Robustness of GR. Attempts to Modify Gravity, part 1 - Nima Arkani-Hamed

okay so um I'm going to be talking about the attempt to modify gravity and why it's so hard um and maybe before uh before we start I should have I should clarify that obviously what I'm talking about is attempts to modify gravity in the infrared at large distances we all know famously that there definitely seems to be needs to modify gravity at very short distances perhaps close at least before or at the Planck length and will actually talk a little bit later about exactly what that means and perhaps more subtle senses in which some kind of long distance breakdown of the low-energy effective theory for gravity might might probably is needed but if but the situation that that's relevant for incredibly subtle quantum gravitational effects well we'll talk about it in in a little more detail um but what I'm talking about are the attempts to modify gravity at large distances that are motivated by by a host of Astrophysical and cosmological observations so why would you want to modify gravity at large this is the most naive reason and you know it's not only a a personal philosophy for how to do physics but I think a good a good guy for how to do physics generally even if you're not me is to just always ask the most naive question in the beginning that's the dumbest possible question first give the dumbest possible answer if the dumbest possible answer is right excellent you've succeeded you can move on to the next question and if the dumbest answer is wrong it's really important to know why the dumbest answer is wrong then you proceed to the next questions and then soon soon you're pretty good but anyway so so we're not going to be shy about asking incredibly naive questions in these in these lectures and the most naive question you could ask is look we seem to have evidence for some breakdown of standard gravity at long distances perhaps galactic rotation curves are funny and of course at the very large distances we have an accelerating universe and normally we attribute these funny behaviors to additional sources of matter that we haven't seen the joke goes we're truly in the dark ages of cosmology we had Dark Matter to explain the Galactic rotation curves and a lot of other stuff and dark energy our cosmological constant to explain the accelerating universe so why couldn't it be that it's actually gravity that's wrong that long distances instead we have to modify we have to modify gravity um so that's that's that's the most naive set of motivations and you know this kind of thinking has a has a glorious history something that you learn if you know anything about the history of physics is that it's possible to choose examples from the past to reinforce any polemic point you want to make and it's not different for this it's not different than this example we need simply go back one and a half centuries to find exactly the same set of conundrums very similar conundrums facing astronomers in the 1800s from very precise observations of the distant planets people realize there was something funny with the orbits of the distant planets um and there are too you could you could ask and they did ask could it be that Newton is wrong you know there seems to be something funny these these distant planets aren't going aren't orbiting like they they should be okay so a back bend could have even that Newton is wrong or that there is Dark Matter dark planet in that case okay and in fact that was a triumph of the French astronomer Lavetta yay actually this is the sad story Level II actually did the whole calculation wrong and completely accidentally made the correct prediction for where Neptune should end up being it's a very sad story it was really done correctly by them by the American astronomer it's so sad that I forget the American astronomers named the medal a got got all the credit for it I shoot I used to know it that's what really makes it extra sad anyway this said I've already a totally flipped out and and his calculations exactly wrong had they looked you know two weeks before after he told them where to look he would have totally he would have been humiliated but but he lucked out anyway but the American guy whose name I can't remember got it right and in any case that was the correct explanation explanation wasn't that Newton's laws broke down at long distances the explanation was that there was a dark planet called Neptune so you see okay but but then of course there was another famous example of funny business with the orbit of mercury and level yay again having having had this triumph with with the Neptune predicted that there should be a planet much closer to the Sun than mercury that he called Vulcan and welcome didn't quite work as well but since I had some American boosterism I should also mention that some American experimental but some American astronomers actually found Vulcan they observed Vulcans in a transit across the Sun so she's American theorists a good experimentalist and you know and of course in that case the Vulcan was completely wrong okay and in that case the answer was that Newton was wrong and you had to modify Newtonian gravity and the answer was the answer was G R as we all know so it's just goes goes to show that you never know ahead of time in physics when you have some major puzzle whether the answer is going to be change everything revolutionize everything throw everything out the window or something relatively conservative in fact almost always throughout the relatively conservative thing is right and also almost and also in all examples that I can think of even though the really radical things never really checked everything out the physics never works that way things things are always subsumed into the more correct theory in a very very tight and logically coherent way so we're never we're never in full revolt mode but if it was the case if it was the case that whenever you encounter a puzzle you it was best to take one attitude or another grad school would be a lot easier you know if the answer was always the revolt when you when you when you all walked in we'd hand you Berets and tell you to go off to South America and and if it was to be conservative all the time well when I used to but I used to I used to say we'd like send you all to Wall Street to become bankers but that that of course ends up being a very bad bad idea but anyway so we just never know ahead of time we never know ahead of time which which way to go it's good to keep an open mind but that as I think it was a famous phrase of Oppenheimer's it's important to keep an open mind but not so open that your brains fall out and so so that's what we're going to try to caution against in these lectures ok so so so with that with that spirit it really is worth asking if it's possible to modify gr at long distances to address any of these to address any of these phenomena but we very happily very quickly have to get down to the business of seeing whether such a thing is actually theoretically self consistent never mind whether it actually explains the via of desired phenomenology I should mention there's actually one much bigger motivation gigantically bigger motivation than simply explaining in a different way dark matter or the accelerating universe I mean those are perfectly fine they're perfectly fine motivations but they're both cases where at least we have an extant explanation for what the phenomenon actually is it could be dark matter it could just be a tiny vacuum energy I think a much much stronger motivation certainly the reason I wasted I mean spent years of my life thinking about modified gravity is the cosmological constant problem this is not something which we have any good mechanistic understanding of the value of the vacuum energy could be as big as easily the weak scale to the fourth power it could even be the electron natural fourth power would make any difference as far as the gigantic mess of the problem is concerned and if it was the weak scale to the fourth power a dead minimum value that we might imagine it being then the the Hubble scale of our universe would be around a millimeter and that looks absolutely nothing like the world that we see around us so of course we all know that it's completely consistent to fine-tune away at the level of one part in 10 to the 60 or one part in ten to the hundred and twenty or whatever some gigantically big tuning it is consistent to tune away the vacuum energy and be left with a tiny positive cosmological constant to explain the observations it does not contradict any physical laws it appears absurdly fine-tuned but but simply as a as an explanation of the phenomenon it remains the the simplest explanation that that we know of but you might want you might desire some kind of mechanistic explanation some some some some reason why the vacuum energy well some reason why is that this gigantic tuning actually isn't there and it seems extremely hard to remove this minimum sort of rock-bottom value of vacuum energy of order the weak scale to the fourth we can't cancel it the way that we the way we might cancel we can't cancel the way using supersymmetry or any other symmetries that that we know of so the kind of the only very - 1/3 or zeroth order idea which which remains is that indeed the vacuum energy is gigantic but somehow gravity gravity is modified at huge distances in such a way that that big vacuum energy simply doesn't give rise the curvature of the universe so that's another very naive very zeroth order sort of idea is it possible to modify gravity in such a way to be gravitate the cosmological constant okay just so that even if we have a very big vacuum energy doesn't give rise the curvature of space curvature of space stuff now that's I think by far the best motivation it continues to be by far the best motivation for thinking about for thinking about modified gravity and as I said it's it's certainly what what motivated what motivated me to think about the subject for for for many many years I think I've actually thought about this subject probably more than any other subject that I've worked on and I've written fewer papers on it and any subject that I've worked on and the papers are by far the worst papers that I've written as well but but that just that that that's fine it's it's it's it's a tough problem but but I can say that that certainly none even of the zeroth order ideas for addressing the cosmological constant problem even get started and in one of in one of these lectures for outside will talk specifically about ideas about the sort of qualitative ideas for solving the cosmological constant and so at the moment I believe that the the best the best solution we have for understanding the the puzzle of the tiny vacuum energy is the multiverse idea and now that does not involve modifying gravity in any it's just standard Einstein gravity it involves a sort of radical modification for what you think lead very very long distance a structure of space-time looks like as you heard about in the lectures on eternal inflation last week so it's not modifying it's not modifying gr but it forces you understand the behavior of this remarkable quantum mechanical multiverse in a way that still remains basically elusive but it still seems like a better bet to me given everything that at least that I've thought of and I know then then trying to mess with maths mess with gravity at long distances as I said as far as addressing the cosmological constant problem goes there isn't even now there isn't in my view there isn't even a beginning of a coherent idea for for how that might happen so but anyway so those are the zeroth order those are the zeroth order those are the zeroth order reasons you might think about modifying gravity and of course the literature is filled with attempts to modify gravity there are a number of simply non ideas I mean they're not even ideas so so we'll pass over them largely in silence and I'll resist the temptation to make fun of them there are a few silly ideas it's better than non ideas but and there are actually some interesting ones there there some interesting ideas in the sense that there seems to be some interesting new structures arising there's something interesting going on and definitely worth exploring I'll try to spend most of my time talking about those those those things but I just want to say again at the outset that that in my view even the best of these ideas have some fatal flaws and part of the goal of these lectures is to try to tell you a why gravity is so special standard gravity is so special a why it's very difficult to modify and then be tell you at least in in a broad class of theories that attempt to modify it in some theoretically consistent way never mind the trying to use these ideas to solve the cosmological constant problem or explain dark matter or something like that just try to understand is it possible theoretically to modify gravity at large distances I'll talk about a whole broad class of theories that includes all the good ideas as far as I know and we'll talk about what we'll talk about them and in as transparent way as we can and I'll try to tell you what the what the main issues are and where we're I think the the the major flaws are so if you like the the slogan for these lectures is don't modify gravity but try to understand it instead there's we don't need to actually the tempted the tempted ways to modify it at large distances are going to have the problem that I'm going to tell you about on the other hand especially if you want to think about the via cosmological constant in this sort of more conservative direction of taking the tiny value of the vacuum energy for granted being forced into this multiverse picture there's really something to understand there about the property of ordinary gravity at gigantic distances trying to understand the properties of de sitter space trying to make sense of this of the small peepers okay so but now let's get get going and let me just dispense quickly with some glib reasons why we shouldn't modify gravity we'll see why these things are our glib one glib reason is that well gravity is a unique theory that's generally covariant and it's beautiful and nice so don't mess with it and this is a really dumb X this is the dumb reason first of all as will stress the number of times general covariance is basically an empty thing we can mess with it or not mess with it as we wish it doesn't really exist and so that's not the issue secondly there is another context where we have an almost identical set of words that you could say say we have normal non abelian gauge theories and they're - there's a beautiful local gauge symmetry the unique theory the local game gate symmetries and so on so don't mess with it but they're for non abelian gauge theories we know very well that we can modify gauge theories in the infrared we can give gauge bosons masses that's a very big modification of gauge theory in the infrared and that's totally consistent with everything so knowledge you're totally consistent we think it actually happens happens in our world so part of the goal is to actually understand so good reasons for things for why things are possible aren't possible and so on are rarely explained by some minutiae of formalism there are some very physical reasons why things do and don't work and so we're going to that's also a lesson that we're going to see over and over in this talk in these lectures what the Lagrangian looks like is basically relevant what what the gate symmetries are is essentially irrelevant what really matters is what the degrees of freedom are and how they interact and trying to isolate them and understand what they do so that's what we're going to focus on all right so so the goal of the lecture today is to to explain to something that probably you many of all of you have heard and probably some of you have seen different aspects of the arguments for this but I want to lay out and I want to lay out a number of different ways of thinking about this statement that gravity or GR is the unique low-energy theory for interacting massless spin-2 particles and associated with that as also the statement that yang-mills theory is the unique low-energy theory for interacting massless spin-1 particles in other words what's special about the theories that we know and love at long distances isn't that they're pretty or that they're generally covariant or have gate symmetries it has nothing to do with aesthetics at all it has to do with it but there's simply no choice given that we have massless particles of spin - or spin one that are interacting in a Lorentz invariant and unitary theory that assume that that those those are the those are the essential ingredients that the theory that governs their interactions is unique at long distances it cannot be anything else there's no choice it's completely locked and fixed by this particle content and the CIMMYT ends and plunk array invariance and general covariance and local gauge symmetries are a very convenient way a very convenient language for describing the interactions of these particles at large distances but they're completely a convenience purely a language they're all in our head they make it they give us a useful way for talking about these for for talking about these these series but the actual interactions are completely nailed so there's nothing we can do about them so this is incredibly satisfying that the sort of two dominant theoretical structures that we're familiar with at long distances are completely inevitable consequences of of relativity and quantum mechanics and that's it okay now so as I said I'm going to I'm going to explain this basics back from a number from a number of from a number of different but very related very related points of view so the different explanations are going to be you know they'll have their certainly logically overlapping but I want it but I want to tell you about all of them because they all shine a difference a different light on the essential issue so all of the drama has to do with the fact that massless particles only have holistic as their as their spin degree of freedom so you're familiar with this that that we have that that a photon which is spin one there's really only two polarization States similarly a graviton just spin - there are only two polarization States the Felicity to spin in the direction of motion and well there's a there's an official song and dance about this involving representations of the punker a group and little groups and Vigna and stuff like that but if you don't know that never mind it's completely equivalent to the intuitive statement that you can't go to a frame if the particle is massive you can always go to a frame where it's at rest and if it has one spin state you can use rotations to sweep out all possible spin States but if it's massless you can never boost to that frame and so all you can have is the helicity in the in the direction of motion so this is a startling fact there is a discontinuous difference between the number of degrees of freedom for massless particles and massive particles beginning at a beginning at well it's not there for spin zero particles but for high spin particles we have this disk there's difference in the number of degrees of freedom for mass and mass of cups so now this has immediately a an important consequence and that important consequence have to do with the second fact and and now now this is just a fact about just a fact about physics in fact about the actual state now we're going to start talking about a little bit how we try to describe this dis physics and a general fact I won't have time to explain in in in any detail but which again you probably seen in one guy's or another is that despite we have fundamentally particles we're trying to describe the interactions of particles but we want to make sure that these interactions are described in a local way we want to make sure that what's going on in Alpha Centauri doesn't affect an experiment that that we're doing here and so what you'd really like to do is build some Hamiltonian that that takes some states of the particles coming in absent some states of the particles going out but you want to make sure that this Hamiltonian is describing local physics so the only way we can do that is to actually take these particles introduce creation and annihilation operators for them the introduction of creation annihilation operators is nothing other than a bookkeeping device okay we have a big hilbert space with a vacuum one particle States to particle States three particle States and so on so what what what we want to do is just make it easy to talk about a Hamiltonian in this big Hilbert space so it's just convenient to introduce creation and annihilation operators to just later' us up and down this the spectrum so there's nothing nothing deep about that so really whatever the Hamiltonian is is a function of these creation annihilation operators but we can't make them random functions these creation and annihilation operators in order to ensure that the physics is local we have to group these creation annihilation operators into fields so if we just have a if we just have particles have spin zero we group them into a scalar 5x and then the Hamiltonian that we write down is an integral of some Hamiltonian density and is local in 5x so the kind of Hamiltonians or lagrangian's that you that you've seen all the time local lagrangian's as a function of the fields so from this point of view the fields are completely secondary objects that it's really fundamentally the underlying particles but we're trying to make sure that we're describing these particles the interaction between these particles in a local way and this is by the way why field theory emerges not only in relativistic physics but it's useful in condensed matter physics useful all over the place simply because it's a good way for talking about many-body interactions in a way that makes locality manifest ok but the trouble starts with spin one with massless spin-1 because in order to describe this these two polarizations I have to introduce a field that has the Lorentz index and plainly there are four of these guys a 0 a 1 a 2 a 3 so there are 4 degrees of freedom here even though we're only trying to describe the two velocities of the photon let's make this a little more explicit if I just write this as a plane wave state then epsilon mu of P is what we would normally think of as the polarization vector but there you go epsilon mu has 4 degrees of freedom 0 1 2 3 whereas there's only 2 in the actual photo now you can try to knock down the number of degrees of freedom you can try to say well I'll only I'll put some the theories Lorentz and rank so I want to put some uh Lorentz invariant constraints on it that's one constraint that I can put on I can say that P epsilon is equal to 0 so that means that there's three degrees of freedom I went from four to three but now I'm completely stuck there's no other constraints that I can put on Epsilon the set of all the epsilon the satisfies P epsilon equals zero some three-dimensional space but I only have two degrees of freedom for the photon what that means is that you might think that there's this nice object epsilon mu let's say for Melissa T plus an ELISA T - okay so you might think there's such a thing as a polarization vector for Melissa t + and Felicity - and we even write it that way and we write it that way because because we think that if I do a Lawrence transformation on this guy but epsilon mu will transform like a four vector until the Rance transformations right and perhaps it'll pick up a rotation phase factor four depending on how much you rotate around the the direction of motion e to the plus I something if it's a plus Felicity e to the minus i something if it's a negative felicity but still you think that's such an object exists that transforms like a four vector under Lorentz transformations but no such object exists there is no such thing there can't be because again there's only two polarization States but the space of these epsilon is three dimensional so let me let me make this extremely explicit let's say that we have a p mu which is a particle moving in the Z direction and let me introduce the polarization vectors corresponding to plus or minus félicité which in a in a sort of standard way that you probably see would look something like that so so the epsilon is is of course transverse to the its transverse to the direction of motion it's only in the XY plane and I've put it in this 1 plus or minus I combination so that it transforms nicely under rotations around the z axis that's fine but if I do this and I now Lorentz transform to another frame in fact if I transform to another frame so let me back up for a sec so clearly if the particle is moving in another direction if it's moving until moving Z Direction is moving in some other direction then one way I could get but the new polarization vector so just take this guy just the spatial polarization vector and just rotate it okay that's perfectly fine that's that's if you like what we define the sort of canonical polarization vectors to be a when two points in the Z direction it's this guy and then we just rotate it to when it's spatially rotated to when it's rotating doin's moving in any other direction that's perfectly fine that defines what the polarization vectors are for all momenta so we just gave a definition what the polarization vectors are for all momenta but you can see that that polarization vector does not transform into itself under Lorentz transformations that's obvious because this polarization vector has a zero in its time component but if I do a Lorentz transformation of let's say boost in the X or the Y direction there's no way that the resulting polarization vector will have a zero and its time component okay very simple so even if you go ahead and try and just define the polarization vectors to be the standard ones that we normally do that's perfectly fine you're allowed to do that but the object that you've defined does not transform nicely under Lorentz transformations okay under Lorentz transformations it doesn't go into itself now what does it go into it's not totally garbagey under Lorentz transformations epsilon goes to lambda epsilon plus something proportional to the to the new momentum that makes sense right that's that's if you like I could have here I could have added anything proportional to e 0 0 e and that continues to satisfy because P squared equals 0 epsilon dot P equals 0 means also the epsilon plus anything times P equals 0 so the only thing it could do under Lorentz transformations is going to itself plus something proportional to P mu ok but still it's not Lorentz invariant . so what do we do we see it's a good thing that I was not the creator of the universe because if it was me trying to invent consistent theories of spin one particle that would just quit right there say ah it's impossible too bad just York our theory in my universe you know but fortunately I'm not God for many reasons and uh what what we have to do is introduce a new idea we have to introduce the idea of a redundancy we have to say that that that epsilon mu and epsilon mu plus anything times P mu are to be identified they're to be declared to be the same state so we can say epsilon mu is in the same equivalence class is epsilon mu plus alpha times P me that's the only thing we can do if we're trying to describe this massless spin-1 particle in a manifestly Lorentz invariant and local way you'll notice of course that back in position space this is a mu and mu plus B mu something up to eyes let me just call it lambdas are to be identified and this is nothing other than the usual badly named gate symmetry you see it's not a symmetry it's just the declaration that these two gauge configurations are to be thought of as describing the same state now that puts strong restrictions on the kind of interactions we can write down for such for such fields because they had better be consistent with this with this with this redundancy but anyway just from counting if we do that for spin one we started with four degrees of freedom in the epsilon we demand that epsilon dot P is zero so that's three degrees of freedom and we have one redundancy that gets rid of one more so we get go down to the two correct degrees of freedom okay similarly we can do the same thing for spin two for spin two we can introduce a field H mu nu we can again we can impose various constraints that that the Lorentz invariance means that we can impose that P mu H mu is zero maybe we can even impose that it's traceless but this knocks us down from ten degrees of freedom in h mu nu - there are four constraints here - one so we have five degrees of freedom still but then we can we're forced to introduce a redundancy that H mu nu and H mu nu plus alpha mu P nu plus alpha nu P mu are identified once again now here there's a there's a little bit of shuffling things around if we demand that this thing is traceless we have to demand that these alphas aren't totally random if that alpha dot P is equal to zero ultimately you could just forget about that and just impose this for all possible alphas but anyway so now we have four redundancies parameterize by alpha mu well except alpha isn't totally arbitrary so it satisfies one constraint so there's three redundancies here so we have minus three redundancies equals the two degree of freedom that we know the massless spin-2 particles should have okay so this is this is linearized gauge redundancy and this is linearized if your morphisms but you can see that both of these things are both of these things are at this point our personal convenience as theorists for describing this physics we're trying to describe these massless particles in the way that makes locality completely manifest and the only way we can do that is by introducing these large amounts of redundancies in our description of the physics perfectly great it's incredibly convenient and it's very very useful but it's important to remember that it's not really there it's it's in our heads okay all right so let's just quickly go through some of the most zeroth order consequences of this some of the most zeroth order consequences well so let's let's first talk about let's first talk about the simplest things that are invariant let me just do it in momentum space directly it's a little easier so for spin one the polarization vectors directly are not invariant but something we can call F mu nu which is P mu epsilon mu minus P nu epsilon mu is invariant that thing is invariant under under that redundancy okay so this is the invariant content if I hand you an epsilon this is the this is the this is the invariant content F mu nu only depends is only me only depends on the topology States the linear combination of some F mu news for position plus and polarization minus okay what's the analog for HM you knew well of course if you know gr you know all we're doing is linearizing is linearizing around flat space so you know that what we should be expecting to find is the Riemann tensor right the the rule on tensor is the linearized rehman tensor is the only thing we should be able to write down which is invariant but I just want to just just for fun I want to pretend that we didn't know any of that and see how might we how might we invent it of course there is my mean at this at this level there's lots and lots of ways of inventing it but I want to tell you a just a fun little way of inventing it so um so you might be motivated by saying okay look there's this H mu nu unlike the other case where I just had epsilon nu which was a vector here I had H mu nu which is a matrix right so so let me let me see if I can decompose H mu nu into a sum well I should be able to uh I should be able to decompose hmm ooh into something that looks like this where there are four where capital A and capital B run from one two three four this is just saying that if you have an arbitrary symmetric tensor you can decompose it as a sum of a bunch of rank one just the vector x vector x times itself okay notice very importantly this is not the veal bun okay this is not the real line I'm not doing this decomposition for the whole metric I'm doing it directly for H mu so this is not something that you're used to seeing writing H mean you with something quadratic okay so we're running H medium itself is something quadratic but I'm doing it for for little reason which is that if we if we imagine that the Delta mu have a standard gauge redundancy so this is some alpha a P mu then you can trivially see that under that shift H mu nu goes under does one of these guys let me put a Zeta here okay so with that with that transformation the transformation of H mu nu is of that form where the Alpha is given in terms of the Zetas and the A's and stuff like that but still the transformation of H mu nu is of that form and I can build any transformation for the h menu that I want out of appropriate choices for what these Thetas are okay all right but then we can very easily come up with an invariant in fact it's clear the only possible invariant we could have you see I could just write down the F mu nu for one of these guys so I could write down P mu a nu a minus P nu a mu a okay that's clearly invariant under this transformation but it can't be the answer it can't be written in terms of the HV to is obviously because these A's and B's are because the hm I knew has a symmetry if you like under doing Lorentz transformations on the a and B indices so anything I do is got to contract the indices for a and B so this camp so this is invariance but it can't be written in terms of the HM you use the only thing I can do is to just repeat this trick and just write another F mu nu here's another one P a a be B minus P be a a oops P theta alpha be a - a B and this guy is now guaranteed to be invariant under the under the under the linearized dips on H but we but we can very easily read off what this is this is P mu P alpha and then that guy is just H nu beta minus P mu P beta H mu alpha minus P nu P alpha H mu beta plus P nu P beta H mu alpha and this is I think up to a factor of two maybe a minus sign which I won't remember this is nothing other than the linearized rehman tensor that's the next way remembering the linearize Ramon tensor and also remembering why it has to have this funny property that it's antisymmetric in mu nu and antisymmetric in alpha beta way to remember it is has exactly the same symmetry properties as an f mu nu times in FF of beta I stress that this is not this is not the usual view lines and in fact this little teeny tiny interesting fact that we can write the R mu nu as kind of the square of the gauge Theory F mu nu is part of a whole as a tip of an iceberg that relates gravitational things gravitational scattering amplitudes ultimately to the square of gauge Theory scattering amplitudes and this little fact is a very very very beginning of that simple of that of that story anyway this was just an aside of course this is not such a hard formula to remember and you could remember and you could do it other ways but I just wanted to point out but if you've never heard of the Riemann tensor you're just screwing around trying to find invariants it's clear why this is a sort of object you're going to get with its standard anti symmetry properties and so on okay so these are the these are the these are the completely invariant the completely invariant the H mu and the epsilon news aren't invariant but the F mu nu and the army new alpha beta are and we can even write down things like the the the equation of motion so so for example for gauge Theory the equation of motion is P mu FP u equals 0 that's just forces P squared equal to 0 so when you here we get P squared epsilon mu that term vanishes because that p dot epsilon is equal to 0 so that's the gauge Theory sorry that's the spin 1 a equation of motion and doing exactly the same thing here it's clear we have to get a P squared it's a P squared that's going to be set to 0 to get a P squared we have to contract some of the indices in this army new alpha beta and it's easy to see that the the index contraction is the usual one that gives you the Ricci feel linearized Ricci tensor so that's time the exercise step that you can do okay so we're starting to see why it is to describe this this physics in a yes sorry just-just-just using just using ada plot space metric everything everything here we're talking about massless spin-2 particles in black space alright now now we can start moving beyond just describing the free particles are starting to talk about interactions and I mentioned I'm going to tell you sort of three different ways of thinking about why gr is the unique long distance theory describing massive spin to particles this way is basically the oldest way and it's really ungainly but it starts off very simple but it becomes rapidly ungainly which is that you try to write down interactions for MU or H mu nu and already at the linearized level you see that it forces the sources for these guys to be conserved so that gives you an idea that HTML to an energy momentum tensor that the aim you should couple to a current but then then you try to bootstrap your way to write more and more complicated interactions and as you write more complicated interactions you find at every step but apparently you can't do it but you get to keep redefining what you mean by this redundancy by higher and higher order Corrections in such a way that you build up uniquely the structure of yang-mills at low energies or gr at low energies um I only going to go a little bit down that road just to remind you how the how the story with the the first interactions with what with matter looks so it's a perfectly fine way to go about things but as I said it's not it's not particularly illuminating as we go to higher not nonlinear level and the other ways of thinking about it which I'm going to tell you about I think are deeper and and and and and more incisive as well okay but the most basic things are that if I'm going to couple this aim you to something or if I'm going a couple this HM you new to something which by malice of foresight I give familiar names to then in order for that redundancy to be satisfied we have to have P mu J mu equals zero P mu T mu equals zero doing it all in momentum space we could say in position space these are conserved quantities okay all right just imagining these is external classical sources nothing else all right now I told you already that having this redundancy is important because the polarization because it allows a theory to be Lorentz invariant okay the polarization vectors don't transform like for vectors but they transform into they don't they don't transform like Lorentz tensors but they transform into themselves plus something proportional to P nu and so if we want to guarantee that the whole physics is Lorentz invariant we had better get we had better ensure that that that under that under that shift nothing changes and that's that's that's a strong constraint for the interactions of these of these particles so on the one hand it's guaranteeing Lorentz invariance on the other hand it also guarantees unitarity so it really guarantees two things what do I mean by that well imagine we have just using these these interactions either for either for spin one or spin two just a process where we exchange these where we exchange this guy at a tree level okay so let's say let's first do it for let's first do it for a for the vector well again I haven't even written down a Lagrangian or anything yet but whatever this is whatever this this this guy is we know that it's going to have if the momentum here is P it has a 1 over P squared we put the plus I epsilon there arm as you learn from want lectures the I epsilon there is telling you something about the choosing the correct vacuum so um anyway but what what we're going to have is a sum numerator factor n mu nu whatever it is I don't know what it is yet okay and I'm going to sandwich this between J mu and J news so let's try to figure out what this mu nu can be well the various there's a very strong constraint on what and we knew is you notice that this amplitude whatever it is when P squared is not equal to 0 I can ignore the I Epsilon so the amplitude is the the amplitude is pure imaginary um but when P squared equals 0 I can't to neglect the I Epsilon and there isn't there's let's take out the i from everything the amplitude is pure real without the i but with the i epsilon there it has an imaginary part and you know that the imaginary part of 1 over X plus I epsilon is Delta of X so there's an imaginary part proportional to Delta of P squared and the imaginary part of the amplitude tells you something about a production cross section that's just the standard optical theory so whatever this n me new is I don't know what it is for a general peak but whatever the N mu nu is at P squared equals 0 I have to be able to interpret n mu nu as a sum over all the states that can propagate all these spin 1 states that can propagate the two felicity's so that means that at P squared equals zero I need to have that n mu nu is the sum of epsilon mu of P our epsilon mu of P and it's a sum over the holistic H equals plus or minus one okay yeah we are in a moment yes yes but so um so let's look at what this polarization sum looks like well this is a reason Edward was asking one one reason you could be asking the question is you might look at this thing and just be absolutely terrified because we know that this thing cannot be Lorentz invariant right we've just gone through saying these polarization vectors aren't for vacuole it can't be a Lorentz invariant so this sounds really worrisome this numerator at least when P squared equals zero cannot be cannot be Lorentz invariant so let's just compute it let's see what it is well remember epsilon mu is like is let's with zero a 4 plus minus 1 over root 2 plus minus I over root 2 0 let's just focus on this two dimensional subspace so I won't write the rest of them so but let's let's look at what that let's look at what epsilon mu star epsilon mu looks like well just as a vector this is 1/2 1 I 1 minus I so this is 1/2 1 minus I I 1 that's for publicity for - olicity it would just switch that sign and so if I sum over the two of them epsilon mu plus epsilon noon plus star plus epsilon mu minus star epsilon 2 - well I just get very simply 1 0 0 1 of course you could have expected that just a completeness relation for this two dimensional subspace ok all right well but that's that's that's as promised that's a complete disaster because this is something that's only nonzero and the in the 1/2 space it has zero components than the time and the three component so it's not Lorentz invariant okay it's not Lorentz invariant but we can write it so we can write n mu nu in another way we can write it as negative 8 mu nu I should have said I'm working with plus minus minus convention we can write it as negative 8 mu nu now negative 8 mu nu now has a 1 and a 1 in these two components right plus 1 on the +1 but then the rest I can write as P mu P bar nu plus P nu P bar mu over 2 P P bar where what is P bar well so P mu is e if P mu looks like that P bar just reverses the time component that's how we define P bar highly non Lawrence invariant operation okay so that's fine that's what the numerator looks like okay okay that's the Lorentz invariant pieces of the Lawrence nando variant piece we expected because the polarization vectors are not Lorentz invariant however the answer is Lorentz invariant the answer is Lorentz invariant because the non Lorentz invariant pieces are proportional to P mu and to P nu and so therefore when they're sandwiched between the two J's the non Lorentz invariant pieces cancel out okay so you see very vividly here what the gauge redundancy is buying us it's buying us as the answer is Lorentz invariant and that it's unitary is that clear to everyone so this is the waving without writing on the Lagrangian or gauge fixing or talking about any of that stuff the structure of the propagator is completely fixed by just Lorentz invariance and unitarity right we just we just did it and we see why we need this redundancy and how it's buying both things for us here the redundancy is buying for us that this is what the numerators got to be for the answer to be unitary but for it to be Lorentz invariance these non Lorentz invariant pieces and better cancel out and that's exactly what the redundancy is buying for us precisely as advertised ok ok but this now tells you something very nice but you should feel free if you want to do any competitions with this intermediate guy you can choose and it's a complete convenience but can choose for example a mu nu to be minus 8 mu nu then everything will be Lorentz invariant throughout your calculation right of course if you do that it looks like you're dragging around these other 2 degrees of freedom but yours but you're not right you're not because this is always sandwiched between the external currents which because they're conserved make sure that these extra degrees of freedom aren't contributing anything this is only one choice I can do all sorts of other things that can add any amount of P mu P nu for example over P squared do anything I want as long as it's proportional to P mu P nu right and and if you know something about the underlying lagrangians and gauge fixing blah blah blah this choice this freedom reflects all the different Lorentz invariants gauges I can choose for this propagator the one where it's minus 8 mu nu is the Fineman gauge it is yeah I guess they're all yeah it's a 5min gauge different values of zero or other Lorentz invariant gauges okay so that's the story for spin-1 let's repeat the X up and let me just make one more comment here just so I mean of course we have the whole propagator so we have the whole relativistic information but just for fun we can see that we have derived from essentially quantum mechanics and Poincare invariants we've derived that like charges repel okay because there's a minus sign in front of that aadum you knew and that so so that means that the for not nonrelativistic external sources where the JMU is you can just approximate by the zero component the Rho component there's a minus sign in that interaction which which traces through two like charges repel okay so let's repeat the exercise we're going to do exactly the same thing now for massless spin-2 so if it's a massless spin-2 so there's some you new here and there's some alpha beta here so again there is an i over P squared and an N mu nu alpha beta and once again we're going to do the calculation of the polarization son just like we did there okay now it's easy to see that there's a particularly nice set of polarization vectors oops negation of a ratio that the polarization vectors for Felicity Plus for gravity is nothing other than the product of the two Felicity plus polarization vectors for spin one if you just think about it it has to be because because when I when I pick up the rotation around the direction of motion I get a one unit of angular momentum for each one of the for each one of the spin one guys so that's going to give me two units of velocity for the other guy alright so this is this is this is now this this was a vector so this is going to be a matrix once again it's only going to be nonzero in those two components the X in the y direction so let's just keep keep track of those so this is a 1 I 1 I I'm ignoring the half okay so this is um so this is a 1i I why okay you knew okay so and so similarly if I do the minus here so now the polarization some that I'm that I'm interested is the summer of the felicitous epsilon mu nu let me just write it epsilon u nu plus star epsilon alpha beta plus plus epsilon mu nu minus star epsilon alpha beta minus that's these numerator factor sorry shouldn't Laura yes it should be thank you alright so so what is this this is a this is this is one plus I plus I minus 1 let me write it like this this is 1 minus 1 plus I 0 1 1 0 mu nu 1 minus 1 minus I 0 1 1 0 alpha beta and I just flip flip the sign of I okay so this is 1 minus 1 alpha beta 0 1 1 0 nu nu sorry okay so that's what their polarization sum is all right let's try to recognize this as the the tensor in mu nu alpha and beta oops you see whatever it is is clearly symmetric in swapping mu nu with each other often beta with each other and the other thing which is clear just from the matrices as written is that it's traceless in mu nu and traceless and alpha beta separately okay so that means that that's this numerator up to a constant it's some constant times the following a mu nu a to alpha beta plus a 2 mu alpha a 2 nu beta state a new beta in a new so these are the only possible tensor structures that I could have okay so there's one that makes up the mu nu and alpha and beta and one we're immuno and alpha beta are together and that relative minus sign is just to ensure that this whole thing is traceless if you take the trace you expect mu nu notice the moon here are just referring to this two dimensional subspace so I just get a factor of two from these traces which compensates for the two that I get from the sum of these these guys okay all right well so so again here this is when mu and alpha beta are restricted to being one two now this I'll leave for you as an exercise to complete this as a whole expression just like we did before with the P mu P bar nu plus P nu P bar mu you can see that this is in fact the numerator is exactly equal to that tensor for all values of mu nu alpha beta plus something with a P and a P bar okay now with two peas and 2p bars in precisely such a way that that extra stuff contracted against the team you news will always give us zero so that's why everything is going to be Lorentz invariant and therefore for all intents and purposes whenever we see a propagator we can replace it by that object okay oh sorry I didn't tell you what this constant is yeah so now we just so this is the structure which is forced on us just from just from the symmetry and the traceless honest and then if you just evaluate left and right hand side it lets say all the all the new new alpha beta are equal to one you discover the constant this one alright so so we've learned that we've learned that in exactly the same way as for gauge theory as for spin one we have this effective interaction and the gauge redundancy allows in this case the linearized if you morphisms allows the thing to be unitary and Laurentian back now a little exercise another little exercise for you these are my homework problems by the way another little exercise for you is to work out what this is in D dimensions and convince yourself that in d dimensions it's just a 2 over d minus 2 that should be completely obvious just from the general structure of that just from the general structure of that tensor requiring it to be traceless forces that negative 2 over t minus 2 and this has a number of musing consequences for example well before we talk about the amusing consequences let's just pause and remark just like before of course this is the whole Lorentz invariant propagator structure um but we learned but we can for example take the nonrelativistic limit in for dimensions we get a plus one plus one minus one for a total of plus one coupling to the matter density that was to be contrasted with the minus one that we got for the spin 1 and so while like charges repel gravity attract so that basic fact is a consequence of Pangkor a invariance and quantum mechanics once again but you'll also notice that if d is equal to three we get a plus one plus one minus two is equal to zero so in two plus one dimensions there's no force at all between two between two masses so it's not and that's associate of the fact that there's no gravitons and two plus one dimensions either okay all right so any questions about this this is probably very familiar to at least a good good good fraction of you but I but but normally sometimes these things are seen in a in a somewhat different direction you start with the Lagrangian you start with a gauge symmetry you gauge fix and so on this is the point of view which is just emphasizing that it's all physics okay it's we're right and everything that we did we did by by necessity okay okay but as I said so that this is for the most baby version of just linearized exchange and everything and if you want to go down this direction and show that the only possible interactions you can have are are the rest of the interactions complete into gr or into yang-mills it starts being a starts being a little bit of work although the essential reason is clear okay the essential reason is we need to have these we need to have in order to have these redundancies we have to couple to conserved currents and when the gauge but one the spin one particles themselves are charged or are because because this has got a couple to the energy momentum tensor because everything can a genuine momentum we have to have we have to have nonlinear interactions I mean and then you have to start building up the structure of the nonlinear interactions in a way that's consistent with this redundancy okay but I want to pass to a second argument again these arguments are very closely related but the just seeing the same physics from different ways and this is Weinberg's argument it starts being a little bit more um so so instead of writing down interaction lagrangian's and checking whether things are whether you can redefine transformations and so on this is starting to get closer to asking for consistency of physical processes and it tells us a little bit more of course we would learn all these things from the other point of view too but it tells us why the only interacting theories we can have for spin one well for spin one it's non abelian gauge theories in general spin two its gr and that you can't have anything for higher spin and so we'll learn that quickly as well and Weinberg asked the following question imagine you have some process let's start off doing it just for particles of what it doesn't matter what imagine you have some some process with a bunch of particles coming in and going out and there's some amplitude for this to happen so this is just a bunch of particles amplitude p1 up to PN now what do you wanted to know is what do you want he what what do you wanted to study was the amplitude for on top of having this process happen having some extra a are emitting an extra massless particle now we can talk about what that looks like for we can talk about what that looks like for any spin spin once been to anything so let's say that that momentum is Q and if I was imagining drawing diagrams then this would be P coming out for any given particle LBP I coming out and so this would be P plus Q in there okay now what he was interested in is what happens in the limit where the momentum of this particle goes to zero okay limit where this emitted new emitted particle is very soft so you wanted to study this amplitude and Q in limit as Q goes to zero now imagine for a second drawing a bunch of tree diagrams you could draw you could attach this P well let's let's let's with a moment just this imagine looking at this subset of everything that could happen where this two is attached to one of the external lines and let's see what what what we get here well from this from this propagator we get something that looks like P plus Q squared minus M squared whatever the mass of this particle is this is I mass squared of particle I so since Q squared is zero this is nothing other than 1 over 2 pi dot Q it's a P I squared is equal to the M Squared and the Q squared is equal to zero so there's a source of an of an interesting divergence as Q goes to zero so we can have some wood so there is some possible singularity as Q goes to 0 and that means that by sending 2 to 0 we can isolate we can try to isolate the possible terms that can become large you see it's actually easy to see that as Q goes zero the only terms that can become large are the terms where this emission takes place on one of the external legs if it took place somewhere inside the diagram imagine inside the diagram is a very very complicated process inside the diagrams by definition all those propagators are off shelf and so you don't get this cancellation this perfect P I squared minus M Squared cancellation you get the 2 PQ plus something else and those don't develop any singularity as Q goes to zero so the only things that can develop singularities as Q goes to zero are the ones where this emission takes place off of the external ones yes sorry yeah we're just talking about we're just talking about some leading water interactions here we're just talking about we're just talking about three interactions here that's that's that's what he was talking about okay now but we're not done now we need to figure out what that we need to figure out what the what the numerator and that interaction could possibly be and Weinberg assumed that that so whatever it is is going to dot some epsilon mu of Q and whatever it is there's going to be some function just of the PI so let's say we have spin one the assumed that we have the most leading possible interaction the biggest possible interaction which just contracts it with a P the peas are big the Q's is small so we have the most leading possible interaction is an epsilon dot P okay again forgetting about writing down the Lagrangian or anything of course you get all this if you wrote down the Lagrangian but but even without doing that this is the most leading interaction that we could have so let's assume that's that's actually the interaction there is all right so so then we have this statement that the M of the P and the Q is just the M of all the peas times a factor which is the sum over all the eyes of epsilon mu of Q dot P I'm you over to piq oh and of course there could be some constant here depending on what the via particle is so let's call the constant e I the stand-in for the for the charge okay all right so so as Q goes to zero this is a this is a nice expression for the amplitude to emit a very soft particle okay now but once again whatever this amplitude is it because of Laurent because of all the reasons we said if I shift epsilon 2 epsilon plus anything proportional to its momentum I need to get the same answer which means that if I replace epsilon by Q I need to get zero right that's Lorentz invariance and unitarity up make for set so that if I replace epsilon by Q a gets zero so let's do that let's replace epsilon mu with Q mu and the factor call this the soft factor s just goes to one it just goes to the sum of e i p IQ over two p IQ which is 1/2 the sum over AI so I learn that charge has got to be conserved so the only way to have consistent interactions for this massless spin-1 particle is sub charge conservation okay now it's very simple to repeat this argument for spin to nothing changes basically so if you don't mind they'll just a race here let's call this instead of E I I don't know let's call it Kappa I and now this is an epsilon mu nu and the most leading possible interaction I can have once again would just put P mu P I knew it's the biggest possible interaction I could have the soft factor here would become epsilon mu nu P I mu P I noon Kappa I everything is the same but now this becomes more interesting if I replace epsilon mu nu with Q mu times anything I should get zero and so F goes to the sum so one of these supplies I contract that pea I with Q mu so that cancels this guy but I've left with the sum sum alpha nu here doesn't matter that the sum of Kappa I P I do so now I have to have this object vanish now we already know the sum of all the moment of inertia's that's momentum conservation of lament or drawn-in coming here okay so we know that the sum of all the moments already vanishes so if the Kappas were generic things then that would be putting another condition on the external momenta other than momentum conservation and that means that for in four dimensions that that the scattering could only happen for discrete angles so that's that's crazy the amplitude should be a smooth function of the external momenta so the only way this is consistent is if the Kappa eyes are all equal to each other okay all the cap eyes are equal to each other then this constraint is automatically satisfied as the consequence of momentum conservation and that's how we discover the principle of equivalence that's so that's a remarkable fact all the Kappa eyes are equal and that's the beginning of again a story that you can now develop further to see that it's that it has to be the interactions of GR and so on but I'll be the most striking zeroth-order fact about gravity the universality of its couplings is just a direct and trivial consequence of understanding its soft limits properly okay let me make a small comment about I make it just a couple comments and then how long do I have myth use I'm done okay so I'll just make a very small comment and then I'm like a very small comment and then I'm done I'm moving a glacial pace that's interesting okay okay let me just tell you how a non abelian gauge theories work in this language everything is the same basically um uh well so let me uh let's let's let's talk about let's talk about a simple process like ordinary Compton scattering if you imagine ordinary Compton scattering writing down just the vertices that we've been writing on the vertices that we've been talking about then there's a Mew here and a new here then it needs to satisfy once again but that if you replace epsilon mu with its corresponding Q that I should get zero now when these are photons are just a single spin one particle if I look at this if I look at this process there are two different things that contribute these two different channels just by exchanging by both symmetry alright and so when when you replace epsilon with Q you get the sum of two terms and they vanish but each one is non vanishing you need the sum of the two of them to vanish okay so that's that's that's all all we need to know to now generalize this to the case where there are many of these guys so let's say we have many of them ok so there's an index a and B so that many of these massless in one particles and we can also imagine there's many mattr fields so here's an eye here's a J here's some particle K in the middle I okay J okay so whatever is at this in addition to the usual factors that we normally have we're going to imagine that we have I K a that here we have something that I'll just call ta I K so in addition to the usual momentum dependent factors I'm going to have something that I'm just calling ta I K all right so now let's look at these these two two pieces you see this piece is now giving me in addition to whatever I got before something that looks like ta I K and then TK t B KJ so it's the product of ta TB the I J component the other thing is giving me something that's that's the other way around TB ta the IJ component and I know that they have a relative minus sign because when it was just one number these two things cancelled each other perfectly but now now you see that that this thing is not going to vanish in general of TA and TB don't commute with each other right this is in fact nothing other than the commutator of TA and TB so once again if I were ruling the universe I would stop and say oh well it's impossible too bad you can only have one kind of massless spin-1 particle but you can try to get around this remember the goal here so just write down consistent local interactions for these guys so you can get around this bite by saying oh there's one more contribution I can imagine having which would involve a self interaction between these guys so if this is a b and c this is still i and j I can say I can give this guy some in addition to its momentum dependent factors let's just call this F ADC and it's just a name and to have any prayer of this working even ignoring how all the momentum factors are going to work out just to have the a IJ structure work out it must be that the commutator of PA and PB is equal to I'll put the I in for fun at ABC PC because here this vertex is at TC I J and of course it works I mean if you put in the momentum dependent factors it all works so ah this is how non abelian gauge symmetry hits you in the face you're trying to break down the consistent theory of interacting massless spin-1 particles no way you can do it no way you can do it without well this is the most general structure that that you can have okay okay a final just incidental comment and then off to leave to a next time the third way of understanding this so a final incidental comment just going back to just going back to the structure Weinberg's theorem is you notice that I kept saying in line breaks there are the most leading possible interactions most leading possible interaction oh sorry very quickly why can't we do spin why can't we do spin three or spin four I sorry that was the point I was going to say that if why can't we do spend three or spin four well if we're going to do spin three exactly the same argument if the couplings are Kappa I will tell us that we have to have the sum of I Kappa IPI mu P I knew was equal to zero and now that's just impossible right because now that battle-ax that's another constraint on the external data which in again in what would force the would force now totally force the amplitudes only be supported at certain funny angles coming up okay so it's just impossible again all of these things have simple avatars in the Lagrangian way of talking about things too right you know you can't find appropriately conserved quantities and so on but it's just another way and very directly physical way of talking about it so that's why the only possible interactions we can have for massless particles or for spin ones and for spin two modulo the following caveat this is the last thing I want to say I kept saying and in this argument we saw over and over again that you can have that you can have in Wiemers argument we forced it to be to be couplings that had all P's upstairs so we contracted the epsilon with all powers of this hard momentum of the particle coming out okay this and I kept calling it the most leading possible interaction you could have that's the interaction of the most number of momenta upstairs that if you imagine nonrelativistic particles on the outside would give you an inverse square law force or in one of our potential if you had any fewer powers of P upstairs you would not have something goes like an inverse square there would be a higher power of the via potential and actually theories like that are a dime a dozen and you can trivially do it for spin 17 if you like so in fact here's a particularly trivial way of building interacting theories of any spin let me give you a different theory for spin one particles for example take that F mu nu that we built beautifully invariant under everything take F mu nu cubed at me nu to the fourth if we go to the fifth write down any old powers of F mu nu that you like if you want a couple of to matter just couple of two matter using F mu nu okay that's perfectly fine those theories are totally consistent however they don't mediate any kind of inverse square law forces that's why we haven't seen any these particles out there maybe there are massless spin 17 particles out there okay but if they have any interactions at all they can't mediate inverse-square law forces so they're not really they're not really visible long-range things similarly for for similarly for us spin to I could take that linearized Riemann tensor qubit take it to the fourth do anything I want with it I can couple to matter with it okay a coupling to matter in the FME new case is like just giving matter dipole moment type coupling stew this stuff and the interactions are dipole interactions as well okay so none of those things are ruled out by Weinberg's argument because they all involve fewer powers of peak in the numerator but but the statement is that anything that can mediate inverse square law force anything with leading to derivative interaction at long distances the only consistent theories are massless spin-1 non abelian a knows massless spin-2 gravity yeah now what i was going to do but which i guess i'll have to start doing next time is give a third argument for this which is in in my view the most invariant and the sharpest and most concise argument which also immediately allow us to take care of many of the other theorems that that you've all heard about of many of you have heard about which is instead of you see here here we're sort of baking in a find me diagrams a little bit of a picture of and writing down interactions with some local lagrangian that's all perfectly fine and perfectly perfectly good but really the very sharpest statement you can make is to just write down the only possible structure that the the scattering amplitude for these massless particles could possibly happen so it's a little analysis which shows that the structure of the possible 3-point amplitudes is completely totally fixed by Pangkor invariants nothing else and then when you start exploring the consistency of for point functions for point amplitudes you discover given the three point amplitudes that are completely nailed by panca ray invariants you discover all the constraints that we've seen already and more discovered there's simply no way and this has nothing to do with the formalism polarization vectors in this that it's just it's a theorem okay you just write down the only possible structure you could have for the scattering amplitude you demand that it satisfies you demand that it satisfies that the to to to scattering satisfies unitarity and lo and behold you discover the only possible consistent theories of massless particles or yang-mills or five to the four theory you know normal scalar theories or yang-mills or or gravity you discover that you can have higher spin things provided there of the stupid linearized R cubed F cubed and so on form you discover that you can't have theories of multiple massless spin-2 particles and so on all of these sort of famous constraints and what the long distance physics looks looks like are all a consequence in this picture of staring it of consistency of two to two scattering amplitudes and that's that's I was finally doing that today but I think we'll have to us start with that next time and then I'll move on to talking about infrared modifications of gravity thanks [Applause]

Channel: Institute for Advanced Study

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Id: WECVq2YBduY

Length: 99min 11sec (5951 seconds)

Published: Fri Jul 07 2017