So I got the part where you were like that's where you were ready to change. Yes you know. You just so thank you for the invitation. Now. Frederick has tried to quote me I can take my revenge a little later on The Talk. I visited Atlanta once before and this was here I experienced my first earthquake. My cutting a little bit out of this one so help. This process is going. OK So so I'll just stand still here for the rest of the talk. So as projects up the hardest things to get the microphones microphones to work. So there was an earthquake when I was here last time and I had another big experience which was that I experienced the banana pudding and that was also a big big part of my experience here. It was it was a very nice visit. So I'll tell you about some work I've been doing recently in the field of studying scattering amplitudes and I'll try to convey why I'm excited about this work. And just to start off. What are scattering amplitudes. Well they're really the objects that the scribe what happens when we collide particles together we create new particles they are basically these interactions and we reach the amplitude tell you something about the likelihood of certain processes happening. So this could for example be electrons scattering with photons electron plus photons scattering to electrons of photon as a person who process. A slightly hot a process to calculate would be glue once interacting to create say a pair of quarks and to walk and so these are the objects that we're after. And we're off to finding efficient ways of calculating these objects and we also after understanding what the structure is so in practice. How do you do you know how do you call light particles Well you smash them together and you see what comes out. So this is the experimental side of the subject. And as you can see the interactions can be kind of messy but what we really have to focus on is what goes in and what comes out. Experimental versions LEP collide electrons and positrons create new particles. Experiments like Fermilab and the late C. collide protons of protons of protons with antiprotons and one of the interactions that happen very often in such collisions are colliding and creating for example all the cool ones but also other particles. Now one thing I also talk about is private on gravity on scattering now here I don't really have an experiment to show this is more like a experiment. So I just have a picture of Einstein as opposed to an actual scattering process. So this is the more theoretical side of the subject through to understand also gravitational processes like this. So the subject is how do we actually calculate amplitudes. So fine man gave us a great tool for this which is calculating Fineman diagrams. The idea here is that we study perturbation theories that the couplings are assumed to be small and we expanded around the free theory. And so as an expansion in small coupling this corresponds to finance diagrams which are then ordered by the tree level one group level which is pie of power and the coupling to look level three looks level and so on and in principle you need the whole series to know everything to arbitrary high precision. So that's a very tall goal to have to know everything at all the order and very often we can describe the physics very well by just knowing the tree level and perhaps the first couple of corrections here. One thing I'll show you is that even knowing the tree level can be quite a daunting task. So what do these abstract diagrams mean. Well actually what finance also says to write down diagrams that show the interaction say electron positron annihilating to a photon and creating an electron and positron. His rolls tell you to translate this type of diagram to a certain mathematical expression. So we have to use these rules plug in can. Late and find what the I'm for to charge that's that's the goal. Now here's a picture of finance vans. It's probably not possible to see it here but they're actually fine with diagrams on the side of it. The license places quantum. OK so I want to show you an example of how the complexity comes in. Even in some of the simplest calculations we can do so here are the final rules for blue ones who want to interact through a three particle interaction and through a four particle like direction. That's something that comes out of the like Ranjan for the theory. Now these interaction vertices are really the building blocks we connect them together into diagrams. So the simple question. I'll ask is how many time grams do I have to calculate at the three level so no close loops allowed at the three level how many diagrams do I have to calculate to get a process with N. and external states meaning say to going to to minus two and minus two so involving a total of N. particles. So for any cause three. This is simple because there's just one vertex I can write down of this one. For any calls for. I'm actually also giving you an extra rule here which is call ordered so I will keep the ordering of the external lines this the node momentum the particle number one particle number two and so on. So keep the ordering of this and not allow any lines to cross each of them. So this is a certain subsector of the theory and is well defined to calculate this. So for particle level I could let one of the four come together into a three vertex and connect that to one to two and three or I could let one and two interact with each other and then go to form three that way. Well I could also that all four come together using the forward six. But one thing I'm not allowed to do is let one go to three because that would allow that would require that I cross lines and I'm not allowed to do this here. So how can I continue count. Diagrams like this. Well to just get an estimate. Let's just ignore the fact that we have those four point interaction and just use the prepared text. So we just get a lower bound on the number of diagrams we need. Well to get from the four point. I have these two diagrams from before. And now I have to insert my fifth line and have to sit somewhere between line one line for so I could attach it to line one or I could attach a two line for starting with this diagram here. I'm not allowed to attach it to the middle here because I'm ignoring the four point interaction. On the other hand from this diagram I can produce three new ones. Because I have to attach line five either to line one to the middle line or to the fourth line. So three diagrams. So I see that while I have two diagrams for any calls for I have five diagrams for an equal five. OK now to continue to get what happens for any six. I have to insert the six line somewhere for this diagram I can insert it in two different ways between line one and five. This one. There are three different ways this one three this one through this one one two three four. So if you start counting you'll see that there's a bit of a branching structure like this I start with one diagram it split into the True for any calls for then for each of those one of them split into two new ones. Whereas the second was the other one split into three different ones. And I can look at each of these diagrams and see that the number of times they split in through basically has the same type of repetitive branching structure the number of possible once you get is the number of places you can insert the lines and there's always going to be one that gives you a possibility of inserting one more so counting all this number of a branch is here. I simply found for any quote six or fourteen and it was seven I didn't draw them all but they're forty two and you can continue that way. Now if I were going on. And equals eight it would give me one hundred thirty two. So now you may look at this sequence of numbers and see if it looks like anything for a million. So the simplest way of figuring it out is to plug it into Google and say what does it look like and you come up and find I hit on the Catalan number it. So you found the camel and numbers well OK so. That's I guess is a nice story actually this this particular. Sprinting structure here was well known so without Googling it you could actually also have found up. There's a mathematician at MIT Richard Stanley who has book of problems maybe more than one hundred or so the WHO to which all the solutions of the cattlemen numbers and this is one of them counting the number of branches. So this was actually this counting problem was a problem that was solved I was curious about the counting so I put it to two Michigan undergrads last winter. Neko Ackman here and William Murdock he was a freshman is a software at the time. And by ground you see my bike. So they actually solve this problem as part of their winter research program. So they also did a bunch of other things. So the Catalan numbers can be described in terms of a close form for a given N.. And if you look at the lodge and limit this growth you see within an exponential power factor here. So what we learn is that as the number of particles I want to scatter grows the number of diagrams grow rapidly. At each diagram. You know from the find minerals that turns out correspond to an increasingly more complicated expression. So we have a problem. They don't have a large number of diagrams and each of them gets more and more complicated as we go up a number of particles. So here is the actual counting this is the first set of numbers I showed you here. The pending on if I want to scatter clue one clue into K. clue ones then for different values of K. I can. See that the trait counting does to try Bale and perhaps what that gives me. Those were the Catalan numbers but actually then into taking into account that I also have this quoting vertex I actually show you the last their actual counting down here you see it goes even faster than that now we do the calculations. Imagine you sit down calculate the mathematical expressions for all these diagrams and you turn the crank you simplify and simplify and simplify and at the end of the day you find that for a process like this one. The answer is a simple complex expression like this. Well simple I mean complex in the complex plane sense. So what do I want to my writing here. I'm looking at a process involving capital N. number of particles. I'll take two of them particles one through to be Clunes with negative publicity. I take the end minus two other ones to be equal ones with positive publicity. And the answer for this entire amplitude say even some of one hundred one hundred fifty diagrams. Is this very very symbol object here. Now what does it mean. Well each of these so called brackets angle one two. For example is simply the dot product of the four moments of the two particles coming in. So and well there's a dot product and then you take a square root. This is true what I said but up to a complex face. So this is a complex fide way of writing the moment so. It's called a spin to elicit a formalism and by now it's actually appearing in textbooks already. So the answer. Turns out to be extremely simple. It has a nice structure the denominator involves a product of seqlock factors. So it reserves the six trucks or. That's not too surprising because the diagrams we wrote had this this this. This restriction that the I didn't allow any lines to cross this is reflected in the sickly factor here and then it singled out the two particles that have the negative because it is they appear here with an extra fourth power. Or in the numerator. So I know this is maybe not a particularly physical way a writing it but it's it's really something that you can translate directly and it's something that people can put into their coach when they calculate a lead sheet backgrounds. So the question that arises is why is the Sphinx so simple. And you know it can't calculate it perhaps in a horrible complicated way. So isn't there a better way to calculate the spring. Given that it is so remarkably simple. The question why is it so simple also has to do with you know what is the underlying structure that makes the symbol. So these are the questions that we're really after. So the remarkable simplicity. And this mathematical structure that it turns out on the lies this that together with the practical relevance this has for particle experiments and calculation of backgrounds is really what motivates these current studies of scaring impotence. So that's what we're after. So well. So in this talk I want to give you some examples of what we do. I want to show you the simplest of the methods but I also want to try to give you an idea of the different areas people are pursuing within this field. So first the different areas that are pursuit I'll try to group into three main branches. One of them is to study at very very interesting theory which is call any course false souping mill spearing it's the theory of the cool ones we just talked about but now we're adding as much supersymmetry year as you can but without getting more spin. That's been true. Which of the spin of the gloom once the theory has partners of the good ones which are cooler winos and it has scalars That's a total of sixteen masters degrees of freedom. And apart from the glue ones this is like nothing what we've seen in the term so this is not our of physical model of what we see in nature but it's a fantastic theory to study because it has an amazing amount of symmetry not just super symmetry. But it's actually it turns out to be a conformal theory. Now there's a certain limit a sect of the sphere that's called Plano that's the one that is scribe what particles are ordered. And the amplitudes. And this sector is also even enhanced with further symmetry. So this is a great very very control theory in which we can develop methods and this is where a lot of the focus in the field currently is because this is where we can really hope in the sense of really having a good chance of it only solving the scattering matrix of the whole scattering problem perturb that simply. There's no other theory that has to do with glue ons and so on in which you can do this. So we have a hope of solving this theory to all the boarders. There is a question of the compact expressions that arise for the amplitudes in this Fieri why are they so simple. We really want to understand what the origin of the simplicity is. Now this theory shows up in a lot of contexts in connection to string theory. So it could be that there is an answer in string theory for the simplicity. That's a second direction which I alluded to before. Which is that well given that we can develop these nice methods for calculating amplitude and then it calls for we're really interested also in understanding more broadly in quantum field theory not just for a very specialized highly symmetric theory but to something like Q.E.D. acuity that is really both useful and is relevant for nature. So the goals of that part is really to get useful insights for new physics and actually create methods that can be used for people to calculate backgrounds for the elite see in Fermilab. Now there's a new method although there's another direction also which is to use in this inside we get here. The standard methods is a very complicated method of calculating amplitudes given that we can find new methods to calculate them. What does that tell us about the underlying theory. What can we learn about quantum field theory itself. So that's one one direction and a. Part of that is to say can we teach. Can we learn anything about perturb of quantum gravity this way. And so it's to say it's a stand out statement that if you have particles gravity particles you scatter them this is not a well defined theory that Virgin says we don't know what to do about them. But perhaps this can give us some insight and I'm going to try to also say a little bit about that in the talk. So maybe I should say the work I've done has sort of well. It covers all the different areas but there are many different. I mean there's a large community of people working in all these different directions. I'll try to just mention a few different things. So one of the so one thing I want to show you is one of the basic ideas of new methods for calculating amplitudes. One of the ideas that is underlying this is the idea of recursion relations. So instead of starting from scratch with Fineman diagrams to calculate a seven point amplitude maybe you could recycle your results from calculating something that involves three bodies four bodies five bodies and so on. If you calculated those already you could recycle them into something like a recursion relation for amplitudes and such a thing turns out to exist and it's been an immensely powerful approach to calculate amplitude as well as revealing new symmetries oppen So I want to try to outline the method that goes into this. So let's think of an amplitude it's something that the Pens on the momentum of the particles that come in on them and so the particles that go out. Now suppose that we take this object here this amplitude and we shift the moment or so with that just means that I replace the momentum everywhere it appears by the momentum plus some complex parameters see time some other vector Q. which could point to something or action but I choose the skew in such a way that the shift of momentum I have for these every match in all the particles and massless so that they stay on shell that the remain masse. Close and that the total momentum is conserved because that's what I have to have for the scattering process. Now I can go ahead and I can regard my amplitude. As a function of the shifted moment so they satisfy the same constraints as the original ones. And so instead of thinking of them as if it's a function of all the moments or now let me just think of it as a function of this complex parameter see. Now put three level amplitudes it turns out that yet. That there are only simple poles in this complex function of Z.. That can't be any branch can't show anything fancy that can't be any double poles that can only be simple poles. Now. So if that draw this in the complex plane I can lend it to point to all my simple poles of this object abstractly and I can imagine surrounding them by a concert. Now if the function as a function of C. which a function I don't know a priori it but if it does such that as a function of C. It goes to zero for large values of the complex problem and then the contour interval like get by taking my function of C. divided by C. integrating all of the country that surrounds all the poles then that thing has to be several because I can the form the contra all the way to infinity. On the other hand this Finke here. Is also the sum of all the residues in particular the residue at the simple point a simple pole at zero is exactly the amplitude I want to calculate with the answer. If that moment. So the the residue at C. equal to zero is just the amplitude with on shift at the moment so the actual object I want to calculate So I I can calculate this amplitude that I want. But it is to be now the minus the sum of all the other residues. Now what happens at each of the simple poles is that the amplitude factor rises into two parts. The simple pulse occur only where there's an internal line in the. Grams was so-called propagator and it separates the amplitude and take it on shell into two actual on show parts. So the effect of all this is that the amplitude I want to calculate is the sum of all ways I can split the diagram for the simple simplest to split the lines into two sectors a left and a right sector here. And then I just have to sum of all this this corresponds exactly to calculating the residues. So what this gives me in effect is exactly a formula of this kind here because if I start with that N. particle amplitude then the number of external lines that can be for each of these two sub amplitude is necessarily smaller. And so this is what gives me in effect this type of formula here. So underlying this whole argument is coaches fear him. So it's really a quite simple underlying reason for this. This was done in two thousand and four by. B. C. and W. And so it's called a P.C. if you method but it applies more generally than what they actually set up. Now you notice that was one assumption here which is that the amplitude that I didn't know has to go to zero as C. becomes large and so proving validity of such recursive methods relies on being able to prove that such a light C. behavior happens. But when I take what you see that I'm doing as I'm really propping a certain U.V. limit of the amplitude. That's something that is very physical and that's something I therefore might be able to say something about without actually knowing the the mathematical formula for the amplitude. OK so in practice if we apply this to our calculation of glue on amplitudes I have to replace the sum of very complicated Fineman diagrams hundreds of them perhaps with simple sums of grams of this form. And it turns out that for this particular set up you see if that you invented. Then there is actually only one van. Diagram only one vanishing residue and so I get this symbol result from just a single diagram in this recursive expansion. So this is an example of where you can go around this idea of having to calculate hundreds of grams. You can get it in one step by by these methods. There are also other methods for how to do this. Like I used to call this the legal approach because you know imagine when you're doing these recursive formulas. You know if I calculate the seven point amplitude I have to know all the ones for less than for seven and then if I want the eight one then I have to go through this whole thing again it would be easier if I had better pick up building blocks like Lagos that I can actually build bigger bigger amplitude out of. And so this is actually what this other method called the C.S.W. method. Does So it allows me to build a different kind of background. They may look a little bit like Fineman diagrams. But the difference is that all the building blocks are actually themselves on shell amplitudes So it's a lot lot easier to calculate. Now that's a question again. Why does this work and which theory is those that work and this relies on this statement that the amplitude has to go to several in this particular U.V. limit. So this is one of the things that were investigated it was shown early that it works for who wants. But that it works for this very useful theory any cause for sleeping most was something we proved in a paper a few years ago now to tell you that's true for this proof but it's really kind of. In the sense that they've relied on some other expression for the amplitude and you have to analyze a lot of special cases but it was a proof so that's at least something but we later found that that's a much much simpler way of using basically simple things a scaling arguments and dimensional and now assists to analyze amplitudes in very very general fourdimensional theories. So they can be theories with or without. Supersymmetry they can be serious with or without massive particles. They can even be serious with non-green on the lives of four couplings and I'll give you an example of what I mean by that and a bit. So we found a very very simple criteria for when you get this large sea fall off of the amplitude this U.V. limit vanishes of amplitudes. So you take the amplitude you look at the external particles. And you're assigned to each of the particles helicity. OK so this could be plus or minus a little before the glue once. So what goes into this inequality is then the sum of these have as it is for the external particles the number of particles enter. And then they see that end up here this is something that really is just dimension analysis you look at your theory you look at what are the Commons and my theory. What are the mass the mentions of those couplings and that number is what goes into this here. So for example for clue once the coupling as I mentioned less so. C. In that case would be surreal. And then what you have to do is calculate this number if it's less than zero recursion works in the sense of these little recursion relations they will work. So there's a sufficient condition for making it work and you can just sit and you can just look at an amplitude if you know which theory it's supposed to be and you know whether or not these on shell methods work. We could also connect them this simple formula to actually give some physical understanding of why it happens to be that way this has to do with invariance of certain operators. I'm not going to go much more into it but I was so there was also a physical understanding of where this comes from. So this work was done with a post-doc at Princeton and the student at Michigan code. Now as special case of this formula shows in one line. Like maybe half a line but this this theory has this construct ability and so that's why I'm allowed to say that there's zero proof is really kind of a way. It's a one line here. So I said here that this also works for Nanami lies about couplings so that might sound like a little foreign So let me given a sample of it. So one of the predominant expected to be predominant ways of producing a Hicks at the late C. is the clue one Hicks fusion The idea is here that you have to go once coming in. They enter actually through a quark loop. And produce a Hicks. So the clue wants and the sentiment of the clue wants to clue ones don't interact directly directly with the Hicks they have to go through the quote loop and the dominant Kwok that can go here this this growth with the mass of the quark So the dominant qua contribution comes from the top quark. Now in the case where the Higgs is lighter than two times the top quark mass which means that you're below the threshold for producing two tops. Then you can effectively say you can effectively replace this one do calculation with a calculate with an effect of vertex and your theory which just says that you effectively let the clue once couple directly to the Hicks and there's a fine rule for that that involves the clue one feels as well as the Higgs field. It looks like that. So this has been used in the literature the power of it is that sometimes it's useful to calculate to loop effects for Hicks and interacting with loons. And if you have to do that then you have well it's a true look perfect but you can replace effectively. One part of the calculation with this vertex that makes the two calculation effectively one two and that makes it a lot simpler. Now we can then show and this was actually shown back here that you can use this Lego approach to calculate amplitudes involving Clunes and the Higgs. And these and the answer for these actually to soar actually also very simple. And there was no proof. However in the literature why this should work. It worked. It was tested and it was used but as fully A simple of way to prove that this work is simply to. Plug into this inequality criterium I showed you on the previous page. Now there's actually a mention full coupling. And it's it has to have mass to mention mine and happens to mask the mention minus one. And then you can plug in and you can find that for type of amplitude of amplitudes like this. The quantity here is negative and one can then therefore use the methods that were used are actually directly ballot. So they give you a little bit of background for this the methods of on show recursion relations and so on were used in many many cases without knowing formally whether they were valid or not. Of course people have been extremely careful and then going back and testing that. But what they get out of using a certain recursion relation that actually gives the right result by taking many different limits and testing it but it's also always nice before you go into a problem and trying you know there's this shift work does that shift work to actually know beforehand so you don't have to just experiment with it. And this is one thing that this us. Now there are other other application of this one is to top quark amplitudes with anomalous magic moments where you can use this type of methods. So some work done recently. That's also work that we did on the so called Cool on branch of any cause for this is a place where some of the particles have mass us. And so we actually showed that there are actually very very simple structures there. And we could actually even show that amplitudes can be with massive particles can be correct constructed from the one with massless particles in this particular theory. Now so far all I've said was about the tree level amplitudes So I argue that this is can even be a complicated thing to try to solve but we have methods to do this now. But of course we'd like to know. Also what happens at the loop level both because as phenomenologically relevant. But also I mean for completeness What is this is this going to say something about quantum field theory to say something about all orders. So what happens there. Well let's first try to take an example of what Luke diagrams are actually good for. So one is the magnetic moment of the electron that a total modern physics that this is true but then there are quantum corrections that come in and change that value to be slightly above true. You can then calculate this theoretically and you can measure it experimentally. And that's a remarkable agreement to many many digit on the on the both the theoretical and the experimental side. Now I get through Frederick. Because he gets some of these calculations is that the loop calculation and seventy four. And he says in his speech from ninety three is that he was invited to Cal-Tech to give a talk. And to go to any other place and say that he and his collaborator had calculated thousands and diagrams to get that the answer is about one time some factor. You know that that's one thing but to do it in front of Fineman. So he's going to ask why. Plus why not. Minus. Why can't it be that how can it be that one hundred diagrams give something of order one instead of one hundred. What is the deal here. While the cancellations like this. What does it mean. So that drug went in fear of Richard Fineman and he went into a deep trance. And he came up that what if you computer instead of all these diagrams just in one big man see group the meant to gauge invariant quantities. Then he found that grouping them instigate in varying quantities even added up to something or a plus minus one health and that indicates that there's something deeper going on than what shows in the fine one diagram. So this reinforces a little bit what we already seen at the tree level that this was already noticed there to see that you know that there is really some more structure than what the final diagrams indicate and the key here is. Really. The idea of the gauge invariance that you have to study their diagrams by themselves are not physically meaningful you have to group them into something. That is now. So so that's the message we have to group things and to gauge in varying Quantas we saw this already at the three level that it's much more efficient to use the recursion relations. At the level they're similar ways of doing this one is called generalized unitary. And the idea here is that if you have some loop diagram. You can basically you can so-called cut it and then you split it into factors of tree level amplitudes. And doing the cuts in many different ways on an answer for the amplitudes that loop level even allows you to fully reconstruct it at that and this can be done even a highly orders. That's also been approaches recently to have recursion relations at loop or level. And and this is much harder because now you just don't have to just have simple poles you also have Bransford Stoppel poles unphysical poles and there's a whole mess to figure out there but the bottom line is instead of thinking about the amplitude that you have to integrate over all the momentum that runs in the loop which is hard when you look at the object that you before you integrate it. That's a nice rational function. It's just a simple polls and you can actually carry through the program. Of devolving developing recursion relations for such things. So this only works in the special sector of this and it calls for theory but certainly hope that we can we can generalize it in fact we're working on a slight generalisation of this right now through something that will work in purely in most theory. No supersymmetry. OK And this actually relies a little bit on some of these methods here. So this is exciting and useful progress to carry on both. But let me remind you that the three different branches here I talk mostly about the first two so far as. Developing methods informal informal aspects and then using them phenomenological. Now one thing I want to switch into is this last subject about quantum gravity. So one of the things that I learned in school at some point was that if you try to do quantum field theory with Einstein's theory of gravity. Then you immediately get in trouble because the fine man diagrams you write down. You look at what what kind of integrals you have to do to calculate this that's one integral for every closed loop you see and then you sort of count how many powers of momentum there is in the numerator in the denominator and how many encircles you have to do and these are all just generically all divergent diagrams. So formally these are all infinite. So looking at the theory this way you say well then you can't have a point particle theory of gravity. It's not a quantum theory and not a quantum field theory and indeed this is one of the motivations for studying string theory that renders the calculation of Gravitron amplitudes finite. Now one of the things we've learned is that we shouldn't put too much weight into the individual diagrams. So one could speculate what if you sum up all the diagrams you regulate them in some way someone up perhaps only divergent pieces would just cancel. So could it be that adding up everything you need to get a good gauge invariant well defined physical quantity. Could it be that that's thing is finite. Even though the individual pieces are not. In other words where we just calculating things the wrong way. So this would mean that the theory would be perturbed actively well defined in the sense of of perturbation theory or of my order everything would have to cancel. Now one thing that's known is that if you add super symmetry to this then you have a symmetry between both hands and permanence in the theory. But since if they come with a factor one in the loop the firm yawns will give a factor of minus one times the same thing. And so those things naturally cancel and it is indeed known that in. Super gravity such cancellations happening in the series. So the more supersymmetry the Marielle So people look at the maximalist supersymmetric theory of gravity and four dimensions it's called any calls eight super gravity because eight supersymmetry is. And the question has then been could this firin with the Maximals amount of supersymmetry possibly be perturbed at simply finite and four dimensions. Now it was known actually for many years. That the one unsuitable order any pure super gravity theory would be finite because of the cancellations I just mentioned. But then for this veery people then started having the techniques unitary methods another other ways of actually calculating explicitly the fall cravats on amplitudes two graphic terms coming into gravitons going out at three low border and they found that it was finite. This was done around two thousand and three and that really reinforced interest in this question of possible purpose of finiteness. That was no explanation at the time of the finiteness. It took about two years and then people also calculate the fall loop amplitude for scattering for gravity terms and that also turned out to be finite. So that's pretty wild. But why should it be finite You know I mean calculating for gravity scattering why about five grab a ton scattering and fall the ball or would that also be fine. There's nothing that implies that a priority in gravity. Six graviton scattering But if you can prove that you have to such Look at seventeen scattering you have to go to order eight loops or something like this. What are you going to do you have to understand first of all the structure of the theory but rather than just brute force and the number of particles in the loop level. So the question we have raised about this is what does the symmetry of the theory even have to say about finiteness is the some way that we can understand why a three and follow orders came out finite just based on the superset of just by. On the symmetry in the theory. So I wait to study this is called to study the candidate counter terms. And the idea here is that if I have a divergence at the elbow order. Then there is some operator which is something constructed out of the fields in the field theory. And it has to have a master mention that is connected to the order of this comes from the mention analysis or expanding in number of couplings powers of mutants constant that comes in. And so I got example of such a counter term here would be at three little border. This gives something of that mention eight. And if I take a Riemann tensor that involves two derivatives already so if I take four of them. I have eight derivatives this is something with mass the mention eight. Now I have inserted you know Riemann tensors has four indices I haven't said anything about how you're supposed to contract these indices. And that's a very complicated problem to know how many individual independent ways you could possibly do this. But this is the idea. This is the type of object that would have to be there hath there been a three loop divergence for scattering of four grandsons this operator would have to be compatible with the symmetries. So when you study these operators it's really really messy. It's very hard to get a handle on them even to know how many independent once you should look at. However one of the things we realized was that rather than studying the operators themselves. We could study the amplitudes that these are these matrix these empathy because study the amplitudes that these operators would create. Using the techniques that been developed in amplitude studies. Now it has to be such that in the theory with certain symmetries if these symmetries also are symmetries of the quantum theory in the sense that they are non anomalous then the operators that correspond to local counter terms also have to respect those symmetries So for this particular any calls eight super gravity theory. This means that all this. The symmetry has to be preserved by the operator. It has to be super symmetry simple. That's something called a global R. symmetry of the theory. It's an a unitary as you ate symmetry that also has to be preserved. And then even worse if you want to say this way. There's also a hidden symmetry in the theory in the sense that there's a global spontaneously broken symmetry. So this Fontayne is the brokenness in the same sense as the Higgs mechanism basically But but in a more complicated section because this global symmetry involves one of the exceptional groups. So if these conditions here. They become constrained They are conditions on what the count of time the operators have to be so they actually also constrain directly how the amplitude have to behave and so we can use them as constraints of understanding the matrix elements of my own sense of the amplitudes we can constrain what counter claims can possibly exist. So here I'm just giving you a huge overview of possible things one might write down so let me try to guide you through it. So what the vertical axis. I show the loop level. At the horizontal axis. I count how many fields are involved in this operator. So here I'm showing things that are constructed out of our Which are the Riemann tensor. And out of derivatives acting on the Riemann tensor in a very very schematic way nothing in detail. So at the three level we discussed that we had out of the four. I could act with some covariant derivative on it that at two mass the mentioned so that takes me up one loop level. And then I can go down the list I can add one more derivatives to make it a higher dimension operator or you could go right and add more hours to make it something that involves more fields. So that's really what you see here. Now the analysis that we did on the amplitudes that these operators create constrain these objects and they're such that actually all the ones that are crossed out here can be wrong. Doubt that's not where the first divergence of the theory could possibly appear. But at this stage we have not that use this exceptional group symmetry. And what that does is is quite interesting actually. So the fact that it's a spontaneously broken symmetry means that it only manifests itself in low energy theorems. These are something that would develop propriety physics in the sixty's. So we are actually now using pi and physics resolves to study gravity and quantum gravity. What it says is that the scalars that this theory has act just like prions in the sense that if we take the momentum to Ciro are one of these pions then the amplitude has to vanish. Then since we can calculate the matrix elements corresponding to the operators we can simply ask do they vanish or not. If they vanish. We can't tell anything because that could be some coincidence but if they don't we see that they are not compatible with the symmetry. So this allows us a direct handle on how to control these operators so how to understand these operators. And so now I'm showing you the same thing again much extended I can cross out now everything below the seven new border. Everything is crossed out here the red ones are not allowed by supersymmetry the green ones are not allowed by this hidden symmetry and so there is actually only one thing allowed for seven new border. And then the rest is ruled out. So without ever having done a single loop calculation. I have ruled out that Virgin says in this theory up to and including six new order. I have also seen that to understand seven loop. I need to only calculate a four cravats on amplitude to actually know whether or not the whole theory is finite. The other thing you see that does nothing cross out and able to border or below the mass no matter what the details of all these operators are the point is I can't constrain them with any of the known. Truths that I have in the theory. So what does that tell me. Well suppose somebody comes and say OK maybe the theory is finite to all the orders. Then I don't understand why you know there has to be magic cancellations. But there's no symmetries that would guarantee these things and this typically just doesn't happen in physics. If there's magic cancelations there's some symmetry usually responsible for there's some reason why these things happen. And unless we find out what that something is then it's very unlikely to believe that the theory really should be finite. So. Let me go go here and that this point and the talk people would often ask what do I think can the SCIRI really be finite or not. And so you know my point of view to this is you know I you know it in some sense it's not my problem if it's fun and I don't not. I just want. I mean it's going to be what it is I can't do anything about it. The answer is going to be. Yes and no we may find out sooner or later I don't know but at least I want to understand what does the physics of the physics of the symmetries tell me and what they have told me so far is that finite in this world requires some structure that we don't know about yet. So if you want to learn about this we should really go back all the way and understand just the three level of gravity on scattering the still a lot to be learned there. We don't know even know you know what sense as much as we do as we do for the say young males theory. So there's something to be understood there. If we could find a new symmetry there we can go back and we can reanalyze the question but it's not really a point in going going on. What's I'm trying to calculate the water by loop order or something like this before we really understand the underlying structure of the theory that the some of the bait about this point in the literature but I think it's sort of are solved in the sense that it's this is it's I think it's fairly clear that you need more symmetry. So people are calculating five loops and there's a good reason for. This because they do it as a function of the mention and this gives you a hint of how things vary as you go up and that mentions things become more divergent but you can try to see if it follows the same patterns and for. Similarly there are some explicit constructions of super counter terms this is a very very hard thing to do too but it also indicates that at seven or eight loop people expect by by now to see at every chance. Unless there's a new structure. So my message really here is here is what the important things that we learn in the process of studying this question is more is you know is more relevant than whether the question directly is yes or no. Yes would be extremely surprising and would have consequences. No would not really get us anywhere. So the problem that is really what did we learn in the process of all this so to answer the questions. What do you think. Well we don't talk about what we think it doesn't matter what we think about it. It matters what we calculate to figure out about the question. So this is what I wanted to tell you about some of the new excitement about amplitudes So to summarize. I think the field is certainly interesting in its own right to regain new insights about the structure of quantum field theory at least perturbs of quantum field theory it's clear that there are new and better methods but what does that actually mean for quantum field theory so. We don't quite even know yet. That's a certain chance that we can solve all orders a probation theory and complete in though the as Matrix a plane or any cause for shipping mills. But there's a little more to it than that than just knowing the answer it turns out that what the structure that has been seen already is very very pretty. There's a geometric picture of the amplitude. That means that the amplitude these mathematical expressions for the amplitudes turn also out to be volumes of certain gear metric object and other spaces. So this is a new friend that has come in the past one and a half years that and. But suits are closely connected to mathematical structures like geometry objects and the volumes. This is relevant for particle physics so there's a nice to all side of formal theory and developments. But that wreck connection. Through what people need to calculate backgrounds. We've seen some fundamental physics applications trying to resolve the question of protective gravity what it does. And that's also been attempts to try to connect things back to string theory is the some string the origin of this story that amplitude to connect the two volumes. I mean that would be an interesting thing to understand. Now I also have some hope. That these new methods where the essence of the new methods is that instead of calculating things in case dependent ways like the fine man diagrams that depend on the artificial parameters that we put into the theory to be able to calculate that we want to do something that doesn't depend on anything artificial We want to be constantly physical all the time and what we calculate and I So that's what I hope also some way of carrying over the lessons we get from amplitude studies to other areas of physics. One might be for example two calculation of correlation functions which has some of the same flavors as amplitude it's and so with that. And thank you. Michigan I'm going to more probably see this couple on call for you Margaret. From Paul thank you very much for part one. Very yes yes. Yeah OK so so for the first question. One of the things that comes up for a background as if you calculate the background to leading order say the one processes scattering processes cool ones to leptons cool ones the cool ones and so on then there's always a question of how good history approximation. So you wait you get to know how good your approximation is by calculating the next two leading order and see how much it moves you. And then what you have that you have to go to next to next to leading order and so on. So what is relevant for the backgrounds is of course you know how good the theoretical the third of all calculations are you need the next two or even all the next to next the leading orders for that. And this involves doing loop amplitudes and the way people do these things they actually use various Unitarian methods to compute the cuts of loops and that involves the tree amplitudes to get the tree amplitudes prove recursion relations. Now that doesn't. In real applications give you the entire loop amplitudes there are certain pieces that are call rational terms that you can't get this way and so one of the key things to know how to construct Now the thing that slow down the program is calculating the rational terms. So my colleagues that do these things. Tell me that this is sort of the key thing that they need to understand now and getting better methods for calculating these rational terms that are not constructed from three hundred two to sixty. Think to try to understand and it's something that we are trying to to approach with some of these methods so that under the first question. So the second question was about other areas of physics the last point here. About the it works in the relativistic theorists get so so I think the key here is not relativistic on nonrelativistic the key here is. Variance. So if you try to apply these methods to something like. A scale of thievery. But there's no H. redundancy in the description in Lagrangian you don't really gain a lot. In fact one of the things you can show is that if you try to do this. Lego expansion of a scalar amplitude it's exactly the same as the find my diagram expense you don't gain anything. So that's what you when you gain is where you have redundancy. And I guess in many cases also a number of sophistic feel free. You would also have some gauge redundancy possibly so. So if that's the case I would think that would be again. But I haven't seen any applications of that that would be interesting. Suffices of the matter when you saw your bow to cover the forest for it's all about why the trails are off by your problem. I wonder if there are problems of bike. I see that would that would be interesting. I mean and. I know one connection where this while there's probably more but the least one connection where this gets into the realm of what mathematicians are interested in and that is that the amplitudes in getting this very special case of a plane there Nichols forces being most actually naturally live in across money and which is the space of K. Planes and an end to mention space. Now studying that since the amplitudes have a six week structure then that's that's something that you would then say to them what do the mathematicians know about six truckers in these spaces of planes and it turns out the not a lot has been known but the optics that physicists wrote down to complete amplitudes in these spaces in a completely different formulation than what I've shown you. Turns out to be of interest to mathematicians. But whether that has any or any craft in the. And so on. I don't know I know that we encountered some number theoretic constructions in terms of you know basic group theory and so on but I'm not sure that's that would connect but graph theory not sure that I know of kind of an interest problem. Yes commentary on science very much about I see. OK To the right. Or away all the structure where you have a sense of what they see yourself. I was told it was obvious you want to translate that very very well I said OK and in the Yeah Yeah that would that would be that would be an interesting connection. So you mention some function you see that. Yes it's possible the functions of this for me all function and so I know yeah that's good good. So that's been another approach so. And yet so so in good. So there has been there's been some connection along those lines. Actually one thing that. OK So let me just say so. The critics what I showed was this the amplitude as a function of Z. and we used the residue theorem to find the residue at the center at zero as you said was connected to the residue of the other ones that what's the recursion relations. So one of the early approaches of hopes was that suppose that you now know the symmetries fully of all these amplitudes can you write down some equation that the estimated cost to satisfy to all orders and then solve it to find the actual answer. So there were some approaches along these lines but now more direct approaches to solving this has actually been. Been more useful. I should perhaps also say that we actually use recursion relations can actually be solved in many cases so we have finite you know for example the three level is fully solved in an equal force with the most one level actually also. But. But but but whether some functional equation has been something that people have thought about but not not fully fully developed yes yes yes. Right. So so in the case of the plane the sector of any calls for stripping mills that sector is into COBOL. And it's very closely connected to the following. So the theory is conformal and has super symmetry. So it's actually super conformal. Now then by studying the amplitude since this was something that was known. OK then by studying the amplitudes it was seen that there was a secret symmetry that you would never have guessed from the point of view of Lagrangian which is active but on the momentum variables as if they were basically space time coordinates in a sense. So this is called the dual super conformal symmetry. Now together the super conformal symmetry in the dual conformal symmetry generate an entire yang in this is the structure that on the lies the interval of the. So one could ask then is this enough to determine all the amplitude and even at that level in the theory. We know that we can write the amplitudes compact Lee in terms of use Yang and then vary and pulling on real someone can write down. Well not. Well the rational functions but you can write them in terms of the special invariant pulling on wheels but that doesn't tell you what the coefficients of these pull in on real Should Be So you have to put in some physics that tells you that these objects have a correct Secondly the right physical poles and so on. So you can get quite far. But it somehow is not everything and the. Asked pink last thing it looks like you need ink is what from the random point of view is that you enforce the quality of the theory. And so that is what tells you where the polls are and that's what terms the road to proficiency. So you can get far but you can't get everything of the stage for everybody in the story. I actually just have to get this here to morrow whole day on the fifty four hours. So if you take the elevator term left you will find us. If you want to talk to me I think we should quote from.