you take one of these extreal black holes to be much heavier than the other you have just the arctan the square root is gone and uh basically you find up to uh change of what you mean by g uh you find that this is basically the same result as the scalar theory just by uh putting this very particular choice of coupling. This is this is the misalignment of the a one angle misalignment of the uh B bps charges of the black holes. And uh here you have the col the colum scattering which you can notice that is just uh lambda minus g² and then a change of name for for g. So if we do now the computation in perturbation theory, we should find something which um you know connect or some structure that connects all these theories and then understanding these structures probably can help us uh understanding the the problem in a more generic setup. >> What is height >> is the scattering angle. So these are scattering of a of a central potential. Oh, >> okay. And here is always a two body system where one is taken to be much heavier than the other. >> So this is a classical >> is a classical computation. Yes. So that I asked to find typos in the presentation. >> Yeah. So okay. So how we how do do we try to reproduce that uh that result? Well, we try to compute loops from trees and uh basically what do I mean by that? I mean that uh let me compute the the scattering amplitude at Loops. reconstruct the scattering ample at a loops by knowing its discontinuity and here I know that unitarity comes and helps me because the discontinuity of the amplitude is given by some um uh unitarity cut sorry here in the T channel so the momentum uh exchange changed channel here you have the four point and uh now here you have to trust me that uh when when I take the prob limit this is some simple power counting but I have no time to explain it in the prob limit this is equivalent to having what is also very intuitive on one side of the cut the where the you have the heavy guy uh a number of sources and here you have the full three amplitude. So in the problem it basically we we mapped the by just by using unitarity we mapped the problem of the central scattering to understanding uh the generic multiplicity amplitude in uh in those theories like three level amplitudes in those theories. [snorts] uh and one one thing you can do is you know since Loops will compute uh some L order in the perturbative expansion you can start inspecting the perturbative expansion of those scattering angles and you find some interesting structure I mean this is just u the electronamic case. And then yes. So here the structure is already evident. At all odd orders you have some very simple term which comes with pi. At even orders you have something which is grow like [snorts] is more and more complicated as you go as you push this to higher and higher orders. Actually this is a feature that holds beyond the fact that the odd orders are simple are simpler than the even orders is a feature that holds beyond the prop scattering and I'm I'm not sure is fully understood but you know we know from this expansion that there is some recursive structure so you know we we need a generic multiplicity three three amplitude and a recursive structure so this screaming BFFW which uh I'm not sure you're all familiar with but uh is basically uh a way of reconstructing uh endpoint amplitudes from lower point amplitudes just by using uh the the fact that the amplitudes the quantum amplitudes like by using three-le unitarity [applause] So the fact that an endpoint amplitude when you go when of the one of the intermediate particle goes near the mash shell blows up and it has simple poles. So the this is a way of using well you just use koshi's theorem and in some variable to reconstruct the three-le amplitudes from unitarity but you know unitarity now can be a weird business in the sense that um uh this is unitarity at the quantum level. Do we find something like this in the classical limit? Does this survive the classical limit? And I mean the answer is yes but not in the form that uh we know here and I mean here there was already some work by Vincent and Julio but let let me give let me give you another way of deriving how three-level unitarity looks like in the classical limit. So in the classical limit you have certain sets of poles where you have he I particles here and I compliment I compliment I this is P in this is P in [applause] the classical limit these two poles will end up on the same point. So basically from a simple pole like the overlap of these two factorizations give you a new factorization theorem where K I is the sum over I of K to M U. Yeah. M U K I² A I A I minus K I. So D in is N U I. Okay. Minus uh Yes. UI zero. So now what what it was a very simple factorization on a simple pole looks like a factorization on a double pole which is basically the sum of like the product of these two. um the sum of these two sorry and [snorts] um a simple pole which looks like a soft theorem but okay now once you have this factorization you basically do well your amplitude endpoint amplitude sorry endpoint amplitude you Just use some cushy stem. So just writing something completely trivial. This is zero. And then this can be written as minus the two partitions of uh S z I basically where the Z I is where this goes to zero when you shift the U with this quantity. minus some res at infinity that uh um is not relevant for any of these theories. Okay. So what do we do? We start computing well the three point amplitude is just the definition of our theory. Let me let me start with the scala is simply this and then you compute the four point. The four point is proportional to k1 k2 u k1 squared. Very simple. Well, if you want the quadratic term I mentioned here, you have to put it by end. And then you compute the five point. And now you start seeing structure. So k1k2 k2 k3 k1 squared k12 squared and then per two permutations of this structure. Okay. [snorts] Now the answer is simple and is simply wrong because now if you compute the six point yes there is 3 4 u k1 2 3 squared and there are 11 of these terms. But this is not enough. You have a new kind of term which is K1 K2 K2 K4 K3 K4 K2 plus three terms like this. Okay. And now again you see a structure a new structure but still you see a structure you can make an answer that uh something like this will always be there and then you have uh you know seven point you have u uh five 25 these numerators and still you miss a structure. So you can compute some higher point and then you need somebody who has encyclopedic knowledge and is very good at recognizing patterns. So [snorts] this is our AI usage. Uh and basically you find something very simple, extremely simple that uh that the two plus n point ampute is a sum over some trees where the vertices of these trees are the momenta of the of the external massless mediator. uh for each tree you you take the product of edges and each edge has its own weight. Now this is very simple. Let let me take a uh five point one two three four five. So the sorry is a seven point. So uh 2 + 5 and I mean among the three graphs you can draw there's this there's this one one two three four well this is this then there is another simple one two three four five. All right. Okay. So, how do we how do we read the weight? Well, you have to choose a root here. Let's let's choose four. And then uh you say, well, the weight of one tree is k1 k3 divided by remove this edge. There are two trees. Now the tree that does not contain the root is the one you want. So here if you remove this there is the root of one. So this is k1 square. Now you do it for 2 three and this is 2 three. These are the two threes. This three here contains four. So you want the momentum of the other one squared. And then 34 again. 3 4 um yep. Uh so here is k1 2 3 squared and k5 squar and actually this this formula does not depend on where you choose the root. So you can check that explicitly and once you have understood this formula you can use basically the fact that you can check explicitly the residue at infinity vanishes you know that it exists a recursion relation. So the the only one the only thing you have to check is that this formula here satisfies this factorization theorem and that's a proof that this is a non multiplicity formula. So and since this formula this formula are quite unique when you take the contraction with the sources here uh the this is since since I mean the three point and the four point which are our uh seeds for the recussion are the same for mod contact terms are the same for the scalar theory. the the electronamic case and in n equal 8 you know that the integrant is exactly the same up to uh up to the coupling in here in front of up to the coupling. So this is this is a proof like that we are close to understanding this hidden simplicity of the classical limit because then the trees that you construct this way you can use it in other context like for example if you want to compute waveforms this is not very different from uh uh the computation we have done before you just add an external leg that is not contracted but you know already the all multiplicity one. [snorts] Um I mean just for some technical uh remark uh yeah where is it? I think it's here. You know for the n equal 8 case you know that uh the pi term there the odd terms are actually not there but still the integrant is u is non vanishing. So there is a very non-trivial identity uh that uh and is not very clear how to understand it in terms of um yeah one t [snorts] The sum of trees trees plus okay here I reconstructed the trees up to Iapsion. There is a very systematic way of reconstructing those IPS as well. So let me say up to distribution terms this is actually proportional to d minus 4 and vanishes in four dimension. So the the these are very non-trigger identity that uh uh doesn't really uh come from uh usual IBPS and you know simplify the computation a lot if one can understand it sematically. [snorts] Uh yes. So the this is I hope this was understandable but let me go back to slides where so this was one way of attempting uh understanding the resumation of perturbation theory in this context and uh let me mention in my last five uh minutes uh something I've been exploring which is very similar to what so some middle ground of resumation something that we can call the high school approach where the only thing we do is uh we uh determine local interactions between the word lines and uh and we determine [snorts] the observables for generic uh word lines in perturbation theory and then we compute the trajectory exactly numerically if you want And I mean this is not very different from what people have been have been doing in uh for example multipolar possian or nrg is just the relativistic version of it. So uh so the the a systematic way of doing the relativistic version of it. Uh so the the point here is that we can compute these uh interactions by integrating out the gravitons in the two body problem without going on shell. So the this you can do from the equation of motions by iteratively iteratively solve the matrix variations and then substitute them back in the Einstein's equations or since we are QFT people here you just compute the in effective action by integrating out the gravitons in a winger kish integral so the winger kish part integral is usually a twofold party integral [clears throat] and uh yeah observe like observe correlation functions are usually computed this way. In the classical limit, the Schwinger Kelish part integral is extremely simple because if you use this uh advanced and uh this the so-called Kelish basis basically you see that at the level of the P integral the coefficient of the of the single insertion of the advanced field is the equation of motion for the uh uh field corresponding to it. And uh so the first if you want to compute the interactions independently of the wland use the word len as sources. So your your word and variable now are the sources of the uh gravitational part integral and then you compute this input integral and the connected diagrams gives you the the effective action and the effective action is once you expand in advanced variables the the linear term is proportional to the the equation of motion and it keeps track of retardation effects and no I don't know. Where is it now? No. Okay. And something very similar can be done for observables like the waveform at infinity. You can compute connected diagrams in the Schwinger Kat integral. This gives you the the expectation value for for the metric perturbation. And once you compute the the like when once you integrate out the the the warning variables, the advanced warning variables will localize you on the uh on the on on the variables in these are basically stationary phase approximation that localizes you uh on the localizes the the word line sorry the waveform on the word lines which satisfy the equation of motions. So basically in this case we compute diagrams with no word lines. So the first term here is just the free particle. Then you have relativistic version of the Newtonian physics with the whole wheel fman interactions. This is radiation reactions. And then uh you know these are second these three are the second possan corrections in the effective action and these the rest is the third poss corrections. The nice thing about this is okay sorry I forgot about this just to show some some results which are very simple you know the the the relativistic Newton Newton you know doesn't have the problem I was telling before you know the interaction is localizes on is localized on the intersection of the two light cones and the radiation reaction uh effect is very easy to compute because we have dimensional regularization which makes this computation extremely uh manageable and this checks with exist existing results uh uh in the literature. Uh the waveform computation in this context comes for free is basically a um a generally like you just take the diagrams you were computing before. You strip off the advanced source, you plug in uh instead of that uh um the the graviton propagator and this gives you the waveform uh for some fun. I mean it's very interesting like uh something that is supposed to be a more complicated computation actually uh bypasses some of the technical bottlenecks like IBP identities which are not needed at any point. And at this point uh we can use the equation of motion to evaluate the trajectory numerically for generic boundary conditions uh like uh relativistic so generic velocities but uh scattering or bound and uh another thing is that you can uh uh have a control of the growth of of the numerical uncertainty. So you can really generate waveforms for bound states elliptic bound states um with some theoretical uncertainty given uh yeah that's it sorry I trash [applause] thanks for questions does any of those three examples process Yes, I think at least the last two. No, I I know about the first two. >> Uh I don't think so. >> So is this simplicity of the classical limit or is it simplicity of theories that have you know sure you you the nonprocessing thing is just a toy model. Uh but uh you can add one over r cubed interaction like something that makes you process and the only thing you have to do is um when you do here the the the product over the edges some edges will have a different weight. So you know this in principle uh can be applied also to gravitational theories even though the structures are a bit more complicated. Yeah. So yeah, it was I mean it's just the simplest playground to actually find find these uh these structures. [clears throat] >> On one your last slides there appeared the infamous minus 113. >> Yes. short term which has the wrong sign of addition reaction and which is actually cancelled because this is the analog of the >> ed electro dynamics 23 of the robot of laurens dra >> but uh this has the opposite sign and it is canled the radiation reaction is higher order in gravity so this term is an intermediate thing that has to be cancelled >> exactly like you can see this from the from the part integral point of view like you though you can use a filter definition to push this uh to the next order. So this will not enter at leading order but uh you will uh so I mean it's a triple derivative and it should be treated as a triple derivative which means push to higher orders via filter definitions or iterating equation of motions. So it will I'm not saying it will end. >> Reaction is more nonlinear. I mean it cancel. >> Yeah. Yeah. You have to keep you have to keep in mind that you have to be careful. >> Uh in the last slide when you say uh there's no need for IBP. Um but uh do you not need IP for doing the uh cage in uh integration because you said you have to integrate out the gravitons. >> Yeah. But integrating out the graviton is computing uh connected diagrams uh uh is just connected computing connected fment diagrams. Uh the the reason why there is no need for IBPS is that you you have an exponential because this is the green fun unexpanded greens function. Um so sorry is is the green function with a generic exponential and you can use that exponential as a generating function of uh tensor integrals. So that that's basically you you can compute higher uh higher tensors just by acting with derivatives. >> Okay, let's thank Stephan [applause and music] >> [music]