 All right guys, what's a fantastic pleasure to be at the ACTP lecturing one of these schools as as always I Apologies, I apologize that due to some difficulties with my scheduling. This is a more truncated visit than I would have Like to have had but anyway, I'm still very happy to be here Just just just to say one thing one of the one of my favorite things when I lecture at these schools Are some our extracurricular physics discussions so that that that I normally enjoy with the students so Apart from the electors and since I'm only here for for the one night tonight. I Want to say if anyone wants to talk about any other part of physics? the LHC 750 GeV particles cosmology anything at all Any collection of you who are interested in having late night talks about this I'll be hanging out the Adriatic a lobby at around 10 p.m. And in the past these discussions started 10 and they go to two three four depending on the Depending on the stamina of the students so So anyway, no no no obligation, but if you want to hang out and discuss other other parts of physics If nothing do these lectures then we can we can do that all right But now to the substance of lecture lectures. I'm going to be talking about The subject of scattering amplitudes and a maximally supersymmetric gauge theories and and a new picture for What these scattering amplitudes are really a new mathematical question to which the amplitudes are the answer? Which involves novel ideas in in math it makes a connection with some Objects and mathematicians have been studying recently and generalizes them in a significant way and I don't want to spend a long time on the philosophy behind this This research program, but just one word about it We have many indications that we have to think of the idea of space time as being something emergent Perhaps I think more and more people are saying it I've believed it for a long time that we might even have to think about quantum mechanics as somehow an emergent concept from More primitive ingredients. I think there's actually very good evidence Or circumstantial weak, but but they're good enough to get excited about Evidence that in fact space time and quantum mechanics will emerge joined at the hip sort of hand-in-hand From from more primitive principles And if something like that and we have lots of reasons to think things like that Should be true ultimately having to do with the well-known difficulties with thinking about a gravity and especially cosmology in in in the usual in the in the usual Quantum field theory framework, but We're going to we're descending way back down to earth from those Lofty heights and thinking about something as concrete and basic as ordinary physical scattering processes Partially because if it's true that there is such a sort of radical reformulation of what Normal physics is where space time and quantum mechanics are emergent ideas. It's very unlikely that that's going to leave the rest of physics completely untouched On the other hand we know that these things work. We know the world is described by by a quantum field theory We know we talked about glue on scattering amplitudes at the LHC and we think about them in this standard way Using local evolution and space time as embodied in Feynman diagrams But if it's really true that there's some way of some different way of thinking about the physics that it stands to reason that there Should be some different way of even thinking about that physics Standard physics where the space time and the quantum mechanics are not playing a starring role But other concepts play a more starring role now in this case. We're not changing anything We're not deforming anything so it's really a matter of reformulation trying to find a different way of talking about This very standard physics where these ideas that we rely so heavily on aren't relied on heavily and we see them come out as derivatives from other things Already it's a challenge to figure out how to do that for completely standard Quantum field theories where nowhere near being able to do that. We're starting with these toy models of Plane around equals four super young mills in order to get started But the long-term hope is that by practicing and learning how this Magic can work even in standard physics It might give us a picture for how to deform away from these ideas in the situations ultimately much fancier situations involving Gravity and especially cosmology where we suspect that that we might actually have to lose these ideas altogether all right, so Now that's a that's a very lofty highfalutin set of motivations for trying to think of a new way of formulating the problem of Really any observable in a quantum field theory, but scattering amplitudes are a great observable to begin with because they're genuinely Lorentzian that first there are the things we measure in experiments almost all the time in particle physics and secondly while the vast majority of our understanding of quantum field theory is really Euclidean Scattering amplitudes force you to ask Lorentzian questions where time matters Okay, so so things go in from the past they come out in the future. So they're essentially Lorentzian questions you can't get away from thinking about time because that's hard-wired into the question you're asking and and of course another indication that there's something like this going on is The famous fact that when you calculate scattering amplitude using Feynman diagrams You get an explosion of complexity beyond the very simplest processes already two gluons to three gluons Two to two we put on problem sets in graduate school But two to three gluons is hopelessly complicated if you open up Peskin and Schroeder you copy out the Feynman rules You get you know 50 pages of completely impenetrable algebra And yet the final answers are shockingly simple people discovered in fact exactly 30 years ago today There's a I just came from a conference at Fermilab that was celebrating the 30th anniversary of the discovery of this incredible formula by Steve Park and Tom Taylor The Park Taylor amplitude which is an example of one of these things were hundreds of pages of algebra collapsed to a single beautiful simple Expression which back then wasn't clear was a tip of an iceberg But today we know it is a tip of a very big iceberg that there is a horrendous complexity in the standard way of Thinking about scattering amplitudes that conceals lots of marvelous simplicity structure a hidden symmetries and and underneath all of it as I said right at the beginning Some powerful and deep mathematical structures that are slowly being They're gradually being unveiled and the story of the amplitohedron is one aspect of these developments but I'm focusing on it here because in the at least in a In this toy theory of plane or n equals four super Yang-Mills. It's complete. It's complete in the sense that I can When we'll get to it tomorrow We'll just start from fresh complete forgetting about physics will define some geometric object We'll define a certain dictionary associated with the geometric object and then studying it will discover that it has all the properties to correspond The scattering amplitudes for low-ons and spacetime the fact that it's local and unitary will be an output from the from the Geometry is not at all something. It's not really geometry. It's an algebra geometry essentially combinatorics underneath all of it But those properties the physical properties of locality and unitary will be outputs and not inputs and at least in this theory We can go all the way and see a complete framework where all those things come out Of course a very big challenge to a generalized away from it. All right, so that's what I want to Tell you about but now this is a large subject It's still developing fairly rapidly It's not clear that we're at the we're at the bottom of it that we know the the best ways of thinking about it yet and So I'm not really going to be able to explain all of it to you in the span of three hours What I hope to be able to do in three hours though is do two things first to Completely define the terms of the objects we're talking about because we're just gonna today We're gonna spend some time talking about kinematics just the how we think about the variables that label scattering amplitudes in the best cleanest possible way and Tomorrow I will Spend enough time to tell you how to think about the very simple geometry involved and By the way, the the the mathematical ideas involved are extraordinarily simple. They're not fancy They're not complicated if you do if you do, you know other parts of a string theory It's it's these aren't sort of fancy Calabi yows and k3s and all these fancy schmancy things it's just simple projective space you just ask questions about configurations of points and planes and lines and It it's very slightly unfamiliar, but it's extraordinarily simple And so what I want to do tomorrow is tell you how to think about this geometry from the ground up in a very simple way And at least completely finish describing and defining what the object is and sketch how we start seeing some of the physics emerge from it Okay So so let's start today with the discussion of the kinematics and let me first remind you of the normal way That you think about scattering amplitudes in textbooks Let's say we have a bunch of Particles with spin one let's say a bunch of gluons then we have a bunch of momentum P1 P2 up to Pn We conventionally use crossing symmetry to imagine they're all outgoing. They're all ingoing. So the sum of the momentum is zero So a is always going to in these lectures always going to be an index that labels the number of particles Okay, and we have polarization vectors in the normal way of thinking about things we have polarization vectors associated with all of these particles So what you actually compute? Let's say using Feynman diagrams is some Lorentz tensor and mu 1 up to mu n which depends on the momentum P1 through Pn but that's not the physical amplitude right the The physical particles are labeled by their holicities. So So in order to get the actual physical scattering amplitudes for any given holicity So the holicities are h1 h2. We're working in four dimensions We contract with some polarization vectors epsilon mu 1 h1 up to epsilon mu 1 hn And so this is the actual amplitude the actual amplitude is not this Lorentz tensor The actual amplitude is an object that has a mixed transformation property When you do a Lorentz transformation on the momenta you get a little group transformation on the holicities Okay, so remember what the little group is if you have a particle a massless particle moving in a given direction There are those Lorentz transformations that don't change the direction in which the the particle is moving They don't change the momentum, but they give you a rotation around the direction of the momentum Okay, so that's those particular Lorentz transformations that leave the the null momentum fixed But do a rotation around it are known as the little group transformations And when you do a Lorentz transformation on the momenta you have to pick up a little group transformation or re-phasing under the Under the holicities. So so the actual amplitude is Has this property. So this is if I do a Lorentz transformation here of H1 through hn. I have to pick up e to the i there are some phase that depends on lambda and p h1 hn So when we use Feynman diagrams were pretending that there was this object called the polarization vector Which is like a bifundamental between the Lorentz group and the little group Okay, so when you do a Lorentz transformation It soaks up the Lorentz properties and it picks up a re-phasing under the little group and all of that's perfectly fine We're talking about massive particles. That's perfectly fine and good, but if we have massless particles It is not as good because these polarization vectors don't actually exist There is no such thing as a polarization vector for a positive helicity photon Well, you say what are you talking about that? I know what it what it is here. It is if the momentum is e 00e Then a polarization vector plus or minus is like one over root two zero one plus or minus i zero something like that Right, so you've seen all seen polarization vectors like this So what do I mean when I say there was no the polarization vector? Well, what I mean is is the following we can just see it from counting degrees of freedom Epsilon has four components right epsilon zero one two three, but there's only two Helicities, so there's too many degrees of freedom in epsilon. How can we go from four to two? Well first we can say that epsilon mu p mu is zero That knocks you down from four to three and if you had a massive particle that would be correct They'll be the correct number of degrees of freedom for a massive particle, but it's not enough for a massless particle There's only two degrees of freedom for a massless particle, and so we're stuck There is no Lorentz invariant way of describing a Giving of specifying a polarization vector for the two helicities of a massless particle This is the key difference between masses and massive and what forces us in introducing gauge redundancies in our description of the Physics what we have to do instead is declare that epsilon mu and epsilon mu plus anything times p mu are Equivalent physical states Okay, and if you go back to position space in position space This is like a mu and a mu plus d mu something Okay, so we see that the need for a gauge redundancy in order to describe the physical degrees of freedom of The massless spin one part now. How do we see it here? The problem is that if you take this if you take this Polarization vector, and I know a booth I now do a boost in the direction in the z direction I do a rotation and then a boost back Okay, so that I take the So I take the p back to itself Then if you do that under that sequence of Lorentz transformations You'll find that epsilon will not come back to this form that only has zeros In the in the in the zero in the z direction. It'll have something non-zero in the zero and the Z directions, okay I urge you to do that the exercise although it's predetermined that it had to be because there simply isn't a Lorentz invariant way of Saying that those zeros are zero and Even if you declare there's zero in one Lorentz frame if you do enough boosts You'll come back to a situation where they're not zero in another frame So that's just very clearly shows you that there's no Lorentz invariant way of assigning polarization vectors to photons The only Lorentz invariant thing we can do is assign this whole equivalence class of polarization vectors with photons And that's why we have to bake in this huge amount of gauge redundancy into our description of the physics when we have scattering from massless particles Already, that's the case for massless spin one even more. We have the difumorphism redundancies when we talk about massless spin two Okay, so our first task then is To figure out and this this should be in the beginning of every quantum field theory textbook Should be in the beginning of Weinberg's book, but it isn't in the beginning of Weinberg's book yet Our first task is to figure out what are the variables an amplitude depend on actually not redundantly, but actually right? They redundantly depend on polarization vectors and momentum But what are they really functions of non-redundantly and that's really going to be our goal in these lectures to come up in this lecture It's to come up with increasingly good variables to describe Describe the amplitudes until we get down to the to the sort of best set of variables on which which most effectively represents the external data and on which the obvious and unobvious symmetries of the problem act as Simply as possible, but to begin with let's talk about spinner-holicity variables so we're going to work in four dimensions and Many of these ideas port to higher dimensions where you have to replace these spinner-holicity Objects with pure spinners eventually, but it's very simple in in four dimensions So imagine that we have we have our our four momentum and we're going to dot it into the Pauli spin matrices the Laurentian Pauli spin matrices in the usual way to get this two by two matrix So I hope you're familiar with this with this two-component vile spinner notation and Okay, so The point here is that if we do if we look at the determinant So if I call this matrix capital P to two by two matrix the determinant of this capital P matrix It's just P zero squared minus P three squared minus P one squared minus P two squared, which is just P mu P mu and So if I do a two by two linear transformation on P goes to L dagger P L for any old L for any L Any L with debt L equals one Then if I do that the debt P well P well first of all go to some P prime But this is the most general Hermitian matrix I can write down, right? So the most general two by two Hermitian matrix I can write down so P prime will just be some P prime mu sigma mu Oh, I'm sorry. What am I doing? Thank you P one plus I P two Thank you very much. Yeah So P goes to some P prime so let's call this P prime which is associated with its own form of Manta, but the determinant of P is Equal to the determinant of P prime and So P squared is equal to P prime squared and so P prime is some Lorentz transformation that depends on L on P Okay, so that's how we can represent. That's the two-dimensional representation of the Lorentz transformations the spinner representation of the Lorentz transformation so the most basic representation of which you can build all the rest of them Okay, so But now something extra nice happens when the one the momentum is null So when when P squared equal to zero the determinant of this P is equal to zero and that means that this P AA dot, let me write it like this as a Two-by-two matrix well, it has it has a zero eigenvalue It has rank one there are many equivalent ways of saying it So I can write it as the outer product of some vector and another two-dimensional vector Now just to be very concrete if you imagine if you imagine that the Momentum is E zero zero E the particle moving the z direction then our P matrix is is two E Zero zero zero and one choice for these lambda and lambda tilde would be like root two E zero and Route two E zero, okay, so these aren't spinners. They're just completely ordinary bosonic They're ordinary two-dimensional vectors Okay, but you see there's something nice has happened because instead of describing a massless particle by saying that it's a constrained Object a four vector whose zero components is constrained to satisfy P zero squared equals the spatial P squared Now I can just give you free objects lambda and lambda tilde Really, okay, and then you can build out of them something which is automatically going to correspond to a null Vector to a null momentum Okay now How to Lorentz transformations act on this guy well the If I complexify everything if I complexify all the momenta everything So I talked about the most general setting then lambda and lambda tilde are independent To spinners But they're not spinners. They're independent two vectors. Let me call them and the Lorentz group is SL2C cross SL2C with a different SL2 acting on the A index and on the A dot index Okay, so the SL2 just acting by two by two linear transformations with determinant one If we're in Minkowski space Minkowski momenta Then that's the case where those P's are real and the matrix is Hermitian So that means that lambda tilde a dot is lambda a star right, so lambda tilde is a complex conjugate of Lambda and the Lorentz group is the SL2C, which is the diagonal of The guys in this complex one Yes Right Sure, sure. Yes, and finally there is one last signature that's going to be very useful for us So We have to learn to be relaxed about reality So this should not be difficulty for an audience of string theorists. Okay, but here I mean Relax about reality conditions on the momenta so We can also imagine that we're in 2 comma 2 signature and the 2 comma 2 signature It's easy to see that what this matrix is would be I'll keep using 0 3 1 and 2, but it doesn't that See in a second. The distinction is more Meaningless, this is how I would write my P matrix. Okay, and so you see that if I take the determinant of P It's just p 0 squared minus p 3 squared minus p 1 squared plus p 2 squared. So this has 2 comma 2 signature and This is just the most general This is just the most general real 2 by 2 matrix and so in 2 2 signature lambda and lambda tilde are simply independent real 2 vectors and The Lorentz group is SL 2 R Cross SL 2 R, but in general I'll refer to the Lorentz group as SL 2 cross SL 2 and then it'll look by contacts It'll be clear whether I mean R or C again. We shouldn't we're not going to be too We're going to be relaxed about About the reality until tomorrow when everything is going to be real numbers and even positive real numbers. All right, so So now now we have a nice set of variables where I don't have to tell you ahead of time I don't have to describe the four momenta as constrained objects. I just hand you lambda's and lambda tilde's Okay Now let me say a little bit more about this so there is so it looks like We said we have Four components of the momentum, but it's constrained. So there's really only three components of the momentum But it still looks like there's four components and lambda and lambda tilde two and lambda and two and lambda tilde so What's going on? What's going on is that? It's impossible to actually uniquely specify the lambda and lambda tilde once you give me a P There's still a redundancy under lambda goes to T times lambda and lambda tilde Goes to T inverse times lambda tilde okay If you do that the momentum of course goes into itself and Again, if for real momenta Because lambda tilde's got to be the complex conjugate of lambda T would have to be a phase Now this redundancy is actually a really good thing Because this is the action of the Lorentz group Sorry, this is the action of the little group. Okay, that's exactly what the little group is It's something which keeps the momentum fixed But it gives a rotation around the direction of motion and that's exactly what this is So this is what we were looking for. We're looking for well-defined objects that transform under the Lorentz group on one side and on the little group on the other side So that we can build invariance out of them that transform properly under Lorentz and little group transformations Okay, polarization vectors are not them or they only all of them redundantly But these guys are them directly and in fact We can now say what an amplitude is so if you give me a bunch of just as a Representation theory statement if you give me an amplitude that depends on an object depends on a bunch of helicities and on a bunch of Spin of helicity variables, then if you take any one of them and you rescale this by TA and that by TA inverse Then this needs to pick up a particular weight For any given particle has to pick up a weight TA to the negative 2ha or h is the helicity M of lambda a and lambda tilde a and This defines the helicity of the particle even tells you what the helicity is Okay, so the helicities don't have to be independently specified The helicities are encoded in the homogeneities that these functions have under this rescaling Okay, so with this we can now write down an example of first example of some scattering amplitude, let's say the scattering amplitude for Four gluons. This is one of the celebrated Park Taylor amplitudes I talked about before so we have an amplitude. Let's say for helicities a particle 1 is plus particle 2 Is minus particle 3 is plus and particle 4 is minus Okay, then I can write down an expression that looks like 2 4 to the fourth over 1 2 2 3 3 4 4 1 oops, sorry. I have to tell you one more thing. Sorry And I have Momentum conservation, sorry. I haven't defined those brackets yet. So let me just hold on to that for a second The last thing we need to do before getting to that is what are the Lorentz invariants? Well, the only the only invariant tensors I have are epsilon a b and epsilon a dot b dot So the only Lorentz invariants I can build is I can take two lambdas for example lambda 1 a lambda 1 b and I can lambda 2 b and I can contract them with an epsilon And so this is so you can call this lambda 1 lambda 2 and sometimes when it's These the a's that are being contracted. This is represented by an angle bracket So 1 2 and the other kind that I can have is The epsilon symbol contracting The other guys So what are more familiar? What are more familiar? Lorentz invariants like for example, what is p 1 dot p 2? Well up to a factor of 2 that I'm not going to be careful about p 1 dot p 2 is just angle square bracket 1 2 times angle bracket 1 2 That's because p 1 is lambda 1 lambda till this is lambda 1 lambda till the 1 lambda 2 lambda till the 2 And if I If I contract these guys the only way I can I get 1 2 1 2, okay? so now Why does it make sense that I have to have an angle in a square bracket? Because whatever the object is if it's made out of peas it has to be invariant under this little group scaling Where lambda goes to t lambda right because the peas are invariant so if we had angle brackets We have to have square brackets to compensate The weight right when I rescale lambda 1 by t lambda till the 1 goes by t inverse Every time you see these objects nakedly not being compensated by square brackets on the other side It means it's carry some helicity way, so it's telling you something about The spin of the particles involved so now let's go back to this expression So it's a it's an amplitude There is a delta function for momentum conservation as always, but I don't have to explain I want to have time to explain where this comes from and what or why it's the correct amplitude at the moment We're just looking at this Just confirming this trivial Group theory scaling property that just by staring this expression I can see that these are the helicities of the particles if I rescale particle one there's two Ones downstairs, so I pick up a factor t to the minus two t one to the minus two which corresponds to helicity minus one okay Whereas if I rescale particle two there's two Four twos upstairs and two twos downstairs, so I get a t one to the plus two and that corresponds to Having helicity minus one So so just by staring at the expression Just it's weights tell me that tell me what the helicities are it also means that I can't have a formula like this This is a meaningless formula Okay, I Can't add these two things. This is completely meaningless because this has a different weight under rescaling as that one does Okay, so we're starting to get there Now a little bit more kinematics Any questions before I go on actually Yes, well you see now instead of so so let me just summarize it again standardly Normally an amplitude we were right as an amplitude that depend on helicities and Momenta right and if we want to build it with Feynman diagrams, we would even Compute it like this with polarization vectors. That's that's usual picture Now we have something else now. I just have a function of the lambdas and the lambda tildes I Don't have to tell you the helicities explicitly the helicities are implicitly contained in what happens to this is a function Lambda lambda tilde with the property that if you rescale It has to have this fixed Be scaling let's fix homogeneity Now what does even turn the finding diagrams? I'll leave this as a little exercise for you. Okay, so let's say You want to see how would you convert you see now what the entire purpose of these lectures is going to be to never have to Talk about finding diagrams. Okay, but but we don't have to jump all the way Okay It's a very nice exercise to take Feynman diagrams even for any thing that you're familiar with you can even do it for scalar QED for this example, okay? and What starts off as polarization vectors and momenta and converts it to these spinner helicity variables, okay? Now exercise for you is to show that the polarization vector. Let's say for a negative helicity Luan is the following an epsilon now because it's a vector I could write in terms of a a dot and this epsilon is actually lambda a mu tilde a dot over Lambda tilde mu tilde Now mu tilde as a completely random Two-dimensional vector you pick any two-dimensional vector that you like and the claim is that if you build this object Lambda mu tilde over lambda tilde mu tilde that this will correspond to a good Polarization vector for a negative helicity glow on and if you want a positive helicity glow on It's the other way around okay, it's lambda tilde a dot mu a over lambda mu Let's just take a few quick looks at it You'll see that this has the correct weight in lambda, right as weight that corresponds to a negative helicity particle here Positive helicity particle there, but you say what is this horrible ugly thing this dependence on mu tilde? It shouldn't depend on mu tilde, but then you remember I told you before there Is no such thing as the polarization vector only this equivalence class of polarization vectors make sense And so a lovely exercise you can do is to show that if you shift the mu tilde By a mu tilde plus anything let's call it row tilde Exercise is to show that this polarization vector goes into itself Plus something times p Exactly the linearized gate transmission that we're talking about okay And so what will happen if you just take standard Feynman rules for scale I encourage you to do for scalar QED just calculate some constant scattering amplitude and scalar QED It's very simple. Okay, just these two diagrams Okay, but now go in and shove these guys in for the polarization vectors use Spine helicities and watch to your amazement as all the dependence on the muse drops out and You're left with a function just of the spin or helicity variables Okay, so that's how and that's of course historically also how people didn't jump immediately to abandon the assignment diagrams This was the in between The the halfway place where you could start seeing these other structures that emerge Okay, so we're gonna now take one more step Um What what what we've done is trivialized the fact by writing each p as lambda lambda tilde We've made the fact that p squared equals zero Manifest but there's something else which is not manifest that the sum of the PA is equal to zero Right, that's momentum conservation Our goal here is to think about the very simplest thing in the world just the external kinematical data We want to think of it in the cleanest simplest Possible way, and it's still not perfect because It's still it's still constrained if you want to I can't just randomly pick out a bunch of I can't just randomly pick a bunch of Lambdas and lambda till this they will fail to satisfy that the sum of all of them is equal to zero Well, let's forget about this for a second and let's concentrate on this for a moment and just to give a little bit of physics a motivation for what what we're about to do we're Ultimately going to be interested in blue on scattering amplitudes and blue on scattering amplitudes have in addition to the dependence on Helicities and momenta they also depend on color indices. Okay, so those are just the just just indices of the of the gauge group Okay, so this now depends on lambdas and lambda till this But there is a or even let me just call it all the Momenta, but there is a standard way of Representing the dependence on all these color factors in a very simple way You can break the color factors into pieces Each one of which depends on on a trace Okay, so let me let me give let me give let me give an example for four particles You can have a piece that's trace of TC one TC two TC three TC four Okay, so that's the dependence on color times some function that depends on one two three and four Now notice that this trace is cyclically invariant under under rotating via T's so this function would have to Have some cyclic action on it as well Of course if we have helicities here It doesn't necessarily have to be cyclically invariant, but these pieces are cyclically invariant Then I can have another piece which is all the different orderings that I could have take C1 C3 C2 C4 And then there should be some function of one three two and four plus and so on So in general this is called a color decomposition of a blue on scattering up a tune MC1 through CN of the P would be a sum over all permutations up to cyclic Trace TC sigma one up to T sigma n of Some pieces M sigma one through sigma n Okay, so these are called a color-ordered or partial Color-ordered amplitudes and while this might not be such an immediate thing to think about when you're doing When you're doing finding diagram calculations Well, actually it is because the way the way it arises when we do a finding diagram calculations Everywhere you see an F ABC when you do a finding diagram calculation You can replace this up to by some eyes by trace of ta commutator of TB and TC Which is a trace of ta TB TC minus Trace TB ta TC and it proliferates everywhere. You have the F ABC's you jam them together in this way And then you use the simple chasm your identity for TC times TC In order to combine the traces into longer and longer traces Okay, and at large n those are the only pieces you have left and there are one of random corrections that have As well so So we can we can always do this at tree level we can always do this and we just have single trace pieces in general At the loop level we can have multi trace pieces as well Down by 1 over n But anyway, we are we're going to be focused on on large n theories in the planar limit And so it's natural to focus on these colored ordered pieces. I should also say that I Mean this is a very important Simplification even just we're doing standard finding diagram calculations as one of the things that very early on in the late 1980s Was imported very very basic feature of perturbative string theory calculations I was imported into field theory amplitude calculations because the color ordering is forced on you when you do string theory Calculations through the champaetons factors. That's precisely how string theory tells you how to do the calculations Precisely in this color-ordered way Okay, so but from here all we want to take away is that for these color-ordered amplitudes There's a natural ordering to the momenta Without in general, there's no reason to order the momenta one after the other But when we strip off the color factors, there's an ordering to the momenta that follows the ordering that we see in the traces okay, but now the particles are ordered and That then makes it natural to do the following thing So so let's imagine drawing the momenta One after the other because they're ordered. So here's P1. Here's P2 P3 P4 P5 P6 and so on okay up to some Pn So I can always draw them in this ordered way and if this is on a four-dimensional sheet of paper then the fact that having Momentum conservation means that the polygon is closed so I can trivialize the fact that Momentum is conserved just by labeling the Momenta Instead by the vertices of this polygon Okay, in other words, I'm going to write each PA mu as xA plus one minus xA Okay That's fine. So now if I say this I've trivialized that momentum is conserved But now I have to stipulate that the edges are null. I have to stipulate that the edges are null Okay, so I'm still not done or I could say that each edge is null But I haven't said that that the sum is all equal to zero yet. Okay But this is a very important object in our business This what looks like a polygon with null with null out with null edges In a in a suitably correctly defined sense the scattering amplitudes turn out to be Um Dualy calculated by the expectation value of the Wilson loop that looks like that with these non-polligial edges This is a strange space. This is a space in which coordinates have units of momentum Okay, so this x space is not the usual space of space time. It's like a momentum space time Okay, but now we're going to spend a little bit of time thinking about this very very basic geometry of just What do these polygons? What's a good way of thinking about these polygons? What's a good way of thinking about these null rays and so on? In minkowski space. Okay, so now we're going to have a little aside here is that we want to think about minkowski geometry conformal symmetry and Twister space so this is an aside and I'm going to tell you in the next ten minutes everything you need to know about twister space at least for the purpose of this subject So well a big motivating idea for Penrose back in the 60s Was actually sort of very very modern in many ways He had a very holographic idea that you shouldn't talk about the points on the inside of a space time because it may be They're all fluctuating but I should talk about things like light cones or light rays that pass through all of space time and make It out to the boundary And maybe you could have a theory for things on this interesting boundary involving the light rays that's fundamental objects and not points in space time and so That's what we're going to start doing We're just going to start by asking the question. Let's say I have a null ray in space time So that's a how do I characterize a null ray in space time? Well, we can be if we're very basic about this What is a space of null rays in space time first? How many dimensional it is a five-dimensional space, right? Why did a five-dimensional space? Well? It's laser beams right all possible laser beams So first I can take my laser pointer and point it in one of two directions on the sky So that's two directions and then there's the where the laser point is relative some origin That's three numbers So there's a total of five numbers that specifies in four-dimensional space time the space of all real null rays it's a five-dimensional space and Since we're complexifying everything you might think that it's naive Complexification would be to a five complex dimensional space a ten real dimensional space Okay, that's a perfectly good. That's perfectly correct And it's not interesting for our purposes you can do that, but it hasn't paid off in any interesting way What twister space is is a much more interesting complexification of the space of null rays from five real-dimensional To six real-dimensional three complex dimensional But we're going to see it we're going to we're going to see it Just by following our nose in a very simple way Okay, so let's say I want to write down a linear equation which is satisfied I mean, it's a it's a straight line. It's a line. I want to write down a linear equation which is satisfied by by x's and Which which which are going to lie on a null ray. Well, let's just try to write down a linear equation What are linear equations? There are the form a plus b x equals zero, right? That's if we just have one variable we'd write down formulas. Let me put a minus sign there Okay, a minus b x equals zero Well, let's just try to write down equations like that except our x's are going to be x's in space time Okay, and because we're we're using these nice two by two matrices in order to encode four vectors Let's attempt to write down an equation of exactly the same form. So so x would have to be some x a a dot So whatever that it's contracting would have to be something that knows about one of these indices So let me put a lambda there and then this is going to be a mu a dot Okay So this is a linear equation That specifies a bunch of all the x's that satisfy that equation are going to lie on a certain straight line in space time Okay, if I give you a mu and a lambda. So my data here is mu a dot and lambda a is just given data Okay, and this is going to be some line in space time Now I claim that's not a random line. This is a null ray Why is it a null ray? Well, let me take two points on it Imagine there's some point x1 on this line and x2 on this line. They both satisfy this equation So I would have mu a dot minus x1 a a dot Lambda a equals zero and mu a dot minus x2 a a dot equals zero So let me subtract these equations from each other if I subtract them from each other. I have that x2 minus x1 a a dot lambda a is equal to zero that means that the matrix x2 minus x1 has a vanishing eigenvector So that means that the determinant This means that the determinant of x2 minus x1 is equal to zero Well, that's exactly that x2 minus x1 squared is Equal to zero. So this is a null ray as I claim. Okay. Okay, so that's that's pretty cool I've managed to specify a null ray in space time by giving two two-dimensional By giving two two-dimensional vectors lambda and mu. Let's think slightly more carefully about it So if I give you so let me group these lambs and mu into a big four vector That I'll call z at the moment. I'm just grouping them, right? There's just random I'm just deciding to group them all together and a moment. We'll see that the grouping has as a Significance, so this capital I runs from one to four. Okay, there's just this four-dimensional vector, but Notice that so if I give you a z I've specified an all-ray But in fact if I take this equation, I multiply it by constant I'm specifying exactly the same null ray, right? I haven't changed what the ray is So that means that it's not z that's in one-to-one correspondence with the null ray. It's z identified with any multiple of z Gives me the same Null ray So what I really have is a correspondence between four vectors up to rescaling And null rays in spacetime Now what is a space called which is just a space of vectors up to overall rescaling? That's just projective space You can think of it as just a space of all rays that pass through an origin in Four-dimensional space right no matter here's a vector two times a vector three times a vector everything is lining up in this along the same ray So the space of all lines that pass through some origin in a four-dimensional space is the projective space p3 so what we have is a correspondence between a point in p3 this is z up to tz, okay And this corresponds to a null ray in spacetime in spacetime This p3 is called twister space and so as as Penrose wanted We have a space now in which the points correspond to null rays in spacetime Now in a minute, we're going to make more of a correspondence between Twister space and the spacetime, but I want to come back now to another one of the important features of Twister geometry Which is we're dealing with light rays. We're dealing with massless particles and you all know that theories of massless particles Yes Yes, yes, that's right Nothing, yeah, I mean that there's no physical significance to a choosing something to be one That's right. Yeah, so so so in a moment There'll be a better answer to this question in five minutes. Okay, because because If you think about the answer to that question in ordinary standard projective geometry we say indeed there was no distinction But if you want to get an ordinary affine space Let's say Euclidean space. Although it's not really Euclidean. It's not a metric It's just an affine space where I have notions of parallel lines But not but not a notion of distance then I have to take projected space and also give you information about a line of infinity Okay, so it's not just a projective space a projective space together with some extra data of the line of infinity There's an analog of that In Minkowski space the there's something called the infinity twister, which is a line in this p3 That as we'll see in a moment lines in twister space correspond to points in spacetime and twister space is the Conformal compactification of Minkowski space So if you have conformal symmetry all patches are equally good Everything is totally fine if you break conformal symmetry then there is a preferred pointed infinity and that is associated with Some extra data of a particular line in twister space. Okay, but at the moment when things are conformally invariant there is a There's no distinction. Well, that's that's just what I'm about to talk about now. You all know that theories of massless particles Have of course, they have dilation invariance, but they also have conformal invariance at least whenever we have relativistic theories and all known examples and so and the interesting part of conformal transformations the meat of conformal symmetry this Really surprising part of conformal symmetry is not the dilations. That's trivial But the surprising part is inversions so So if you there's a very So we have conformal symmetry The big surprise is inversions and that's a symmetry under which x mu goes to x mu over x square and You know, of course, you're all fancy Quantum field theorists, so you know all about these things, but you encountered inversions even before When you're in kindergarten or whenever it was in college where you solve this problem of the electric field outside a conducting sphere, right you've all solved that problem in school and And you notice that this problem is solved by the smart method of images, right? You you find this clever place to put this image charge with some q prime Okay, such that kind of miraculously when you work it out when you put it in the right spot the the the sphere is it in is a Equipotential surface now. Where do you put it if this is radius little r and this distance is capital r Well, if you remember you put this thing exactly at little r squared over capital R Okay, so you put it at the inverted location and so that's why this method of images trick works is because Is because Electrostatics has inversions as a symmetry is a special case of the conformal symmetry Okay, you know, but this is a very surprising symmetry. Okay, it relates big and small. It's and and as you know if you if you do a Translation if you do an inversion a translation inversion back Then you get the infinitesimal version of the conformal symmetry, which are the special conformal transformations. I Mean that's already just a little comment. I Won't I'll just take another five five minutes here Before ending just just a little comment is it might have bugged you a little That we have this big conformal symmetry even before you enter conformal symmetry even when we just talk about Poincare symmetry It might have bothered you that that space-time symmetries in The usual way of thinking about how they act on space-time. They all look rather different. We have translations Which are ddx roughly? We have rotations Lorentz Which are x ddx and we have a special conformal Which is xx ddx? All right, and yet of course they all form just one big Algebra none of the generators are special to any of the other ones and yet by demanding that they act on local space-time coordinates in the standard way We're sort of obscuring the basic similarity between all of them Okay So now so let's keep that in mind and now instead of asking how do those symmetries act on points in space-time? let's ask how do those symmetries act on light rays and That's that has a beautiful answer that acting on light rays these symmetries are Treated on a completely equal footing and the action of all of those symmetries all of the conformal transformations are nothing other than four by four linear transformations on these on these twister variables, so let's see. Let's see why that is and I'll end with that before picking things up again tomorrow So To begin with let's imagine that we have two points in space-time x1 and x and y and Of course the distance between them x minus y squared under inversions It's of course is not even it's not even variant under dilations and it's also not in variant under inversions under inversions This goes to x minus y squared over x squared y squared under inversions or what you'll Clearly notice is that if x minus y squared equals zero then this goes to zero under inversions, so Having two points null separated is a conformally invariant notion The distance isn't conformally invariant, but if they're not separated that is a conformally invariant notion Okay, and so we can then ask how the Since the light rays are now associated with these muses on lambdas we can ask How these how the symmetries act on the muses and the lambdas so So let's see how do Lorentz transformations act on Lambda and mu well, this is obvious. Okay, it's just their act is an SL 2 As an SL 2 cross SL 2 So Lorentz transformations are SL 2 cross SL 2 What about translations well if I take x goes to x plus delta Then I just have exactly the same formula, but my mu is shifted mu a dot goes to mu a dot plus Delta a a dot lambda a so that's the action of That's the action of translations so So you see Lorentz is mu goes to l mu lambda goes to l prime lambda That's a linear transformation on mu and lambda Lorentz trans translations are a linear transformation on mu and lambda just mu goes to mu plus something times lambda Finally, what about inversions? Well, if you just take this equation and we divide by x squared Then then then we find that lambda is equal to lambda minus x over x squared mu is equal to zero So inversions is just lambda swapped with mu also a linear transformation on mu and lambda so That's the beautiful thing that when we group the land is on the muse together Into this big four vector z and now this four vector has a point to reason to be called a four vector Is that the conformal transformations? Just act as z goes to some big L z where L is in SL 4 Those are the complexified conformal transformations, okay and so and Who are so again in this big four by four matrix? There is an SL 2 for Lorentz here. There's an SL 2 for Lorentz here This guy up there is the translations. I guess this guy down here are the translations and These guys up here are the special conformal transformations Okay, but everything is treated on a beautifully equal footing a final thing I'll say quickly before ending is just to finish this basic correspondence between Between twister space and spacetime So we saw that a point so here is this p3 of twister space st There's a point z in p3 and this is associated with a null ray in spacetime st Null ray now now let's say that we take here two points in twister space Let's say a point z1 and a for z a and a point z b Okay Well, this is associated with some null ray. That's associated with some other null ray and It's not so trivial to a to visualize But we'll see it in equations in a moment. Those two null rays intersect at a point Why do they intersect at a point? Well? Let's look at all the x's that sit on the the line z a well They're all the form there's some there's some mu for a I'm suppressing the indices now minus x Lambda for a is equal to zero and then let's ask is there a mu for b minus the same x Is there a some x common to both lines? Lambda b equals zero can I solve these equations and of course I can because this is two equations That's two equations four linear equations in total for four unknowns the four x's Okay, so there's four equations and four unknowns and so I can solve for all the x's and this is a tiny exercise I'll leave for you. You can solve the x a a dot is Apologies for the mixed notation here, but it's mu a Lambda b Minus mu b lambda a Over lambda a lambda b. Okay, so given two points in twister space. There is a point X a b Associated with them in space time. All right, but just like before there. This is actually very redundant Before we could multiply our equation by any constant and we got the same null ray Now we can do much more we can take any linear combination of these two equations Any linear combination of these two equations will give me exactly the same point x So that means that I can take any linear combination of a and b of z a and z b and they will give me exactly the same point x Now what is that space which is all possible linear combinations of a and b? What is that geometrically? Well, if you think about that space in a in the four-dimensional space in which the z's live What are all the possible combinations of two-direction z a and z b? That's a two-dimensional plane Right, there's a two-dimensional plane that passes through the origin that contains the a and z b Right, so in this four-dimensional space of z's there's a two plane in four dimensions Right, and now what happens when I look at this projectively, okay? This looks like a line in the projective space Okay, so a line in this p3 a Line in p3 corresponds to a point in space time And so that's the most basic twistorial connection is that points in twister space correspond to Null rays in space time Lines in twister space correspond to points in space time Now what is the beauty of all of this? The beauty of all this is that when we think in twister space All the symmetries have turned into SL4 Especially when we're doing everything in a complexified way all the symmetries have turned into SL4 And it's just projective geometry There's no notion of distance. There's no metric. There's nothing fancy or complicated We just have four by four linear transformations. The only invariant symbol we have is the epsilon symbol and geometrically We we can never talk about the distance between points or anything like that the only things that we can talk about are for example Any two points are on a line Maybe you can have three points in general. They won't be on a line, but sometimes they might be on the straight line together Or if I take one line and another line in general, they won't cross Maybe they can cross each other. Okay, but so there's there's nothing about a distance or anything else, right? Just these simple questions of incidence whether whether points and lines and planes pass through intersect through each other or not In particular On this and this we'll pick up Next time, but I just want to leave it for you as something to a play with So all of Minkowski geometry all of conformal geometry reduces to just thinking about planes and points and things like that in three-dimensional projective space and and Thinking visually about geometry and projective three-dimensional projective space is exactly like standard geometry so when you say a line in p3 you just think of it as Just the line in a three-dimensional space and you will never get anything wrong when you think like that. Okay We'll have a little lightning review of projective geometry next time But but it's there's nothing. It's completely familiar geometry The only novelty is that every all lines intersect they might intersect at infinity but all lines intersect That's the only small novelty. This was worked out by people in the 1500s. So it can't be very difficult Okay, so But but the beauty of it is that you don't have to keep track of metrics You don't think about like cones, you know, you just think about lines and how they intersect each other. So just to give a Concrete example if you have two points in space time now You can ask take these two points. What is the distance between them in general? It's not conformal invariant But if the two points are Null separated What does that mean back in twister space? So now now I'm back in space time. I have a point here x x1 I have some other point x2 and Let's say there are null separated. What would that possibly mean? well Let's think in twister space. What could it mean? Well x1 corresponds to some line Here's here's the line that corresponds to one and x2 corresponds to some other line Now what could it possibly mean that they're not separated? The lines intersect that's all in general two lines in three-dimensional space will not intersect But if they intersect that means that the corresponding points are not separated Furthermore, what is the interpretation of this intersection point? That is now point in twister space. So who is that point in twister space? What should it correspond to? Or respond to a null ray. Which null ray does it correspond to? That null ray, right? The one that passes through x1 and x2 right, so this will be the basis of Our beginning our story next time To come up with the most ideal set of variables to describe the scattering process But for now, I just want to see the question of whether things are like separated gets turned into little geometry questions about whether lines Intersect and I want to leave you with this homework assignment To try to solve the following geometry problem in complexified Minkowski space, okay? Let's say you're given four points. You have four radio stations, right? You're given four points in Minkowski space a different space time and They want to send signals out and To be all four of them Not separated from the sum fifth point so So you want to they're all they're all sending signals out and you want to be somewhere Which can receive signals light signals from all four of those points simultaneously, okay? So that's the geometry question given four points in Minkowski space given four points in space time Can you find a fifth point? Which is not separated from all four? Okay, that sounds like a typical egghead nerd math problem, okay? But actually it shows up very concretely in these kind of problem and big generalizations of it show up very concretely in Scattering amplitude calculations, but anyway, you can spend a little bit of time thinking about that just in straight Minkowski space It's not that hard in straight Minkowski space either, but you have to visualize a little bit like gold spheres Things things like that. I want you to translate that problem into twister space and try to solve it in twister space So first can you find any such points? How many of them are there and so on if you're super dumb? You might think that you're solving Four quadratic equations, right? You have x5 minus x1 squared equals zero x5 minus x2 squared So if you're super dumb you might think well you're solving four a couple quadratic equation Maybe there's two to the four solutions. There's 16 solutions. There's something like that, okay? Well, that's a little too dumb. There's actually two solutions There's precisely two solutions in general But I I want you to see if you can understand what where those two solutions come from from this language and In this language it turns into a very different seeming problem given each point in Minkowski space turns into a line in space time So now the question is I'm I give you four lines. Sorry four lines in the twister space. I give you four lines in three space Think think think so four general lines Can you find a fifth line that intersects all four? Now that question is the classic very first question that started the subject of algebraic geometry in the mid 1800s in the hands of mr. Schubert and it's the first of the first classic question If you take any undergraduate course in algebraic geometry you'll run into this very very early on it's the most basic problem in the Schubert calculus, but in this case It's really extremely simple and you can visualize it and see what the answer is okay, and and if you want to think about it I I encourage you to take this picture and specialize the lines a little bit. Don't keep them totally Skew you can specialize for example to the case where two of the lines intersect Another two of the lines intersect in some other way and in those more special situations you can count to see how many solutions are what they look like and See if you can see why the answer is two What's nice is you'd see not only that the answer is to but precisely what those points are because you sort of build build the line And you get some very nice intuition for it. All right, so that's your little to a story of exercise and tomorrow we will We'll pick the story back up and finish describing the kinematics before moving to projective geometry and the Positive geometry of the amplitude. Thanks a lot