Nima Arkani-Hamed and Thomas Lam in Conversation

all right well it's really a wonderful to be up up here with my friend and collaborator Thomas lamb who I've known for five years now and I think Thomas and I are up here to illustrate yet another aspect of this of this grand structure out there this mathematical structure in the universe which is perhaps exemplified by the fact that if you were to ask either one of us I don't know five ten years ago whether we'd be up here on the stage as talking about the relationship between mathematics and physics I think we'd both tell you you're totally nuts this is definitely not something I ever imagined I would be doing with my life when I first came to the Institute around eleven years ago I just spent two years where my dominant collaborators were talking the collaborations were talking with people like file experimental particle it was something I imagined I would be doing more and more when I came here I suspect ten years ago you didn't think Thomas that you'd be spending much time talking to a physicist neither one of us are professionals at this mathematical at this interface between mathematics and physics I'm not a mathematical physicist by any stretch of the imagination of a physicist who loves mathematics that's not not the same thing and and yet here we are my main collaborators over the past five years have been mathematicians so I think what we'd like to talk about is what it is that drew us together just as a vignette as an example of life at this math physics frontier and just to talk kind of abstractly about what we're going that the points we're going to illustrate any such interaction and especially the sort of interaction that that brings people who didn't think they would be doing this to be interacting with each other I should say another interesting aspect of this is this is an example where neither physics nor mathematics is particularly had just independently it's visited mouth we found ourselves running into virtually identical objects from radically different points of view so it's it's it's a kind of an amazing and delightful shock to both of us that that this has happened and it's fruitful to a pursue but to begin with something has to bring you together and the thing that brings you together is the observation that some structure a that somehow connected describing the physical universe is connected maybe in a surprising way to some mathematical structure X in the universe of mathematical ideas and this is already interesting to try to understand why this happened and to establish this link better and better but once you see that this happens it's the beginning of a kind of fruitful interplay that you can have between the two subjects because so the physicists a never comes alone structure a as part of something is supposed to describe the physical universe and so those structures and ideas can be related to other things they can be related to bigger ideas and structures be or even smaller things see but what she knows that there's a link between a and X given that a is related to B and C on the side of the physical universe it strongly suggests that there should be something on the other side some correspondence between B and C as well on the side of the mathematical universe so that's something that that that that the physicist understanding and familiar with what the physical universe has to offer the mathematician but of course it goes the other way as well the mathematician might realize that structure X is connected to mathematical structures y&z and that suggests that there should be some analog of those connections on the side of the physical universe as well so that's what we're going to illustrate in a number of examples in this discussion and so let's just jump right into it the physics that we're talking about is very basic physics it happens in the real world all the time and it involves the collisions of constituents of the nucleus of the atom the collisions of particles like protons so this happens gajillion of times a second with protons in in cosmic rays banging into protons in the Earth's atmosphere it also happens when we collide protons at the Large Hadron Collider a picture that you've seen a number of times already now protons are not elementary particles they're matter of smaller constituents called quarks that are held together inside the proton by imaginably named particles called gluons and so what we're really interested in when protons smash into each other is studying the collisions between these underlying sort of point like elementary particles the quarks and the gluons and this is something that's actually practically important to do for experiments at the at the LHC in order to compare the predictions of theory and experiment now there is a time-honored way of proceeding to figure out what happens when you collide let's say two gluons inside the proton do going on inside the proton come along all sorts of things can happen but you might be interested for the probability for two the ones that come in the probability or the so-called amplitude for two kula wants to come in and three gluons two a go out and what you're supposed to do it you learn in textbooks is draw these little pictures that go back to retro Steinman that represents all the different ways that two cool ones could have come in and hit each other and produce three out this way that way that way millions of different ways and you have to add them all up together a crucial element of these pictures I want to emphasize is that these blue one sort of come in and go out actually the experimentalist don't see real blue ones coming out we see jets of strongly interacting particles going out but anyway this is the sort of basic underlying process but inside here are things that that are sort of more of a theoretical fiction we call them virtual particles they're not things that sort of propagate in and out from long distances they're virtual particles that occur as intermediate steps in these calculations and the presence of these virtual particles means that despite the fact that the rules for writing out each one of these pictures is incredibly simple the final answer turns out to be tremendously complicated looking naively anyway so a result of a brute force calculation for this process it's kind of 30 complicated pages that looks like this and then around a little over 30 years ago some theoretical physicists realized that in fact if you add up all of these things the final answer turns into just one a single term I'm not going to explain what these symbols mean they're representation of the energies and momentum of the particles involved but everything collapses to a single term and so something many people have been trying to understand for a number of years is where this simplicity comes from why is the answer so much simpler than you would have naively expected and you go back to the kind of simplest possible processes that could involve these gluons so you imagine the very simplest possible interaction between elementary particles is were three of them meet at a point in space and time the gluons have a property called spin or polarization and so there's two kinds of ways that that three of them can meet where two of them have negative polarization or negative olicity and one as positive or the other way around these turn out to be incredibly simple and so you can just begin by asking if I instead of imagining I have virtual particles I forget about virtual particles but I glue these most simple possible processes together where I eliminate the idea of virtual particles and just play around what would happen and you start playing around and you realize it if you glue these little pictures together with black and white vertices just this single picture just a single picture which is eliminated the notion of virtual particle gives you that simple final answer instead of the sum of 30 complicated pages of algebra so that's something that a number of us ran into these sort of fascinating pictures with black and white her disease and the beginning of the interaction with our mathematician friends is that they had also seen these pictures in another setting and maybe Thomas could tell tell us something about that yeah so let me first echo Nima and say I'm not trained as a mathematical physicist I'm I'm a combinatorial list which may be unusual for this room I think Karen Willem back said many mathematicians don't understand quantum field theory or find it hard to understand now I'm one of them so so let me say something about my area to explain a bit about combinatorics this this picture gives a very simple one of the most basic common Ettore objects so this is a picture of a permutation so a permutation years is a reordering of a bunch of numbers so in this case the numbers went up to five so we can think of it as over here they're ordered one up to five and then we send one two three two two five and so on and that's a permutation it turns out that you can get a permutation from one of these pictures that Nima described so one of these pictures were black and white dots so so the rule to get this permutation is you label the sort of external legs with the numbers 1 up to n in this case 5 and then you have to follow you follow a path through this graph and you follow these things called the rules of the road the rules are if you're at a at a black vertex you always turn right and if you add a white vertex you turn left no I'm so if you follow if you follow these paths so this purple this purple path starts at 1 and turns right at the black vertex left at Y right at black left it white and it goes to 3 so the first part of data in the permutation is 1 goes to 3 and you repeat this for every vertex and you'll get a permutation so this is this is a basic thing in combinatorial that sort of relates to it relates to basic things in cognate oryx permutations and these by colored graphs although actually this particular construction is actually not not that old it's maybe the last 15 years so one thing in combinatorial set we we do with such a by colored graphs or a picture with these black dots and white dots is we want to build a space out of them and the way we build a space out of them there's a there's an intermediate step which which is that from one of these pictures we'll build a matrix and the analogy the analogy is suppose I suppose I've got a I want to describe a triangle and I'm trying to describe a point inside the triangle then we often think of the point as some sort of combination of the vertices of the triangle so it's like maybe a third of this one half of this one and a six of this one um and we can write down sort of working working projective ly what we what we think as we draw we draw sort of coordinate axes for each of these vertices and we think of think of this pink purple purple point in the center of a triangle as some combination of these coordinates and in this case one seven thirteen some kind of combination and the what these by colored graphs do for us is that they allow us to generalize this construction to something called the positive cosmonium which is some similar kind of construction where instead of getting just a row of numbers we'll get this matrix of numbers and the rule the rule is extremely extremely simple but also surprisingly recent define only very recently last last 15 to 20 years years you you take a matrix and you just ask that all them all the determinants are positive so 1 so 1 times 3 minus 2 times 0 is positive 2 times 4 minus minus 5 times 3 is positive and so on and this describes some kind of space um and and and there's a non-obvious question which is what does this space look like but if you did this if you did this construction with sort of 1 by something matrices you would get a you get a triangle or some simplex okay so so Nima explained that in this basic glue on scattering picture there were two simplest kind of interactions which correspond to this black dot with sort of three three legs and another one with a white which is a white dog with free and sort of so these are our most basic objects and and sort of by definition we assign to the black dot this particular matrix which is which is sort of has two three parameters X and y and you can think of this as a triangle this and assigning to this white dot we assign this 2x3 matrix also with two parents there's an also a triangle but I do kind of triangle and then there's a there's a game we play with when you give me this big graph there's a way to glue these glue as we glue the vertices together we can also glue the matrices together and build a big matrix and - and this is the an example so these ABCD are sort of parameters assigned to the edges here that written so we can glue up big matrix you know out of this graph and one thing to quickly say is that the permutation the really basic combinatorial operation I said earlier which is assigning a permutation to a by coloured graph this is actually an operation that's probably familiar it's it's essentially giving the information of how to do role reduction on this matrix so when you apply row reduction on this matrix you get someone's in certain positions the pivot ones and the location of those is equivalent to the information of this permutation in technical terms it's it's the the permutation describes the bruja stratification okay so you might think that that that it's just sort of coincidental that both sets of people find these funny graphs with black and white vertices but in fact as we as physicists and then in interaction more and more with the mathematicians studying these things realized the proper way to do this physical calculation that actually gets you these simple final answers the really correct way of understanding physically how to move these things together directly leads you to this way of gluing should be thought of this particular way of gluing the basic matrices together which have this positivity property into a more complicated matrix that continues to have that any property so these things aren't they're not capriciously related to each other they're deeply related to each other they're in fact end up being exactly the same thing so this is an example of the structure a related to glue on scattering in the real world out there connected to something rather mysteriously connected to combinatoric sand these funny spaces that are generalizations that of triangles into into somewhat more abstract places so that we've established our surprising link between two worlds but now we can give an illustration of the example that that that that the physicist knows that there has to be more going on and that's if we go I gave you the simplest process with two particles coming in and three particles going out if it turns out if it turns out if you go all the way to the next most complicated process well let's say three particles going in and three going out or two going in and four going out that that is not represented by a single one of these pictures so single one of these pictures are associated with simple geometric spaces but the actual final answer that you get in leading order of approximation ends up being the sum of two pictures like this now to our mathematician friends this plus sign didn't make too much sense it was not so obvious how to combine these two different pictures together into one bigger object and yet it's physicists we knew there had to be something there because physics cares about the final answer only and not about a particular way of breaking it up into more elementary pieces so that's why we were confident that something had to exist to extend these geometric ideas to something larger that kind of took as elementary building blocks these little pieces that we had before but somehow glued them together and and after a while a sort of picture began to emerge that we should think of these total processes as being associated in a sense with the volume of some geometric shape that's determined by the data specifying the energies in the momenta and the helicity z-- of these gluons so you give me the energies of momentum the Felicity of gluons I draw this geometric shape and I need to compute its volume but you see one way to compute its volume is to break it up into two simpler pieces that little tetrahedron at the top and this little tetrahedron at the bottom and each one of these turn out precisely to be associated with one of these pictures and in fact you could do it another way you could do it in the sum of three tetrahedra going around and that would be yet another representation of the answer in these interesting terms so that suggested that there had to be some general some generalization of these of this idea of the positive rest manya and indeed just like triangles could be generalized into this more abstract space in the way that thomas described to give us the positive respondent then there's another natural generalization of the notion of a triangle you can sort of glue triangles together into polygons or you can glue tetrahedra together into into into shapes that are called polyhedra or polytopes more generally and if you extend it if you generalize these ideas into this more abstract space in exactly the same way as we did with triangles then you get an an interesting new mathematical object and there's one more procedure that you have to do to it in order to get yet another geometric object that incorporates the more subtle effects of quantum mechanical of quantum mechanical corrections to these processes so this was an example of something that was that was new to the mathematicians I think maybe Thomas you can say something about at least one how natural one part of this generalization actually is right I mean if we so this picture here going from the triangle to the Pentagon you can get any end gone out of the constructions they have extremely natural to Nima and in physics but back in sort of in Greek mathematics there was a study already the study of sort of three-dimensional solids for example things like tetrahedra cubes octahedra there's a there's a natural higher dimensional version of these objects and it's curious that the constructions constructions from physics only leads to sort of one particular one particular kind of shape in sort of high dimensions kind of a positive shape and there's a there's a natural there's a natural analogy to search for to search for different different shapes in in higher dimensions which are analogues of these ones motivated by physics as far as I know these grassman polytopes them we don't know what physics goes with them yeah and I think I'd say that this is another example of this phenomenon this this object I think is extremely extremely natural I think it's a complete historical accident that we ran into a version of it before before the mathematicians that but here - physics keeps going there is an there's a further richart's extension of these things which I think still is not something so obviously you guys I've talked about but but having started exploring it a little bit together they seem to be interesting let's talk about another set of examples illustrating similar themes but in in these examples it'll we'll see more of the inspiration from existing structures in mathematics telling us to look for corresponding structures in physics and actually a lot of back and forth so we'll just really go over this part a little more quickly this is also to just give you a flavor of what we're actually doing this is related to things that Tom Ennis Thomas and I have been working on just over the past year or so while he's been visiting us here at the Institute so let's go back and talk about another set of very basic processes now not involving the gluons inside the proton but involving other kinds of particles actually very very general sorts of processes but for the sake of this discussion you might imagine it's sort of scattering that move all the Higgs particles or other particles that we're familiar with in particle physics called a pilot and one very basic aspect of what these collisions look like is that special things happen if you have sort of two particles going in and four particles going out then a physicist is very used to thinking about the more special situations where the way this happened is that really secretly four of them came in and made a fifth particle that propagated some distance before decaying into two of them or three came in and produce a middle one that propagated some distance before decaying back out the three of them and then you could keep going down and seeing the same thing happening here so these four things producing a fifth while something more special could have happened by realizing that that occurred when first to produce a third one that the Cait to this that the Kaede to this guy and that guy that decay to this guy and that guy that propagated some distance and decayed in this way so this way of organizing more and more special processes is something that sort of mothers milk and bread and butter for physicists but something that I think none of us know physicists certainly expected is that exactly these same pictures can actually be organized in a totally different way and this is something's been known to mathematicians since the 1960s don't you want to say something about this picture Thomas yeah so associate Hedra is this is this again again polytope where you have in this this free dimensional object where you arrange all these all of these graphs as vertices and the edges joining them if they if they differ just by this kind of local transformation where you have you take an edge you contract it and then you expand it out in a different way and you draw all these edges it's interesting that when this was first discovery was discovered in in topology it was described just as sort of as a collection of adjacencies rather than being embedded into space which I think's makes it interesting for the next slide yeah and just just say again I mean the fact that you should care about how things are related by these by going from a general to a more special process is definitely something that physicists care about but it never occurred to us to imagine drawing these things in a shape and even seeing that they're realized as as some that this network of relationships of more and more special ways in which these complicated collisions could occur can actually be organized in this in this geometrical way now this as the seda has been known just abstractly as a graph like this has been known since the 1960s but but once it sort of penetrated our heads in the correct way and given the story that we just saw with the collision of gluons it was clear that there should be some cousin of this that this this older and more familiar mathematical object should be describable in the same kind of language that we use to describe the story of the ample tehy Droon and gluons and that led some some of us a number of years ago to a very specific realization of this basic combinatorial shape with a you see it as this is a very specific shape some of these sides are parallel and it's not it has exactly the same kind of it has the same kind of shape and same kind of combinatorial rules as this one but as a more specific it's a more specific version of it and in fact this way of constructing this object that was motivated and inspired by physics is not exactly one of the standard ones that the mathematicians knew about even though this object has been studied by mathematicians for for quite a while but now the story goes back to mathematics because mathematicians look at these pictures that are super duper familiar to physicists and they do something else with them which is not familiar to the physicist so if you want to say something about yeah so so the basic the basic combinatorial description of these of these edges here use this move here where you sort of contract an edge and expand it out in a different way um and and this move sort of again in sort of somewhat recently last last twenty years has been um has got in the name of mutation and it's part of a big subject now called cluster algebras where here's a here's a picture of where the usually the this operation of mutation is described on this pink graph rather than the black graph and starting with some what's called a quiver on the on this pink graph um we can sort of mutate at a particular vertex and then produce a new one and this leads to a theory of the algebra switch is a which is an algebraic object that you build out of these notations and there's a whole zoo a really really big zoo of these cluster algebra yeah in fact maybe you can say something about the classification of these guys well I mean but I guess in the last 15 years or so it was realized that that that this zoo comes in two two types a sort of finite type where the algebra is finite and everything else were the where the algebra is infinite and these pictures will be extremely familiar to both physicists and mathematicians we've seen these pictures and many many other places of before this finite set already comes in two different types a kind of a general type sometimes known as the classical type in some a few exceptional cases and so so these objects exist now we already saw that there's a connection between the simplest type of of the picture and the physics of the collision of Higgs particles in this simplest possible way where you get pictures that look like trees but so now we have an example though of the phenomenon that that a is related to all these other things in the mathematical universe and so that that means that surely there should be some role for the rest of these things in physics and so you can go and look and see if it's true and indeed it's true that if you look at these other classical type cases anyway that they they are also related to the scattering process in ways that are surprising to us that we would not definitely not have thought of ourselves without being nudged in the correct direction by the existence of these cousin structures and then there's lots of open questions again going back and forth all these exceptional ones do they have any role to play in physics we don't know on the other hand in physics there are even more complicated processes and involve more and more loops here and it's clear they should have some kind of connection with the world of cluster algebras but if they do it has to involve these more mysterious infinite type ones there's an infinity there that the cluster people know and love but the animal that infinity does not exist on this side so it suggests that if there's some connection here it should give us some way of caming or at least thinking about these infinities in a new way and it just keeps going and going I'll just just give one final example something else that physicists love to do is to start with the picture of particles and actually think of this as a special case of a case of looking from very very far away of thinking in terms of strings instead so the you can imagine the particles of little loops of string and if we look at the so so this is a very natural generalization in physics well when you start with this mathematical structure on the side of the cluster algebra it has it has a very natural there very natural object associated with it actually directly give you this picture of open strings first from which you have to take a limit to go back to the picture of particles so that's extremely interesting and given that that exists on the side of the of the cluster algebra as you wonder what it means for all these other guys so they're somehow associated with the generalization of these pictures of the particles in strings that we're trying to understand on the other hand there's this doesn't stop from the side of physics we go from this little picture of open strings to a picture of little closed loops of strings colliding with each other and that that does not so obviously have an existing and existing understanding in the world of cluster algebra that in fact their existence suggests that there should be some new geometries in the land of cluster algebras and that's something that indeed we're finding and are starting to explore so this is an example of the interplay going back and forth now a number of times right where you just jump back and forth something in physics you see something in math there's something in math is connected to a few things so there should be something in physics is there yes there is but then there should be that too is that there on the other side this exists that doesn't exist and so we just pop back and forth back and forth and this is all about some extremely basic physics and as I hope you see also some extremely basic mathematics and this is a sort of force of of this mathematical structure out there in the universe that brings people like that non-professionals like thomas and i into a collision on this kind of in this kind of area I just want to end with a with a few general a few general remarks first maybe Thomas you could you could you could share your your sort of broad perspective on this interaction between physics and the combinatory it's into the in historical art of your own subject yeah so so I I work in what I think of as the structural side of combinatorics so algebraic combinatorics geometric combinatoric topological combinatorics and what's surprising to me is how broad this interaction is and how many different parts of the community works that I study actually appears in in these stories that we've been talking about so we've already mentioned polytopes which is one of imeem goes back to the Greeks and another subject that appears yes Shuba calculus which comes up because we have these matrices which describe points in ingress manion's and elm and this leads to a geometry of Shuba calculus but also also other facets that come up are extremely new parts of combinatorics so two of them we've mentioned really really quite new sort of movie in the last twenty years a cluster algebras were discovered around twenty years ago and it's it's coming up though it's also cluster algebras appear in in physics in in a number of places and then total positivity which really sort of developed in combinatorics the definition has been around for a long time but total positivity came in sort of really developed in combinatorics in a lot in the last twenty years so it's really exciting to see that actually a lot of old old ideas and a lot of new ideas coming together yeah and I'd say it's it's not obvious where these ideas that fit in the in in the in the world of physics actually we keep seeing them in various isolated places there are some how deeply related to the very basic things about physics there they're deeply related to processes and phenomenon that we normally talk about by making the principles of space-time the principles of quantum mechanics completely manifest but their way of talking about those principles and phenomenon where in in somehow a different way using more more primitive underlying structures and just I'd say in the last sort of five years or so we're seeing a few more examples here or there of the same kind of ideas showing up in very different in in very different parts of physics as well and I'd like to just end by just saying a few things about what the day-to-day interaction looks like between physicists and mathematicians there's a kind of a cliche about how fizzes and mathematicians talk to each other that that physicists are these non rigorous people who don't sort of don't care about proving anything and on one side and mathematicians are obsessed with rigorous from Anousheh on the other side and I've found this to be not true at all I've found this not to be characteristic of our interaction at all in fact that the sort of data that the culture is extremely similar I think there's there's one interesting difference that's that that's that's maybe worth saying a little bit about which is that mathematician well once you've figured out there's something interesting going on you're trying to establish a more of a connection between the subjects pretty soon the mathematicians are interested in giving precise definitions to the kind of objects that you're talking about and the physicist is nervous to do this and resists making precise definitions of anything partially because there's always something bigger that that that's going on in physics and so you're you're nervous about committing too early to some some final definition of something but this this difference in attitude is a source of a kind of a fascinating creative tension between us in in collaboration maybe you can say a little bit about what our attitude is good for for you guys and okay yeah absolutely so so Nima likes to keep things fluid and and well well I like to know node on the definitions precisely right away and and I think I think physicists Elam that I've talked to they're very sensitive to interesting mathematical phenomena so so when I talk to NEMA he knows that something's going on he can sometimes he can't sort of formalizing in mathematical language exactly what is going on but there's something interesting going on and so when I talk to business I think is extremely fruitful when we discuss phenomena that there's no existing mathematical formalism which is actually the case for some of these new kinds of politics that came up earlier they're the formalism doesn't didn't exist but the phenomena were there and I think that's the part where I think the interaction between physics and mathematics is extremely sort of useful and also it's something that surprised me I had the sort of physicist chauvinist pig attitude about the relationship between our subject but something that really surprised me over and over again in this interaction is the fruitfulness of coming up with precise definitions and one reason is that that the other reason the physicists to resist giving things precise definitions is that were a little bit scared of very singular situations if you give a very precise definition as to work at every possible place in every possible setting and physics is filled with things that always go wrong and somewhere or another if you're too too precise and yet it has happened over and over again that actually committing and and giving precise definitions for things was helpful in fact crucial in many cases for understanding the effects all right so I think we can leave it at that Thanks thank you [Applause]

Channel: Institute for Advanced Study

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Length: 36min 47sec (2207 seconds)

Published: Thu May 30 2019