Nima Arkani-Hamed | UV/IR and Effective Field Theory

well i'm really delighted to be able to join um uh uh you um even if virtually this really seems like it's been a terrific uh uh series of talks and um i'm talking about the sort of general subject of uh the uvir connection and effective field theory um and uh uh many of the things that i'm talking about um are are contained in sort of two papers that came out in uh december uh one with the uten and jimmy huang uh on the so-called eft hedron um and uh and another one with monica pete anna maria roccario and andy strominger on um on celestial amplitudes and its connection uh to uh uh to the to the uv uh i'll focus on the aspect that has to do with the connection uh to the uv and um i should uh i mean i'll preface uh all my remarks um by by saying that uh you know since uh since my earliest days as a graduate student i've been a red-blooded effective field theorist like almost everyone who came up in the field thinking about physics beyond the standard model that was the uh the the picture of effective field theory um both the sort of uh uh the the kind of roughly speaking the harvard picture uh uh associated with steve weinberg and that i learned mostly from uh howard georgia um and the wilsonian picture uh these completely transformed my understanding of what fundamental physics is really about um and so i was an absolutely hardcore sort of wilsonian effective field tourist for much of the early part of my career but starting around 15 years ago um i've been thinking more and more about things that are somehow in tension with the uh with the bosonian paradigm um and that's what i want to talk about today um of course everything in tension with the wassonian paradigm one way or another has to do with this uh with this uv-ir connection so what i want to do in the talk today is say a few sort of general things about this uvr connection and talk about um uh what i see is a number of interesting frontiers in in exploring it um so i'll talk about sort of three connected but uh broadly connected uh frontiers and exploring the the uh uh the connection between the sort of uv-ir uh entanglement uh and how we think about effective fieldfood but let's just uh begin uh with the sort of glory of this wilsonian paradigm um is the absolute decoupling of uv and ir physics and you know uh the slogan here is that you need a microscope to probe short distances um so if you try to figure out what's going on uh uh with physics uh on a scale with separation between two points um well when that separation between the points from the distance when x minus y squared is tiny uh you need very high energy you need a microscope to probe short distances and uh and the standard picture of effective field theory sharpens this and turns us into a powerful engine for both thinking about physics conceptually as well as doing calculations um but i want to stress two aspects of this uh first this picture is fundamentally euclidean uh and everything that we're talking about here this notion of short distance is a short euclidean distance and secondly the kind of uh at least some of the origins of this picture from wilson's thinking um came out came from uh condensed matter systems and uh and both of these things are in tension with the real world um uh a slogan that i like that i've said many times is the world is not a crappy metal uh and there are a number of aspects in which it isn't the crappy metal uh first um and most basically uh the physics is laurencian and not euclidean and even in field theory uh even in field theory where in principle of course we can get the laurentian answers from analytic continuation from euclidean space we've seen over the past 15 20 years of course going back further to uh dispersion theory but sort of picking up again around 20 years ago uh that the laurentian aspects of the physics um uh highlight things that are much more obscure and difficult to see from the uh euclidean point of view for example uh uh in laurentian's signature you can have two points that are close to each other quote unquote close to each other where x minus y squared is close to zero but that close distance with x minus y squared uh close uh approaching zero can actually be probed at macroscopic scales as long as they're close to the light code so um uh one place where this uh shows up is in the bounds on the coefficient of higher dimension operators that we can have in ineffective field theory again in the standard picture of effective field theory the only way you can probe uh or the the most effective way of probing higher dimension operators is if they break some symmetry of the low energy theory and if you had a situation with like just a a scalar field with a shift symmetry um and a leading interaction like some constant d phi to the fourth then that doesn't break any symmetries that's like a garden variety higher dimension operator so you think there's nothing much you can say about it and and that would be at least naively true in the euclidean picture but uh but we know that that's not the case when we think about the actual lorenzian physics and the reason is that uh there is something which is critical about the two derivative theory and the what's critical about the two derivative theory is that uh signals um propagate exactly at the speed of light and so uh so the the two point function is exactly supported on the like code and therefore if you turn on some homogeneous background for example a background where phi dot is not equal to zero then it's possible that uh that you can have corrections to that they can either push you inside or outside of the light cone and you can be sensitive to it from a macroscopic experiment that you can sort of measure a time advance or delay at humongous distances and yet be sensitive to the coefficient of a higher dimension operator and then that's where we learn for example the coefficient of these higher dimensional outputs have to be positive in order to avoid super luminality and and and we can also see that that positivity arises from uh dispersive arguments which i'll review in a moment uh quickly and talk about uh extensions but that's the the first crucial thing is that the physics is laurentian and so this sort of decoupling between short and long distance that's sort of trivial in euclidean space is more subtle in uh laurentian signature secondly the world is gravitational and this is a much deeper sense in which in a very famous sense in which there's the ubir ir correspondence and there are many aspects of this but one of the simplest and most famous ones to talk about is what happens when we collide with particles that energy is much bigger than m plunk in a garden variety non-normalizable theory like let's say the weak interactions um before we knew about w's and z's if we just imagined uh um what could happen when we scattered electrons and neutrinos against each other and energies a thousand times above the fermi scale we'd have no idea what happened we really need the sort of uv completion in order to be able to even roughly say what would happen if we did those collisions but that's not true when we have gravity and that's because that energy is much much bigger than the plug scale we actually produce a large black hole and we roughly know what happens that large black hole evaporates by hawking evaporation so this is kind of an amazing thing that the hardest to understand uv series uh gravity uh actually at zeroth order we know what happens you just produce big black holes and they emit soft radiation and evaporate and so um so we know something about ultra high energy states and the ultra energy states actually become larger and larger um uh the rays these black holes become larger and larger and then there are consistency about what very large black holes have to look like there's the consistency of the lawns of horizon thermodynamics for black holes for example which then enforce infrared which which enforce uh consistent conditions that we can understand in the ir uh now um this has been understood for a long time of course but uh i think um probably most people thought that the that the uh the the novel consistency conditions associated with the fact that we know ultramassive states are big black holes would only be sort of used for very esoteric questions for example um there are holographic bounds which tell you the number of degrees of freedom grow like uh the area in plank units rather than the volume but but the questions about this and the questions about black holes and the information paradox and so on naively involve uh effects that go like e to the minus some entropy so e to the minus one over d newton um and naively i think most people thought that uh that whatever these uh roughly holographic or ubir aspects of gravity were they're only relevant for these very very precise questions or these non-perturbatively small effects and i think over the past uh 15 years or so there's more and more evidence that that's not necessarily true and there are actually things that we care about which are impacted by these uh by these uh considerations for example another famous old uh consequence of this uh of this type of thinking is the fact that there are no global symmetries in quantum gravity but the idea there's no global symmetries gets sharpened in the weak gravity conjecture to a statement that does not involve exponentially small effects a statement that says that you can't make uh sort of a u1 gauge coupling arbitrarily weak uh you wanna get gauge coupling g arbitrarily weak without bringing some kind of ultraviolet cutoff of the theory uh uh down as you hold them fixed and that uv cutoff scale would scale like g times m so that's something that actually has teeth um uh and so those things aren't aren't proven they're still conjectures but they're sort of more and more and more evidence that that that things like that are actually true uh so that's another aspect of the of the deeper aspect of the uv-ir connection is something uh uh gravitational uh so uh the the so just to stress the the uh a is not in conflict with the uh uh you know well certainly an effective field theory i mean it's a fact even it's about field theory but a is just saying that there's more in the lorenzine world of field theory that meets the euclidean eye and that it's uh and it's very it's very useful to to to to think in this uh lorenzian way b is something that just says that the whole picture of uv-ir decoupling is wrong when you have gravity ultimately very very high energies uh turns into long distances again and of course uh just to mention quickly um uh see uh that that there's another indication not just at the punch scale but perhaps at much longer distances that there's something wrong with the sort of this basic bosonian picture of the world which is the the spectacular failure of the wilsonian notion of naturalness which works beautifully you can have metaphysics of course that's where it came from but the failure for the cosmological constant and the uh and the higgs masquerade but of course it fails dramatically for the cosmological constant we don't know for sure if it's failed entirely for the higgs masquerade yet but as i like to say if you turn into a pumpkin at midnight uh given the lhc results it's around 11 45 pm for this notion of uh uh naturalness for the weeks here okay so these are various reasons um uh uh these are and and of course uh c also suggests uh at least a failure of naturalness one thought that it suggests is that perhaps there's an explanation uh uh a natural explanation for these parameters that involves some kind of uh correlation between the uv and the ir although that that's that's very vague and i won't say more about it here but uh but uh uh but uh i just wanted to give this quick mention to another indication that there's some something wrong with the uh these wilsonian ideas not just at the park scale but uh perhaps uh closer to home uh another comment um is that at least the the a and b the uh the lorenzian uh nature of physics and uh and the fact that uh that we have gravity uh those aspects of uvir are clearly related to each other for example uh having causality or the absence of superluminal propagation uh is crucially related to the uh consistency of horizon thermodynamics um it's an absolutely critical part of the of being able to talk about the entropy assigned to horizons that the notion of a horizon is universal um so everyone has the same speed of light and uh and even if we turn on interesting backgrounds in those backgrounds we shouldn't be able to have different species uh propagate uh with the different speeds if we could it wouldn't be clear which area uh uh we could have different horizons for different species and it wouldn't be clear which one of them was assigned the entropy to so protecting the light cone which a is all about is crucially related to the consistency of uh of uh of uh horizon thermodynamics and the issues in b and you actually we actually see this in many places uh um i i won't spend too long talking about this but this is one context where i'm especially fond of thinking about it again going back to the weak gravity conjecture um one way of talking about the weak gravity conjecture is the following you imagine looking at the spectrum of elementary particles and of of objects in nature particles in nature with some charge as the mass gets larger and the charge gets larger and uh and the claim is that asymptotically we have enormous very massive objects uh with huge charge but with the massive charge ratio equal to one in clock units and those are extremely black holes and if you just imagine plotting what the spectrum looks like uh so those guys have been mapped way bigger than m plunk if we look at uh we look at wider objects in the real world we have things like electrons whose mass is much smaller than our plank and this massive charge ratio is minuscule much much smaller than one and the weak gravity conjecture uh a sort of strong form of the gravity conjecture uh would actually say that if we look at this plot of the the the spectrum of the particles of mass m um on the horizontal axis and plot the massive charge ratio on the vertical axis then it asymptotes to one that's what extremal black holes are but that incident but this curve should uh be approached from below so that corrections should push you down to push the massive charge ratio down um uh that's again that's one aspect of this uh weak gravity conjecture that allows extreme black holes to decay the lighter objects uh and it's associated with gravity being the weakest force i mean these are all uh equivalent statements um but it's interesting to see how we can see uh this pushing down of the curve at asymptotically enormous masses uh there are corrections for the effective action uh to the two derivative effective action that are calculable corrections uh coming from ir logarithms okay so just the logarithmically enhanced graviton loops they give us various higher dimension operators and four dimensions that give us the f to the four operator and that's calculable that gives you a populable correction from super infrared uh physics to the mass to charge ratio and you can show that that correction always pushes you down the sign of that correction is that indeed it always pushes you down on the other hand when you're closer to the point scale uh there are effects that come from integrating out massive string modes or whatever is going on in the uv and uh part of the evidence for the weak gravity conjecture is that in every situation where we've seen what those higher dimension operators are we always find that that the signs work out in such a way that the massive charge ratio gets pushed down um so uh down here uh around damn funk or on uh and the string uh this has to do with what looks like details of uh of uh the uv completion of gravity um up here at enormous masses it's something totally calculable from the effect of field theory at long distances and we see they're continuously connected to each other so that there's there's a basic fact that this curve is pushed down but you ascribe this fact to something which is calculable effective field theory generating operators of the correct sign so you can't get super luminality um at asymptotically enormous masses and you ascribe them instead to these uh perhaps stringy higher dimension operators that come from consistent uv completions of gravity um at uh at masses uh in the neighborhood of empire but they're continuously connected to uh each other okay so so i think um there is no to in my mind there is no very sharp distinction between the issues a and b uh and they're and they're continuously connected to each other actually part of the reason they're continually connected to each other is precisely the uv uvir correspondence that tells us that these enormous masses we are talking again about long distances where we can calculate many things by factor field theory all right so that's uh that's a rough a summary of um uh of of some of the uh sort of standard famous ubir uh connections what i want to talk about um in the rest of the talk are uh three frontiers in the uvir eft uh connection uh first is an extension of this sort of basic idea that we alluded to a moment ago that there are positivity constraints on the coefficients of higher dimension operators and effective field theory coming from causality um uh unitarian causality dispersion relations and so on okay so um now uh this is a subject that uh i think over the past uh few years there's been more and more papers about i think i've seen that there's been a number of uh talks just in this series uh on this general kind of theme so i won't go through this part in uh in great detail um what i want is what i'll do instead i mean i'll give the sort of basic framework um i'll review the sort of basic framework where these things come from what i want to stress instead is the geometric aspect um uh to this problem uh that there's a there are two origins of positivity um that we standardly talk about the positivity of the energies and the positivity of probabilities uh and these two positivities and causality um uh force the coefficients of the higher dimension operators in effective field theory to actually lie inside a certain geometric region and i want to stress the geometric aspect of this problem this geometric region we call the uh eftahedron um and i'll so i'll tell you just something about the the basic non-trivial facts that give you this geometry and just flash some of the results but i i i suspect that these things are are probably more familiar uh to the audience of this lecture series next i'll talk about um another aspect of the uh irub connection um that that that that that theories with interesting infrared physics um are precisely the ones that are difficult to uv complete in a simple straightforward way uh and and if we ask this question about uv completion just a tree level it very naturally leads us to thinking about strength um and uh so so this is a second interesting aspect of the ir uh uv connection um and uh and here uh i'll i'll i'll set up a question to see whether we can sort of discover uh strings from the bottom up as the unique answer to the question of what could tree-level uv complete gravity um uh it's a very well posed question and i'll give some uh first uh explorations in in this in this direction um the problem is uh as we'll see that the problem is is is beautifully posed but is uh uh but it's a bit more non-trivial than you might think at first but i still think it uh uh that this is an area where probably a lot could be done um also in part one i'll i'll sort of uh sketch what we know about the uh yep the hedra but i also want to stress what some of the important physical uh uh ideas are that we still need to incorporate to do even better than than we're doing in part one and finally in part three um uh i'll i'll switch to uh talking about uh what looks like a different subject but actually comes from uh uh very similar motivations um we'll talk about uh uh the so-called cholesterol scattering amplitudes and so um these are scattering amplitudes that that are uh that are the most natural if you think about uh a theory living on the uh celestial sphere that surrounds any point in minkowski space um and uh and the the physical point is that uh celestial scattering amplitudes are not scattering uh amplitudes in them in momentum eigenstates like we normally think about but are actually it's natural to think about scattering boost eigenstates uh and uh scattering boost eigenstates completely destroys the wilsonian paradigm and i'll give you some examples of that um i think this is extremely interesting because uh uh i'll give you some examples of that i know also some intuition about what's what uh what fossil amplitudes look like and how we can translate things we know about effective field theory um uh into properties of uh celestial amplitudes but the sort of zeroth order point is that when you think about scattering loose eigenstates you can't even go to a low energy limit there's no such thing as a low energy limit because you're scattering things of arbitrarily high energies and uh and if we want to figure out how to go beyond the smithsonian paradigm more sharply uh it seems like a good idea to begin with a situation where you don't have the comfort of being able to go there to begin with um and that's what scattering boosts eigenstates does and i'll spend some time telling you again some of the most basic things uh about that all right so that's the plan uh for the rest rest of the talk and so um let's get going okay so first let's talk about uh iaf dahedra um and uh and it's been long appreciated uh going back to the 60s that locality and causality and unitarity imply um uh analyticity properties for scattering amplitude that's the uh that's the it's really the hard part of the s matrix program was figuring out what does causality look like um you know it's not obvious if you just measure amplitudes far away at infinity it's not obvious what property of the amplitude encodes the fact that it came from what you can ascribe to causal evolution in the interior of spacetime and from uh from analogies from what we know about sort of two-point function propagators what we learn about injections um uh for index of refraction and so on we know that it's somehow related to the uh properties of the amplitude analytically continued in the complex plane but what precisely that meant is still not was still not perfectly understood by the s matrix theorists in the 60s we still don't exactly understand it today i think this is an amazing both embarrassment and opportunity that we don't know precisely what causality means at the level of the analytic properties of uh amplitudes but we don't roughly need some kind of analyticity for amplitudes and uh and uh a unitarian means some kind of positivity of a probability so it's been long understood uh that at least the arrow in this direction has been a long long understood now over the past 10 years or so we've been seeing something like the opposite of the s matrix program um which instead of trying to slavishly derive the rules of uh it slightly drives the amplitudes by imposing causality and unitarity um on the result um which didn't work anyway uh in in this approach we take an opposite pack and we ask whether there's some kind of question that lives directly in the kinematic space that defines the amplitude just in the space of end null momentum for example if i'm scattering in particles and and we've been discovering that there are some fascinating geometric structures that live in this kinematic space uh they are associated with the words amphitheatre generalized associated cluster algebra polytopes and so on and so forth um uh and these have an abundance of life of their own they have a definition in life of their own um and you learn to ask particular questions of these guys and the answers end up being interpreted as amplitudes that have locality and causality and unitarity so it's not like locality because reality united are inputs with their outputs from these more abstract very simple but more abstract structures that live in uh kinematic space from this point of view the word positivity is incredibly important again and various kinds of positivity and notions of so-called total positivity which involve looking at matrices where all the determinants are positive for example that's a very common idea that arises that's the kind of star and uh and these physical principles like locality causality unitary or outputs from that so we've been seeing that more and more examples and having seen that um we are motivated to go back to these old considerations uh to see if there's any more positivity there after all this in this second way of thinking the positivity some a different kind of positivity is is more fundamental and it gives us these other things which we normally associate with giving us positivity to begin with so really motivated by the existence of these positive geometries we went back to see whether there was somehow more positivity than met the eye in the in the story of a factor field theory and indeed there is there is infinitely more hidden positivity and as a as i mentioned in a in a sharp sense all the coefficients of higher dimension operators in um uh at least in the simple case of two to two scattering that we're just about to talk about actually have to live inside uh an analog of these geometries that we've been uh seeing elsewhere um uh uh that are associated with strikingly similar notions of positivity um that that that are these uh efd okay so how does this work uh so i'm just going to give you the sort of basic structure of the uh of the uh uh of the argument as i said uh there have been a number of talks on this type of thing in this series so uh so i will really restrict myself to um talking about uh uh what the origin of this uh of the of this geometry is um but to begin with we we talked about the a basic process a b to a b scattering and imagine uh and and uh the first important point is that this has a dispersive representation so uh uh just for simplicity to begin with imagine that we're just integrating out massive particles it could be a tree level it could be that loop level could be strings okay but we're just integrating out the massive particles and we're not looking at the massive loops in the low energy theory yet um in this uh discussion then that means that in this approximation uh the the the amplitude is analytic around the origin uh and and gets uh cuffs uh uh massive cuts or poles far away now um the crucial point is that uh if we fix t uh to be much smaller than m squared then uh then it's very it's easy to argue and you can argue this uh both a little bit more abstractly and fancifully by looking at landau equations or even very directly by looking at the schwinger para showing or finding loop integrals that um that t doesn't have to be zero you don't have to go directly to the forward limit you can already just keep t much smaller than m squared and all of the all the branch cuts here are associated with physical particle production okay this is not true for general t if t is big then there are there are there are you know infamous things like anomalous thresholds and even worse things that can happen those are branch cuts whose discontinuities are not associated with physical particle production um uh so that's that was another one of the big problems of the s matrix program was how do we determine those okay but we're not talking about that we're just dealing with two very small compared to m squared um uh so first of all the analytic structure in the s plane is simple and uh secondly you can argue that the amplitude is bounded basically by s squared um now in detail it's a little bit different if you have a theory with a gap versus if you have a gravitational theory i won't go through this in uh in in detail um but uh but uh but the intuition is that since we're close to the forward limit um uh we don't know what the the analytic properties of amplitudes are for totally general s and t but close to the forward limit it's close to a two-point function and for two-point functions we do know for two-point functions we know that uh if i think about the two-point function in frequency space causality in in the time domain implies that the fourier transform is analytic in the upper half plane in the fourier domain so we do know something about the electricity and boundedness of uh of uh of amplitude from causality and unitarity for two point functions and that's inherited for a statement about the uh the four point function again as long as t is small enough okay so the amplitude is bounded um and uh and there's a somewhat different argument when when uh when when you have massive gravitas gains and the amplitudes are dominated by it by gravity but interestingly the final answer is the same the amplitude is dumb is bounded by s squared and that allows us to write using cohesive theorem a divergent relation for for the amplitude again t is fixed and we think about it as a function of x okay so here it is uh the fact that it's bounded by s squared means there are these two subtraction terms i won't uh i won't talk about them anymore but here is the interesting part we have a sum or an integral over a spectrum and then i just have a partial wave expansion so the sum of uh with some weights of p s of m squared and some spins s um the legend polynomial or in general dimensions they're called giganbar polynomial of cos theta that are now rewritten in terms of uh s equals m squared and the propagators okay so and here i've written things in principle we could have the s channel and the u channel okay so uh so uh the bottom line is that uh is that uh fixing t uh causality uh fixing t we have the simple analytic structure where all the branch cuts are controlled by uh by particle production um and uh causality tells us the amplitude is bounded by by s squared as you go to infinity that allows us to write this dispersive representation and this defensive representation has two important positive numbers in it one is these probabilities are positive ps of m squared is positive two the spectrum is positive the m words are a positive okay so what we'd like to know is what is the implication of these two positivities on the expansion of a f and t uh at low energies okay so um so the effective field theory expansion is just uh is just expanding um uh uh expanding this expression at low energies uh in this way and we have these coefficients a d comma q um uh and uh and so let me just organize all of them in a table a 0 0 a 1 0 a 2 0 and so on so uh the first number is the kind of dimension of the of the coupling the second number q is uh the sort of expansion around the uh uh around the forward element okay so we have this infinite table of numbers that come um that we have a different table of coefficients uh and the claim is that these infinite uh set of coefficients are actually forced to lie in an interesting geometry okay um now this this sharpens uh some some intuitive things that we expect uh perhaps just from naturalness on the coefficient of higher dimension operators in effective field theory for example you would expect that if the coefficient of dimension sticks operators suppressed by a tv the conversation of dimension eight operators the ten operators should not be suppressed by the planck scale you would roughly expect that they're all suppressed by the same scale um uh that would lead you to expect that there are interesting nonlinear relations between the couplings um uh they have to be that in order to have uh just by dimensional analysis have to be they have to involve products of operators of different dimensions um in order to compare them with uh each other but you would expect there are certain non-linear constraints on the coefficient of higher dimension operators now you might think that the consequence of naturalness and perhaps you could find some uv theory and fine-tune in some crazy way in order to make it so that a operators are suppressed by a tv and the dimension eight operators are are suppressed by the by a million tv but we're going to see that there are very precise things that you cannot do okay so there are there are sharp things there's nothing to do with the naturalness um there are just fair precise things that you that you can't do um uh and similarly the uh uh we can compare uh operators of the same mass dimension with each other and here they're just linear relations between them that you would expect they all have the same mass dimension but there are linear relations uh uh between them and in fact these are just two extremes of a general story that mixes these linear and non-linear constraints between the uh uh between um uh on on on on these coefficients okay okay so um now uh so so so that's that's the claim now uh just to give you a zeroth order idea where these things uh come from um uh every time in physics you run to this uh situation where you have some interesting physical quantity that's represented as a sum of objects with positive coefficients every time that happens to you you should think the word polytoke okay so that's uh that's what's going on in all of these in all of these uh cases um so here's just a quick crash course on on on thinking about the polytopes so here's the sort of simplest example if you imagine you have a bunch of vectors v1 to vn and you want to know what all the points are sort of in the interior here of the sort of convex hull of all of them then i would write this point a as a weighted sum of all the vertices okay w1 v1 plus wn vn over the sum of all the w's where very importantly the weights are all positive now it's actually better to think of this picture projectively and actually often in physics this is where uh how it's actually given to us um that that means that i make a one higher dimensional vector out of this by putting a one upstairs so so a is now one x and these vertices i also put a one upstairs here and i think of now everything up the overall uh rescaling then if i do that then all the points a are just of the form w1 v1 plus dot dot wnbn and i don't have to divide by the sum of the w's that's uh that's implicit in this uh in this uh projectization so i go from having this polygon in this example to have a cone in in in one higher dimension i have to live somewhere in that cone and if i slice the colon with any plane i'll i'll get the the picture of this polyp okay so that's often the way of some problem is handed to us we have some some interesting object that is naturally written as a positive sum um uh what's what's uh non-trivial is we'd now like to sort of characterize all the a's that are of this form for example you had me in a i want to know is it inside or outside um and from that uh perspective this definition in terms of the uh in terms of the convex hall is not very useful what you want is a dual description of the uh of this uh of this polytope which in the polygon is very familiar you want to find what are all the faces of the polytope and we can cut out the entire polytope by imposing a bunch of uh inequalities okay so um so that's the non-trivial problem when you're learning all the vertices uh how do you determine all the faces and uh um uh because it's the face description which uh concretely tells you how to cut it out uh with inequalities and lets you check easily whether a point isn't is inside or not and um and it it's a very simple fact about linear algebra but i don't have time to uh to uh prove here um uh but again uh this is this is this is the first thing you do if you ever run into this kind of problem is once you figure out what all the vertices are that the facet structure is completely captured by the pattern of of positivity or zeros of the determinants that you would make by uh uh by by putting together all these uh vertex vectors this va1 through vad roughly speaking if va1 if va1 vav is positive it tells you that a1 is on one side of the plane spanned by the other one if it's negative it tells you it's on the other side if it's zero that tells you that they're all uh they all lie at the same time and so that pattern of pluses and minuses and zeros completely captures everything that you want to know about the facet structure and you can actually determine what all the facets are once you know all of these signs now there's a very special class of polytopes in any number of dimensions which actually naturally generalize the polygon um uh which are actually have an ordering so imagine the vertices have an ordering they're ordered in a natural way and uh and the pattern of signs is especially simple is that all of these uh all these uh determinants are actually positive so long as the columns are ordered when that happens the matrix is set to be totally positive or in another language to live in the positive grasmania uh if i look at the convex hull of the vertices that look like that they're a remarkable polytope that's known as the cyclic polytope and uh the stickley polytope in turn is the sort of first simplest example of this object uh that's uh relevant for gluon scattering amplitudes and maximum super symmetric theories the the the uh k equals one amplitude um in general the amplitude is a fancier object with curvy boundaries but the very very simplest case is actually polytopin of the same gun so so this is a very canonical and basic and interesting object and uh and it's a it's a beautiful fact about it but that we actually know the equations that cut out all the faces ahead of time okay so and it's the following very simple pattern that uh that uh uh that says that i consecutive vertices v i v i plus one v j v j plus one v k v k plus one is the determinants of all of these guys together with a being positive this is linear in a and so it gives me the the equations that cut out uh uh all the constants of the single equality all right so so that's a very general thing that you can do um uh if you have any problem where you're taking the the convex solve a bunch of the config of a bunch of vectors um you should figure out what the vectors are you should look at all the all the signs of all the determinants and that tells you what the facets are the surprise in our particular problem see our particular problems of the same sort we're gonna expand uh we're going to expand this expression in t and in s okay if we expand it in t and s clearly all these coefficients are going to be written as a certain linear combination with positive coefficients of something okay um uh and the surprise is that both in the propagators as well as in the gigan bowers uh there's a hidden total positivity there's a structure that has all determinants positive and therefore ahead of time we know what all of the uh what all the faces are and so we know what all the inequalities are that uh cut cut this guy out so let me give you uh uh just uh just a a a flavor of this um so let's say uh i'm going to do just just just to illustrate a sort of simple toy example first to illustrate some of these nonlinear constraints imagine that we had a function that was just the sum over m squared with some positive weights one over m squared minus s okay and then i just i'm just going to found that out in powers of s so i want to know what does it take for these coefficients a0 a1 a2 etc to be in the convex hull of something that looks like one over n squared one of m to the four is one over n to the six and so on well um uh it turns out that that set that curve one x x squared x cubed and so on it has all minor and positive it's known as a moment curve if you take any endpoints on that curve all the determinants are positive and therefore we know what all the equations are that cut out that cut out its facets and in this case you can translate it to the following extremely simple statement uh about uh about this set of coefficients so you start with this vector a1 through a1a2 and so on you actually build out of it the so-called henkel matrix a two by two matrix whose i j uh entry is a i plus j and so here it is um and uh and and and the the positivity above of both m squared and the probability p of m squared translates to the very simple statement that all the minors of of this angle matrix are positive okay so for example um the easiest case of all the that means all the minors i mean every k by k minor that that that uh you can make yours got to be a positive so the easiest case is all the one by one minors that just tell you all these coefficients are positive that's the very old statement that the coefficient of higher demand this leading higher dimension operator has got to be positive um but all the rest of them have got to be positive too so so the sort of simplest case you can also sort of trivially see by hand would involve something like a2 a0 minus a1 squared where you can just very easily see from the koshi schwarz inequality that you have to lie uh in this region where a 2 a 0 minus a 1 squared is positive okay so so this is this is an example of a nonlinear constraint between power demand for operators um uh but it goes beyond that goes beyond that to all all the k by k minors of this matrix and this is an if and only if statement um if you happen to know the gap if you happen to know that everything starts from m star uh uh that there's nothing in the spectrum of m star and above you can actually say a little bit more uh in that case if we work in units where m star is equal to one um i can build not only the henkel matrix uh for a zero a one a two but also for all the discrete derivatives so a1 minus a2 a2 minus a3 the double discrete derivatives and so on and the claim is that all of these vectors have corresponding ankle matrices uh that are totally positive so if i go back to this quadratic example if i know there's a gap uh then then i cut off this picture with a line uh here and uh in the limit where i don't know where the gap is so m star could be arbitrarily low this this line goes up uh and you cover you go back to the uh initial case again of course if you're alone the effective field there's you don't you're learning the observer you don't know what the gap is so so this sort of second statement is a little bit more academic but it's theoretically interesting but we do know something more about uh what what cuts out the space if we know uh something about the gap okay so this is an example of the nonlinear constraints that i was uh talking about now what about the linear constraints well um let's go back uh to this uh um uh to this dispersive expression again uh just for simplicity i'll do an example here where i only keep the expansion in the f channel um then uh when i expand this out uh as a function of t i have the well i have the powers of f but i also have powers of t that pull down derivatives of this uh gegenbauer um uh polynomial and if i now restrict myself to those coefficients that have a fixed mass dimension then from here you can very easily see that these lie in the convex hull there are given by positive combination of the following interesting vector um which is just given by the derivatives of the of the giganbarrow polynomial evaluated at the uh at coset equals one evaluated at the forward limit okay all right so again as promised we always get this uh situation where we get the we we get the sum of a bunch of uh vectors in this case the vectors are labeled by the spin um just by the partial wave expansion the spins are naturally ordered and the remarkable fact is that if you actually just make this matrix of the derivatives of the gigabyte polynomial um uh you just stare at it if you compute any determinant of this matrix it's positive all of its minors are positive so that means that the convex hall of the gigan bauer of these ganga bar vectors is actually a cyclic polytope and so we know ahead of time all the equations that cut out uh the space that's allowed by these by these a's okay so this is a it's a very it's a very cool and unobvious fact uh it's actually equivalent to a result known since the um uh 1960s uh that's true for any orthogonal polynomials which are orthogonal respect to a positive measure so in fact there are sort of a three periods uh in in the history of mathematics where total positivity was discussed uh began with chebyshev in the mid 1800s then again in the 1960s um in the context of this kind of question and then in the 90s again and the mid mid uh 2000s um but anyway uh this uh so it's not obvious but it's a it's a beautiful fact that uh that there's this connection between the total positivity of of of derivatives of orthogonal polynomials with respect to a positive measure um and that's true for any such orthogonal polar it actually turns out there's even more hidden positivity in this gigan powers than is sort of manifest from what was known in the 60s and it's useful for the story of the ef the hedren but um but uh let me not say anything more uh about it now so um uh this is just meant to illustrate these two things so so just from the expansion of the propagator we get one set of uh objects where all the minors are positive and we can uh which allows us to uh characterize uh all of these uh everything which comes out of a positive sum of propagators with positive coefficients in terms of matrices with all minors positive these ankle matrices and then the other one is a similar story for the expansion of graduate polynomials it tells me about the key dependence these are both objects with total positivity properties um which are then which then allow us to to predict ahead of time and calculate what what the passage structure of uh of the corresponding polyjuices and uh so we did this very systematically in in our in our paper it's 120 page paper i don't want to spend more time talking about it but i want to stress these things really apply to the real world so if we're talking about the coefficient of higher dimension operators for example for photon or photon graviton scattering these spots are a little hard so you can't see them but just at least impressionistically uh you can see that in some two-dimensional uh parameter space for photon scattering or graviton scattering you literally lie in these finite close small bounded regions it's extremely interesting no matter what's going on in the uv um uh these ratios uh um this is an example of the giganbaro constraints um uh these ratios have to lie inside these finite bounded bounded regions in fact uh the giganbaric constraints just tell you about unitarity um if you do a dispersal away from the s channel sometimes uh you you also get information uh you have uh you have a sometimes you can have a uh a symmetry for example if we have a real scalar we have an fpu symmetry or for particular velocity assignments for photon and graviton scattering we can have uh we can have uh uh we can have uh both symmetry for the uh for the external guys as well which in which imposes further constraints on the coefficients of the low energy uh effective theory so that on the one hand they have to lie inside a polytube on the other hand they have to lie in a plane that uh that uh forces uh that forces the both symmetry uh crossing constraints and so the intersection of this plane as a polytroph that we talk about gives you even smaller regions so that's what these very tight boundary regions look like here are some examples when we have three coefficients if i go to the next order in the effective field theory again they can't lie anywhere they have to lie inside these very particular uh three-dimensional shapes and anyway it goes on uh from there in general um i just want to quickly say something about the the structure these objects are simpler cousins of uh amphithehedra um uh and uh and uh and slightly loosely speaking all these coefficients a b q are are given as a particular kind of positive combination of these gigan bower vectors now the gigabyte vectors have this total positivity property and the coefficients themselves have a certain non-linear ankle positivity uh condition on them if we just had a vector is c1 v1 plus c2 v2 plus dot dot as i told you uh that just defines the polytope here it's slightly more interesting because the coefficients themselves aren't just random positive numbers but they have to satisfy certain nonlinear positivity constraints the story of the amplitude is very similar um uh i won't say anything about the context but uh just just to make the comparison we also have some sort of external data in that case it sort of specifies the momentum of the gluons of the scattering process and again there is a interesting region that we want to look at which are uh which are associated with the uh but where the coefficients instead of being positive are associated with the non-linear sort of grasmanian uh positivity and again the external data is positive in exactly the same way as the gigan bowers were possible um so i think this is sort of amusing that in the past a few years we've been seeing these interesting non-linear uh geometric objects uh make an appearance in so many different places in uh in physics um some of my mathematician friends um told me that uh gelfand would go around in the 90s to uh people working on polytopes and say you probably thought people were doing trivial things you should generalize polytope synthetic grasmanians and that's exactly what we've been seeing um uh so uh so so they're really generalizations of these polytrophs the eftahedron is a sort of little generalization of polytopes the amphitheatre is like you know two levels deeper than than than this and has a lot more things that going on but uh but there are some interesting uh similarities between them and certainly this notion of encountering sort of objects with total positivity in the sense of matrices with all positive uh all positive minors is a common theme in um all of this stuff all right so um uh i'm i'm being very very slow so uh but um let me just make uh a couple of more uh comments so that's just the that's the the review of the story of the uh the hedren um i want to make uh a a a a couple of comments before uh moving on i guess i'll talk about um uh one of the other two topics uh quickly uh first i um i want to mention that that we began this discussion about uh the coefficient of higher dimension operators by thinking about um uh super luminal propagation in some background and we then switch to talking about um constraints on amplitudes coming from uh unitarity and uh and causality from the dispersive representation and uh something which uh which always bothered me for a moment from a long time ago is that it looked like this that it looked like the power of those two things was not the same for example super literality only tells you naively that the coefficient of d5 to the four is positive at least that's what we thought a long time ago whereas we seem to get much much more from the uh from the uh dispersive representation and uh i think it's an it's an interesting question um whether in fact we can match the constraints more perfectly with each other um and a step in that direction is the following a step in that direction is to actually realize that uh that causality tells you more than simply what we thought certainly what what we thought before um uh in order to in order to see why that is let's try to sort of match uh in more detail these two pictures that we talked about we started with this picture we turn on a background for a phi where phi dot is non-zero and in that background uh we see that you can have a super luminal uh propagation for the wrong sign uh the point being that no matter how small the coefficient of the higher dimension operator is c is if i make this sort of blob where i turn on five big enough in size i can get a macroscopic time delay that i can a time advance or delay that i can sort of measure at very very long distances okay so so one argument is turning on this blob the other argument is about two to two amplitudes okay so how are these related well obviously they're related because we think about the propagation through the blob as a sequence of scattering processes of this form okay so um and and this is important for the following reason um uh whenever amplitudes are under control they're small um and so we would like to uh so in order to be able to get a constraint we need to in the language of amplitudes we have to go to uh some situation where despite the fact of the basic sort of the the fundamental amplitude is small um uh something is arranged so that at the amplitude instead of being one plus i small it actually exponentiates the e to the i of phase and that phase can become large if that phase is large then we can see whether we get a time advance or or a delay from it that means that you have to arrange a scattering situation where despite the fact that the basic amplitude is small uh something happens so that you can exponentiate in this case what we're doing is uh is uh uh is uh is arranging for that by imagining that we're scattering our hard particles energy e off a soft background that corresponds to a huge number of particles with small uh with with small momentum and then if we just add up all these diagrams of course i get the enhancement in the propagator from the softness of the background so despite the fact that this basic interaction is small it does exponentiate and so this amplitude actually goes like uh e to the i some phase delta where if i rewrite it in laurentian variant terms is e to the i a a of s over s multiplied by something which is the overall size of the condensate that i uh uh that i turned on and therefore in order to have a prime delay we learned that the derivative of this a of s over s uh has to be a negative a of s over s has to be perfect so that's interesting that's more than what we strictly thought before which was just about the coefficient of s squared in the amplitude at low energies that's the coefficient of the d5 to the fourth term this tells us something more it tells us that the function of s that uh that negative a of f of over s has to be uh has to be monotonic and it's cool that this uh more fundamental fact about causality actually follows somewhat not slightly not really from the dispersive representation okay so so if you take this representation and take d by d s a of s over s you find that it's the integral of something manifestly right so at least this is one thing in the direction one step in the direction of trying to more accurately match uh what we know from causality considerations and what we know from expressive uh uh considerations um i don't know if they'll if they'll match perfectly but it'll be fascinating to think of more clever and interesting backgrounds perhaps that could allow us to access um things that we don't have access to uh well like that can first for two to two scattering uh uh perhaps connect to what we've seen already for uh from from the dispersal considerations but we also know that there are things that could be true for three to three scattering for any point scattering from turning on uh from uh uh uh positivity constraints on on these uh super lonality constraints on these effective theories so in the other direction um uh causality tells us more than just things about two to two scattering that at the moment we don't have good access to from the uh dispersive part so that's i think an interesting direction to try to figure out how these things match with each other better second point i think this is a a much bigger point a big missing ingredient in these arguments is that um we use fixed t dispersion relations um and that means that there's a constraint that we're not inputting um uh and you can actually see a hint that that that there's even more constraints so so we get this huge number of constraints on uh infinite infinitely many constraints and higher dimension operators but when you look at what healthy uv theories do they still don't seem to populate the entire region available from the constraints that we found they still sort of they still congregate in corners uh and and it seems that's because we have not yet input um we've input the constraint of uh uv of uh causality uh which is again in the dispersive representation that this reggie limit where t is fixed and s is arbitrarily big um but we have not input the crucial constraint of uh uh sort of traditional uv completion which is uh the behavior of the amplitude the softness of the amplitude at a high energy fixed angle scatter um so amongst other things this shows up in the following way in all sensible uv examples the contribution to the discontinuity of the amplitude or if you like to the partial wave expansion from higher and higher spins is more and more suppressed as you go to high spin okay so um uh this is there this is there in in for example screening examples where you have towers of higher spin particles even at tree level you see it in in field theory loop level where in the partial wave expansion you still see higher spins that correspond to the all the different sort of angular momentum of the particles that could run around the loop but clearly uh their their contribution is the annual mental gets bigger and bigger and the particles get further and further apart is more and more suppressed to the to to the amplitude but nowhere in our that and that's crucial for the for the uv uh healthiness of the of the theory but that's not reflected in our in our in our analysis so it will be fascinating to figure out how to put in this extra information about actually uv completion uh um uh for uh which will very likely put even more constraints okay now um i've already gone one hour um so uh uh perhaps i can ask the uh uh organizers um can i maybe have another 15 minutes uh i guess yes uh yes please okay um i think uh maybe what i will do is um uh is uh skip this part um all right i'll skip this part uh if anyone's interested i'll be happy to talk about it in the in the discussion um well let me quickly switch to this uh uh uh to this other topic of uh uh celestial amplitudes and um and uh and what you can think of is an anti wilsonian uh paradigm okay um all right so let me just jump into uh thinking about the celestial sphere um and one one natural motivation is just thinking about no momentum in uh um in minkowski space so um if you have a null momentum uh the associated two by two matrix uh when you dot into the uh sigma matrices has vanishing determinants and so we can write uh p alpha alpha dot as the product of lambda alpha and lambda tilde alpha dot where lambda alpha and lambda tilde alpha dot are two vectors these are the famous spinner helicity variables okay and the spinner eliciting variables very naturally uh give you something that is parameterized by a coordinate on the on the celestial sphere after all of course a null and null ray points in some direction of the celestial sphere that that surrounds you and uh and this representation of the null momentum makes that very manifest so if i take uh these are two vectors um if i take out an overall scale out of both of them which i'll call root two omega then i can write one of them as one z and the other one is one z tilde and z and z tilde are just complex coordinates uh z and v bar sorry z and z bar are really complex coordinates on the riemann celestial sphere okay and so therefore when when just labeling the uh null momenta in terms of this frequency oh my god that the energy of the ray and the direction of the ray uh naturally uh gives you variables that label the celestial sphere okay now the lorentz transformations act on lambda and lambda tilde by sl2c just two by two complex transformations and so from here we can see that the v prime uh under a lorentz transformation is just a mobius transformation of z so that's uh that's just the uh that's just the usual sl2 uh uh mobius or conformal transformations on z and omega changes uh in this way um and and uh and i also pick up a phase that i can think of as the action of the little group okay so so anytime and you know people who work with massive amplitudes are using these variables all the time and so they're automatically uh can be thought of if you write the lambdas and land matildas in this way as variables associated with points on the celestial sphere and energy and so so far there is nothing new but uh the interesting new thing is is to consider the scattering not of momentum eigenstates but of boost eigenstates and so uh so a boost eigen state is going to be an integral over all energies omega raised to some weight delta of omega and z and then you can see that under lorentz transformation a general florence transformation this guy goes into the mobius transform z and it picks up some overall weight and in particular if you do a line transformation in the direction of z it just picks up uh it's it's it's it's an eigenvector it picks up uh z stays the same and it just picks up some uh overall weight okay so these are these states or eigenstates of boosts in the z direction and uh and if i scatter these things instead of momentum eigenstates the objects that i get transform under sl2 exactly like uh correlation functions of a primary field in a conformal field okay but that's the but uh at the moment i'm not talking about any conformal field theory anything like that that the relation to conformal invariants is just trivially that the lorenz group is sl2 which is also the conformal group uh on the sphere okay the physical point is that we're scattering boost eigen states instead of momentum eigenstates and so um how do we go from a blue side instead to uh uh m minus these two boost sentences well we just take the ordinary uh amplitude um momentum space amplitude as a function of omega and z and i simply integrate them with the weight d omega over omega omega to the delta so this is a melon transform okay so i go from something that is a function of an energy in the directions d uh and z to things that are a function of a weight uh and z okay right this is especially simple at four points um if you have four point scattering the sl2 invariant tells you that that the amplitude which is just lorentz again i remind you it tells you that the amplitude can only depend on this cross ratio of the four points z which is z1 minus e3 z2 minus e4 over this other combination uh and um another nice fact is that we know that the amplitude uh uh we know that by momentum conservation two and two out the amplitude lives on a plane just lives on a on a two-dimensional plane that two-dimensional plane intersects the celestial sphere on a great circle and so we learned there's something interesting about this thing uh that that that this melon amplitude has to vanish unless the four points lie on a common circle okay the constraint that four points lie in a common circle is simply that the cross ratio is real so that means that uh that uh by momentum conservation there's an overall delta of z minus z bar um uh in front of the whole thing and after we factor that out uh we have this extremely simple um uh expression that says that the amplitude in uh sorry this is the amplitude in in melon space that that should have been a function of uh deltas here is something that depends on the validities of external particles this this delta of z minus d bar that just tells you the four points are uh coplanar with each other and what's left is nothing other than just a melon transform of the ordinary uh amplitude as a function of the center of mass energy omega um weighted by this parameter beta uh which is just the sum of all the weights of the external particles so after all of this song and dance uh when we scatter a boost eigenstates for a two to two scattering all we're doing is taking a melon transform of the conventional two to two amplitude at fixed angle at fixed angle but a melon transform with respect to the energy all right now the study of uh celestial amplitudes um over the past uh five six seven years especially from uh uh andy stranger and his friends uh has already clarified unified and revealed many aspects of infrared physics ir physics soft amplitudes fascinating connections between weinberg soft theorems and memory effects and uh and uh symmetries on the enhanced symmetries on the celestial sphere okay so there's a lot which has been done and continuing to do and we even talked about some more of them in our paper on infrared aspects of the story but what i want to focus on here is ultraviolet aspects because i think this is actually something really novel and uh interesting because boost eigenstate scattering is a perfect probe of the uvir connection um uh when you scatter bluestacking state it maximally violates what's on intuition because we're scattering states with arbitrarily high energies and as we'll see effective field theory amplitudes don't even make sense in this context you can't even talk about uh scattering boost eigenstates without having uv complete things uh from the get-go uh some people might think uh this is bad and if you're a conventional effective filter this is bad but if uh but you know if we're trying to understand how to more precisely think about the way in which the wilsonian picture of the world is wrong especially when we have gravity it's good to be forced into a situation where you don't have that as a crutch to uh rely on okay well let me just give you an example so i think that so this is a good thing this is not not a bad thing uh but let me give you a simple example let's say you have a two to two amplitude like gravity that goes like energy squared omega squared okay well you'll just see what the problem is how do i melon transform this if it goes like some power p how do i melon transform integral d omega over omega omega to the uh omega to the beta plus p okay so uh this doesn't make sense it's ill-defined for any p bigger than zero i don't i don't even know how to uh how to make sense of this uh integral okay all right so um so let's then begin with by looking at something which is sort of healthier in the uv and just to start getting some uh intuition um and what we're going to do here uh in in the entire discussion i'm going to be focusing on the behavior of the melon amplitude as a function of beta that's because beta is like the total boost weight okay so uh beta is the sum of all the electrons from the particle beta is a total boost weight and so uh of course there's a lot of interesting things in the z dependence as well normal angular dependence of the amplitudes but but but all the at least the zeroth order novelties of scattering blues eigenstates are seen in the dependence on beta so the rest of uh in in rest of this discussion i'm just going to tell you various interesting facts about the the about the uh about how the amplitude in in on how the celestial amplitude depends on data okay and develop some intuition for it in some dictionary with things that we understand uh uh better and and also connect to to uh affect the field theory okay so let's do our first example um where we imagine that we're integrating out something a tree level like a phi cube theory okay um uh so uh imagine we had a phi q theory we entered out a tree level at low energy would have some effective phi 4 coupling um so uh so yeah so here i'm integrating out something of mass m uh but in the full theory this is what the amplitude would uh look like okay so in terms of the cubic couplings this lambdas g squared over m squared all right so that's the that's the tree level amplitude uh in momentum space what is the melon transform of this okay in this case it's easy enough to do this transform and this is the expression we get first if you just want to do it analytically it has this interesting structure we get this uh the the dependence on the mass is just m to the beta and it has an infinite sequence of poles okay it doesn't uh it has this one minus e to the i pi beta that means it has uh it has infinitely many poles both on the negative axis and on the positive axis and i'll explain a little bit more where these come from uh to begin with but uh it i'll say a little more generally where they arise but we can already see it in this example what are all these poles at negative beta okay all these poles are negative beta well where do the poles come from in this expression when uh uh the only possible pull come from uh when the integral is going close to omega goes to zero or omega goes to infinity if we look at the part where omega goes to zero i have d omega over omega omega to the beta and around omega m zero let's say that the amplitude uh had a power series expansion in omega then i'd have an integral of omega to the beta plus n that would give me a pole that looks like one over beta plus uh that omega to the 2n it's powers of omega squared so i get poles and beta plus 2n from the infrared part of uh of the expansion of the m okay so they're pulled at negative beta that are reflecting uh the expansion of the amplitude uh uh uh in the law energy effective theory okay and the coefficients the residues of these poles are literally the coefficients of the higher dimension uh or the or the coefficients in the in in that expansion if you like the coefficients of the higher dimension operators are literally the residues of these poles what about on the other side well on the other side this is just a garden variety field theoretic amplitude and then also the power law follow-up with energy at large energies so we just have the inverse of the phenomena that we just talked about the residues of the poles here on the positive data axis are just telling you about the about the various uh power law falloffs with energy of the amplitude in the ultraviolet okay now already this example shows you the anti-wilsonian nature of what's going on when we scatter a goose eigen states instead of momentum eigen states if i have the ordinary momentum amplitudes if i just take this expression and stupidly take the mass to infinity i just go to the lower energy expression smoothly so the longest expression for a 5 4 coupling would just give me lambda b minus lambda here um uh so okay so uh that's it um but that's just not true when i scatter the boost i can stick see the entire dependence on the mass is in this factor m to the beta so nothing nothing smooth happens as the mass goes to infinity um and so there's no sense in which this expression sort of turns into the expression that i would get from melon transforming the 5 4 amplitude of the low energy theory see for the 5 4 amplitude in this case i would get the integral of d omega over omega omega to the beta and uh and there you can interpret that as delta of beta if you just uh uh switch to an exponential representation for omega you can interpret that as delta of beta okay so there's some kind of uh roughly reasonable formula for the melon transformers of um of the of the low energy of the effective field theory by four amplitude but that object is not analytic in beta it's just this terrible delta of beta and it does not smoothly connect to what we get in the full theory okay the full expression is perfectly inaudible but it's perfectly analytic and there's no sensible there i mean there's a formal sense in which one expression can be interpreted as the other is m goes to infinity but but they're they're totally different from each other one of them is infinitely many pulses the analytic the effective field theory one is none so so that just illustrates uh what what what's happening why is this happening because we're scattering goosebumps and things that have arbitrarily high energies in them all right so just saying again when i said a uh a moment ago about where these polls come from um we get both ir poles and uni uh we get poles when the low energy amplitude has an expansion and powers of omega to the 2n now coming here from at low energies and we can also get poles if the high energy amplitude has a power law fall off as we expect in in field theory that gives us these false one over beta minus two n okay so that's the that's the uh that's the sort of zeroth order picture is that in the m plane we expect to have poles the negative data poles reflect effective field theory expansion the positive beta poles are there uh uh reflecting power law fall off of amplitudes in uh quantum field theory and that actually already uh indicates a radical difference between quantum gravity and field theory so if we have uh if we have the melon transform of amplitudes in quantum gravity um they're they have a qualitative difference from what we just saw in field theory uh and that's because we know something about two to two scattering of high energy about high energy two to two scattering gravity i'll i'll review a little aspect of it in a second but again we have this picture that at very high energies two to two scattering basically produces large black holes sorry uh uh the scattering high energy produces large black holes and therefore the amplitude for two to two scattering exclusive to the two scattering should be exponentially small it could be given by roughly e to the negative the entropy of the black hole of uh of energy oh my god then four dimensions would go like e to the minus g newton omega squared so the height of the amplitude is exponentially soft in um uh in in any theory of uh quantum gravity ultimately because you make black holes now if you have a weak coupling or a scale beneath the plunk scale like the string scale then that exponential process kicks in earlier uh we know that strings are already an example that exponentially stops but ultimately gravity makes it as as even softer okay so regardless of whether or not we have streams in the intermediate uh energy range asymptotically and high enough energies um uh the amplitudes are exponentially soft and that means there are no contributions uh there are no poles from the omega goes to infinity limit okay remember all that's going on is i'm doing these melon transforms so i can get a poll uh from from the omega goes to infinity part if they're the power law fall off with omega if clearly if i make the exponent beta big enough i should have a divergence but if the high energy behavior is exponentially soft i don't care i can make better as big as i like and uh and there are no singularities so nothing extremely interesting qualitative difference between uh again uh reflecting this anti wilsonian business that if you have uh the if you have the celestial amplitude for a field theory and for quantum gravity there's a fingerprint of it being gravitational which is the total absence of any of these poles on the positive beta axis okay now actually we can go further than this um we know that in the lawn of the effective field theory um at tree level we have an expansion of powers of omega squared but at loop level we get logs we know at loop level the massless loops generate logs and so the real expansion is in powers of omega and let's say i have gravity i'm just illustrating the case of gravity powers of omega powers of g newton only gets squared because uh uh that's the the effective uh dimensionless coupling for gravity and some powers of uh of of uv logs okay um okay so this is this is the uh expression that we have so we we not only have these powers in omega we're modulated uh with the logs and uh and that's very simple because the the contribution uh to the melon transform at low energies from powers of omega modulated by logs see the logs i can just get by taking derivatives with respect to the to a and so what i get uh what i get from here are um not just simple poles in beta which is what i had before but uh from if i have r logarithms i get not just simple poles but double poles triple poles higher poles are poles in the beta all right so so here's the summary then if we're talking about some a gravitational theory there are no poles at all for positive beta for negative theta uh i have polls at uh negative integer uh values of beta the the residues of the simple poles are the coefficients of the higher dimension operators and the revenues of the double and higher poles are the logarithmic running of the of the effective field theory coefficients so i find this quite beautiful because um uh in on the celestial sphere uh the effective field theory coefficients and they're running uh turn into something very canonical you know we of course care about them in momentum space because we're stuck at low energies and we're trying to measure these coefficients but they have a more invariant meaning on the celestial sphere is residues of pulse um as a function of the booster and these are of course exactly the objects that we just talked about satisfying all these infinitely many positivity conditions so at the very least we're learning some consistency conditions on amplitude and on the celestial amplitudes somehow we've got to produce these interesting functions with residues only on negative data if we're talking about gravity not a positive data and the coefficients have to satisfy all these interesting positivity constraints all right so so much for the analytic behavior in the beta plane i'll just say something quick and then end about the behavior at large beta so first as beta goes to plus infinity uh uh this melon integral is dominated by large energy and again we know what we know roughly what the amplitude looks like at large energy uh it goes like e to the minus g newton omega squared that's just this uh that's the uh that's the uh the the black hole production picture so so the two to three amplitude dies exponentially with energy as we go through high energy but uh that tells us that in melon space uh the amplitude mod squared grows exponentially in fact it grows like gamma beta as you go to uh high energy so that's uh that's one uh interesting qualitative thing is that the physics of black hole production isn't a certain exponential growth at large positive data there's there's a more a more uh i think a very interesting but uh detailed point um which is that uh while we say that the two to two amplitude dies exponentially at uh at high energy in gravity um it's not smooth uh here is a simple model for what the uh for what the uh two to two uh what for what the amplitude involving black holes looks like imagine that we have some basic interaction between let's say the n states and the out states and uh and the jth black hole microstate okay so we have some basic interaction here now uh the claim is that their order e to the s uh black hole microstates and the claim is that each one of these amplitudes is of order e to the minus s e to the minus the entropy of the block over two but maybe modulated with some some phase now this picture explains why the total production cross section for the black hole is one because i just mod squared uh i much squared this basic uh three point and sum over all the black hole states and i get the e to the minus s from here and i multiply by an e to the plus s of the number of black hole states to get something of order one but the into out amplitude is a sum uh uh over e to the minus s black hole from the from from this product if i just glue these two things uh if i glue them in the out together but now i get uh when in and out are not the same i get the sum over e to the s what i call random phases and there's the usual root n cancellation there so when i take the mod squared for the into out amplitude for the exclusive amplitude then i do get something from this argument that goes like e to the minus s black hole but it's very chaotic so there's some envelope that's dying like e to the minus s y call but princess coming from uh from averaging all these e to the plus s y call random phases it's very very widely and chaotically however that's not what we expect for the amplified melon space because exactly because it's doing uh an integral over uh because it's a boost eigen state it's averaging over all the energies and so we expect and you can see this very you can see this in simple toy examples that the melon amplitude is growing exponentially and is smooth so all the physics of black hole microstates and all that stuff is somehow encoded in energy space in the two to two amplitude that dies exponentially was chaotic wild fluctuations but is instead seen on the celestial sphere as a certain smooth exponential growth uh at large beta that's a qualitative difference that i think is very interesting and worth uh thinking about more finally um let me tell you what the large beta behavior looks like as being minus infinity uh and uh and and here we can also see it from our simple uh toy example uh as beta goes to minus infinity uh this amplitude just it it goes like m to the minus beta and it has all of these it goes like m to the beta and has this infinite series of volts um and you can actually argue that on general grounds that uh that the large beta behavior actually goes like there's always infinitely many poles and it goes like m star to the beta where m star is the gap to the first massive states so at tree level if we had only poles would be where the first pole is uh if we had a branch cut in s it would be where the branch cup begins and even there's a modulation of the power of beta that reflects the threshold behavior near where you start seeing the first states okay so so uh so this is this is uh interesting we have this function it only has poles um if it's uh quantum gravitation only has poles on the negative beta axis the revenues on the poles are effective field theory coefficients and we know something about the behavior of infinity we know what it's supposed to do with plus infinity is uh at least roughly it has to have this exponential smooth accidental growth associated with black hole physics and at minus infinity uh the leading behavior is has to do with the gap to the light of states um okay so uh so uh all we've done here is sort of translate uh familiar facts from uh momentum space to statements about the structure of the um uh of the scattering of bootstrapping states uh the hope for this program uh is that perhaps there's the the the question of determining the amplitude as a function of uh the perhaps the question might be easier or at least differently posed um on the celestial sphere than it is for the usual uh amplitudes in uh momentum space so just summarizing what we've seen so far we have this uh interesting function only with poles and beta no more complicated analytic structure the poles and beta reflect the effective field theory couplings they're constrained by all the eftahedron positivity constraints uh beta goes to plus infinity is prescribed by black hole physics the leading behavior beta minus through infinity is controlled by the gap so supernaturally you think well we know a lot about this function we know about its pole and its leading behavior of infinity um what else is needed in order to uh determine it could it be that that this information is actually enough to uh uh to determine what f of data is and so it appears that something more is needed um for example just just to give you a a dumb example um let's say i take a of beta and i shift it by some constant mu to the beta where this mu is bigger than the gap to the left of states m star then uh this is it essentially changes nothing in the analytic structure it doesn't change the poles and it's subdominant to the known behavior of both the plus infinity and beta goes to minus infinity but if i translate this back to frequency space it's horribly non-analytic in as a function of omega or the function of s i get something that's shifted by delta of s minus mu squared now so basic analyticity and frequency or an energy which has to do with causality um we need to figure out some way of uh of translating uh basic analyticity and energy to statements about uh the statements about the celestial amplitudes uh in order to go further um okay so uh uh that's it i had a third subject that i wanted to talk about but i've already gone off over far too long so uh maybe i'll just leave it at that and see if there are any questions thank you

Channel: All Things EFT

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Id: IEpWK8T9YM4

Length: 91min 11sec (5471 seconds)

Published: Sun Jan 31 2021