 the first speaker. Thank you very much. I just wanted to say I mean I think some of you have already been here in the previous school so you don't you know the format but just to remind just to tell the new people who have just come for the first time here you will notice that most of the days there is a discussion session in the afternoon. Now that's really part of the school because many times because the lectures are quite condensed in one hour the lecturers do not have time to you know answer many of the questions in detail. Okay so detailed discussion should be postponed to the discussion session and indeed in the previous spring of schools students have learned a lot from the discussion sessions. In fact sometimes even more than the lectures themselves. So I would really encourage you to attend all the discussion sessions and take part actively. With that I would invite the first speaker Professor Hartman and he'll be giving lectures on the average non-energy condition and care. I guess it's working. Okay thank you to the organizers and to ICTP for hosting this. It's great to be here. I look forward to meeting the students here from all over the world. So please do come introduce yourself to me after the lectures. Please interrupt during the lectures. I'll probably postpone a lot of questions so that we can get through lots of material but don't hesitate to raise your hand or just shout out since it's a pretty big room. Just stop me and we can clarify things as we go. So the topic of these lectures is the averaged full energy condition. Quite big is this big enough? Okay I'll probably start writing smaller but stop me and I'll start writing big again as soon as I remember. Okay so when you learn general relativity you usually come across some energy conditions. They show up in all the GR books that go by names like the strong energy condition, the weak energy condition, the dominant energy condition, the null energy condition, and these are usually introduced in textbooks as ways of proving theorems. They're assumptions about local energy density produced by matter that allow you to say things about space time. So all the facts that you're familiar with like the fact that black holes have to have singularities or the fact that there has to be a big bang if we trace our universe backward or the fact that we can't build time machines, all of these statements about space time, the structure of space time, and what it can and cannot do are based on some sort of energy condition. So these are great for proving theorems but all of these energy conditions are wrong. Okay so they're all violated. The strong energy condition I won't get into too much about what these are since they're wrong. The strong energy condition is just not true at all. It's very a massive scalar field to violate the strong energy condition. The weak and dominant energy conditions are violated by negative cosmological constant. The null energy condition is also wrong, violated that is, but it's in a little better shape so let me tell you what that is. I'm going to pick conventions and this will last throughout the lectures that u and v, null coordinates in Minkowski space, so u is t minus y, v is t plus y, then the null energy condition states that t u u is everywhere positive or non-negative. Of course it's can be phrased covariantly. I'll pick u, u just to keep the notation simple but the statement is that the stress energy density dotted into any null vector would be everywhere positive. So this one is in better shape because it's obeyed by ordinary classical matter. Any reasonable classical matter you can come up with obeys the null energy condition but it's still violated by quantum effects. So let's see why that is. So for example if you have a free scalar field then classically is up to some factor just d u phi squared. The g u u component of the metric is zero. There's no null-null piece so this is just obviously positive classically because it's a square of something but in the quantum field theory t u u is an operator now so it's the normal ordered d phi squared where phi is now the field operator and because of this so the thing the thing inside is still something times its conjugate so it would be positive if it were just that but the normal ordering mixes around some of the terms it adds some new terms to this and this is just not the square of anything anymore. It's easy to cook up quantum states where it's not positive so it's not a perfect square it could go negative and indeed it does. For example if you take a state like the zero particle state a superposition of the zero particle state and a two particle state then what happens when you calculate this you plug in so you plug in for the field operator in terms of a and a daggers you do the normal ordering to pick up some extra terms and you can easily check that in a state like this those extra terms dominate the the null energy the only contribution and can come with either sign depending on how you pick epsilon. Another easy example to see where the null energy condition is violated would be in the chasmur so in the chasmur effect you have two conducting plates and between the two plates there's just a constant negative energy density so obviously the null energy conditions can be violated in that region so again that's quantum effects coming in and you could trace it back to these normal ordering terms. All of these local energy conditions and in fact this isn't just a failure of imagination there's no local energy condition that's true in quantum field theory so all these are violated now on the other hand total total energy is positive in quantum field theory otherwise we'd have a problem it wouldn't make sense it would be unstable so the total energy integrated overall of space is positive the local energy can go negative but actually and this is the statement of the average null energy condition there's something in between which is the statement that taking this null energy tuu and integrating it over a null line u from minus infinity to infinity it's rid of those offending terms this gets rid of those normal ordering terms that led to negative contributions in the scalar and if you calculate if you plug in the the quantize scalar field into this expression then when you do the integral you'll find something that's like integral dk ak e to the minus ikx ku mod square so in the scalar field we've just you can just prove by direct calculation that once you integrate this positive this is a positive quantity more generally we could ask whether this is true in any quantum field theory and that's going to be the subject of these lectures so this is the anac the average null energy condition is the statement that this operator e is non-negative in any state in other words it's it's a positive operator yeah so this is this is a definition so i'm defining the integrated null energy to be e and all i've said so far is that if you if you just explicitly calculate this for the free scalar you can check that it's that it's a mod squared of something and and check that it's positive so at this point this is just an interesting observation that hey we found negative energy but it turns out we don't have to integrate it three times we don't have to integrate it over space we only have to integrate it once so the total energy through this the energy flux on this single null line is enough is enough to get a positive quantity and then the subject and then we're going to try to understand why that's true in general that's the that's the goal that's right that's right so in space time um you know there's in space time over here there's a transverse direction which sort of comes out of the board but we've just picked a single null ray say through the origin and uh integrated the stress tensor along that null ray no not no they're violated no i don't think so they can be constant and they they can be violated in a constant way by a cosmological constant so i think not all right at this point it's just a coordinate i could i could define this in a covariant way by by saying that this is an integral of a null ray of t dotted into the uh we could have an affine parameter here and then dot into the usual normal uh but i'll just i'll just use the u coordinate good so so um this so this condition was introduced in gr and it was a statement about curved space and understanding things in curved space but i'm going to focus exclusively on minkowski space so uh just the state just the question of whether this statement is true already already it was motivated by gravity but already the question of whether this is true in ordinary quantum field theory and ordinary flat minkowski spacetime is already a very non-trivial question about quantum field theory and that's why i'm going to stop um there's a proposal for something that's true in curved space but it's uh but no there's no proof of it so the outline of these lectures um is i want to describe the proofs or derivations of in interacting quantum field theories including strongly interacting quantum field theory in five minutes we did the proof for the free scalar uh and other free quantum field theories were considered in the 90s and it was checked that this seemed to be a true statement but it's been understood in the last few years that this is actually a fundamental property of quantum field theory and that's been understood from several different angles so the different angles are of all causality so this will be based loosely on a paper by me sandpan kandu and amir tijini from a couple years ago the second method and these are out of order i've reordered them for the lectures these are not historically ordered uh but the second method that we'll talk about is holography this will be based loosely on uh paper by kelly and wall from i think 14 although i didn't write down the number uh and the third will be from the point of view of quantum information and this is based on a paper by faulkner li barker and wong from 16 so i'll just come ask me for the for the full references or i'll post some notes at some point so that you have the references uh but they all all of the papers have no energy in the title so they're not too hard to find okay if we're gonna prove the ennec three times then we better have some good reasons to do that so before we jump into it i'll give a few words of motivation about why we want to understand the ennec uh so roughly roughly three uh so first of all this is a fundamental property quantum field theory we're really we're not talking about anything exotic here so as i said we're going to be working in flat spacetime we're just considering general quantum fields so the ennec uh is is a consequence of unitarity the very fundamental property of quantum field theory but it's a it's a uh repackaging of unitarity in a very special way that seems to have lots of interesting consequences it applies essentially to all well actually to all reasonable quantum field theories including everything from say the standard model to to stat mech models like the ising model in fact there are predictions that you can make using the ennec predictions for for quantities in the ising model which were new predictions from the ennec and have been confirmed numerically so these are these are things that in principle you can even go out and test the second reason is the fact that there seem to be so many different ways of thinking about it just in itself is very interesting so uh what we're going to see is that this is related to understanding quantum fields in the highly boosted limit makes sense we're talking about integrating over a null line so this is going to be connected to properties of quantum field theory at null separation and what we'll see is that quantum field theory at null separation even in the strongly interacting case is especially tractable okay so when things are strongly coupled qft is hard but in the light cone limit it's not so bad so for example we usually think of ads cft is something sort of mysterious and difficult and hard to map between the two sides but in the light cone limit everything is solvable it's a solvable limit of ads cft where you can explicitly map bulk and boundary and beyond that um it's a limit where holography is actually even much more general so ordinarily holography is a statement about a certain class of conformal field theories i have the large n and a large gap in the operator spectrum but in the light cone limit uh that's actually not even the case actually holography is essentially just a derivable exact statement about conformal field theory as long as you're restricted to the light cone limit reason and i won't really get into this in these lectures but is a connection to gravity quantum gravity so uh you know it's not it's not a coincidence that the anek was first discovered or discussed in the uh gr community this came from relativists uh it seems to be a fundamental a fundamentally important fact about how quantum fields couple to gravity and uh so it turns out that the anek is essential for for getting things like causal structure and curve spacetime to work it's essential for getting black hole thermodynamics to work uh and is it also underlies some of the more recent work on uh thermodynamic and and tropic and other relations and quantum gravity things like uh quantum focusing the quantum null energy condition various other things that you may have heard about okay so um the first so that's that's uh sort of the introduction and uh the first one that i'm going to talk about now is the causality based argument for the anek and i really want to sort of i really want to start at the beginning um and set up the background here so we're going to spend a while doing that it'll be a while before we really get to uh the argument um but let me just draw a picture giving the uh sort of the strategy idea of what the strategy is uh is we're going to look at correlation functions in minkowski space it's working in real time so this is my minkowski space again and we're going to look uh we can think about four point functions although there could be higher point functions and uh four point functions are related to causality uh because you can think about them as sort of an experiment where two of the operators create some kind of background say they create some excitations here so they create some background and then you can ask whether in that background created by those operators you can ask whether you can send a signal from here to there you can think of some of the operators as a background and the other operators as a probe and then you can set up an experiment to test whether that situation is causal to test whether uh signals can bend out of the light cone in the presence of this background yeah i i don't mean much by it i just it's just words for for the question was what does background mean uh that's we'll get into that um but i just mean in the very loose sense of this is some kind of excitation that this probe has to go through so you can think of this these operator insertions as creating some cloud of junk or particles or something and then this probe has to interact with that stuff uh as it propagates so just in the very loose sense and we'll see precisely what's meant as we go along uh so we need to understand correlation functions and especially uh we need to understand correlation functions in this highly boosted limit so that's going to be the first topic uh and i'm going to sort of start from the beginning and discuss both of these things just general properties of correlators in qft including some things that that that are important but not usually stressed in qft courses and especially focusing on the light cone limit i will mostly assume that we're in a conformal field theory for this for this part one that we're going to be in an interacting conformal field theory most of what i say uh also applies to conformal field theories in the uv that are deformed by some relevant operator uh but it's important that there be an interacting fixed point in the uv since we're going to be working in the null limit uh that's that's a limit where things are effectively close together uh so it's effectively controlled by the uv fixed point and relevant operators won't get in the way but just to keep things simple i'll assume it's conformal okay so the first thing we want to get into for background is properties of correlators in the null limit the essential tool to understand this is going to be the operator product expansion so the operator product expansion uh says that if we have two operators psi of x1 psi of x2 inserted say here uh you can basically that you can zoom out okay if you zoom out if you have two operators inserted and you zoom out on this picture you look at that from very far away uh then you'll just see some local excitation or some local operator rather and uh so that means that you have to be able to at least approximately write this in terms of the local operators of the theory so there's a sum over the operators of co psi psi of x12 i've expanded uh in operators at one of the points chosen arbitrarily but that's not it doesn't really matter which one or where we could expand uh about the center or anywhere else nearby so dimensional analysis already tells us quite a bit about what this sum has to look like so uh on the right hand on the left hand side if suppose that psi has mass dimension delta psi uh then we need the mass dimensions on the right to add up to two delta psi so the only thing that we can write let's say let's say we have a a uh an operator showing up in that sum with a bunch of indices mu 1 to mu l those are the Lorentz indices for a spin l operator then we need to uh contract those indices with something so the only thing available to contract indices is going to be the distance uh the separation x1 x1 minus x2 so there's a x1 minus x2 to the mu 1 x1 minus x2 to the mu 2 all the way up to l what else can we stick in here uh well we need to get the dimensions to add up right and the only other thing we can stick in is x1 minus x2 um the x is the distance from x1 to x2 and uh just counting up dimensions on the two sides we need to get two delta psi that gives us all over the right mass dimensions so now we have to get rid of everything else so there's a plus delta o to cancel off the dimension of this one uh then there's a minus the spin of o l uh to cancel off all the dimensions from these um and finally there could be some constant coefficients c psi psi o so in a general q of t uh this is just the most general thing that we can write the most general sum we can write down that makes sense by dimensional analysis uh i've i've written this last thing as a function uh if there were if you could make if you had a theory with a mass scale then this could be a function because you can make a dimensionless function in a theory with a mass scale by multiplying m by the length uh but in a cft there are no masses uh there is no dimensionless function that you can stick here the only thing you can put is a constant and that constant is the ope coefficient so this is what the ope looks like in a conformal field theory yeah psi here is a scalar that's right but there are two nice things that happen in cft one is that this is not a function it's just a number uh the other thing is that we understand quite a bit about when this converges and when it doesn't converge the statement is not quite often you hear the statement that in in c in cft the ope converges that's true uh under the right circumstances it's true in euclidean signature when there are no other operators nearby it's not true in lorenzi's signature in the kind of situation that we're interested in so we're mostly going to not be dealing with convergent ope's but they still make a good asymptotic series so you can still use them to approximate these operators well sort of uh the anek and euclidean signature is is captured by unitarity which has a euclidean statement that i'll talk about but it's very hard to get at the anek from euclidean signature it's a it would if from the point of view of a euclidean quantum field theorist the the kinds of quantities that will write down can all be stated in terms of the euclidean theory but they would just be they would seem like totally crazy things to talk about in the euclidean theory and they're very natural things to talk about in the lorenzian theory so in the euclidean ope limit by which i just mean that uh we have so let's put our operators symmetrically uh we have an operator psi of u v and another operator psi of minus u minus v the usual ope limit that people talk about is one where we just move the operators in the in the ordinary direction that is we send u and v both to zero simultaneously and in this limit we can think of all of these all these position dependent things here are scaling to zero at the same rate so it's easy to figure out which terms dominate and uh basically there's a x one minus x two to the minus two delta psi plus delta o the minus l's here cancel against the l factors of position showing up here uh so each term comes with distance to this power so the lowest delta o's the lowest dimension operators are the ones that dominate the expansion this is often what makes the ope useful at short distances is that uh if so say say you have a conformal field theory or someone hands you some data about a conformal field theory but they don't hand they don't tell you everything about the conformal field theory they just tell you the dimensions and uh maybe the dimensions and ope coefficients of some low lying operators then you can already get a very good approximation by looking at those operators in the ope what we're interested in is not that limit it's the light cone limit the light cone limit means uh so here's my u axis and v axis again we're going to start with our operators placed symmetrically but now i'm going to take these together in the in i'm going to take them to zero distance but while maintaining a null separation so we can do that by sending this one up here and this one down here this is the limit v to zero uh u held fixed the limit v to zero with u held fixed um we get so the the distance squared x one minus x two squared is just minus u v so we get a factor of minus u v to the minus delta psi plus delta o minus l o over two uh and then what we get for those other things depends on what indices show up on our operator in the equilibrium limit all the different components of the operator we're showing up at the same order but now that which component you're talking which clarence component you're talking about is going to matter so say we have an operator with a bunch of u indices and a bunch of v indices with h of these and h bar of those there could be other there could be other transverse directions as well those won't affect the conclusions here um if we have h of these and h bar of those then that means we get a factor of u to the h v to the h bar from those contractions okay so from this we can see two things and the question the question here is what are the important operators so what are the important operators in the light con limit well um if we have v indices down here then those have to be contracted with v's and we're taking v to zero so one thing we see is that anything with v indices is going to be suppressed the good dominant contributions are the things without any v indices okay so v indices are suppressed and the second thing is that the combination now that's showing up as so we're taking v to zero let's say we get rid of all the v indices then we have a power of v to the delta minus l over two okay so uh the expansion is no longer controlled by the scaling dimension the expansion is now controlled by uh what's called the twist means dimension definition of twist is just dimension minus spin this means that very high dimension operators can be very important in the light con limit as long as they also have very high spin okay so in the ordinary euclidean limit things were dominated by low dimension in the light con limit things are dominated by low twist so what are the low twist operators in conformal field theory well the twist like the dimension uh it can't be too low it can't be too low because uh there are uniterity bounds which say that if things have dimensions that are too small you get negative norm states okay so unitary bound these come from demanding that two point functions are positive and these tell you first of all that scalars have twist greater than equal to d minus two over two where d is the spacetime dimension this is just saying that free fields have the lowest twist so like a free scalar has dimension d minus two over two and a scalar that's not free can only have a twist that's higher than that so it's saturated higher spin operators um have a slightly different unitary bound that just says tau is greater than equal to d minus two and the way to think about that is that the lowest the saturated case tau equals to d minus two is it corresponds to a conserved current so conserved currents have minimal twist and anything that's not conserved has to have a bigger twist so for example if we're in d equals four and we ask what are the low twist operators in the theory well then maybe if we have a if we had a spin if we had a conserved current uh say from a u one global symmetry then um that would have spin one obviously there's one index in four dimensions it would have scaling dimension three so tau we also always have the stress tensor so the stress tensor exists in any conformal field theory any quantum field theory and uh is always conserved so this is something with spin two scaling dimension equal to the spacetime dimension which I was saying is four here so again has twist two uh the lowest twist operators in the theory are the conserved currents uh but we're not quite done there are more because uh we can also start taking derivatives so take a derivative of t mu nu when you take a derivative the spin goes up by one so now now you can make a operator of spin three uh but the dimension the scaling dimension also goes up by one when you take a derivative so delta is equal to five and again the twist is two this wouldn't happen in the euclidean limit in the euclidean limit you take derivatives the dimension goes up those are sub leading in the larancy and limit the the leccon limit uh derivatives and it's of course we can keep doing this d alpha d beta t mu nu also has twist two I'm going to ignore okay so so if we we ask what are the low twist operators the lowest ones can be the scalars I'm going to ignore the scalar operators in the ope we can come back to them in the discussion or people have questions they won't affect anything that I say uh just because they have spin zero they don't uh lead to the kinds of terms that we're interested in and they can all be dealt with but I don't want to carry them around and then and and have to write them all the time so I'm just going to ignore the scalars uh now generically the only conserved current is the stress tensor if the theory is interacting then the only possibilities are the stress tensor and things that are spin one uh the spin one things the spin one currents also like the scalars won't affect anything they'll say we could keep them but I'm just going to drop them so this leads us to a key conclusion which is that dominant contributions count ope as v goes to zero the following operators so it's just the stuff that you can make uh out of the stress tensor and we said remember we said that we wanted things that are low twist and that the v indices are suppressed okay so the the dominant contributions are just t u u and the u derivatives of t u u expand as v goes to zero these are the dominant terms in ope yeah let's see that I guess um since I've only I've only put my scalars in the uv in the uv plane then the transverse directions you can just absorb into so I haven't even written the transverse indices you can just absorb those and those create higher dimension operators so those won't come in yes I'm about to make that comment yeah that comment too yes so oh spin one currents would also be on this list um I'm I'm gonna put I'm just gonna assume that the theory doesn't have a spin one current but it wouldn't affect any of the conclusions that we're gonna reach eventually okay so two caveats which were the two first two questions um this would be false in um a free theory in a free theory uh you can you can make a tower of arbitrarily high spin conserved currents so take a free scalar for example if you have a free scalar you can make operators like this uh that are conserved by the by the free scalar equation of motion uh and this would be a so by by adding terms like this together you can make uh a spin for conserved current and this would spoil this claim so these would also contribute in the light cone limit at just the same order of those as the stress tensor terms but interacting theory uh these conserved currents all go away so there can't be higher spin conserved currents in interacting theory their twist gets increased by the interactions that the other caveat uh is that this would be false in a two-dimensional cft uh the reason for that is a little different the reason for that uh is because the twist of the stress tensor in two dimensions the twist uh is d minus two okay so the so the stress tensor has twist zero and the fact that it has twist zero means that you can make new twist that you can make an infinite number of other zero twist things just by multi multiplying together the stress tensor so you'd also also have to include things like t u u squared t u u to the 10 v u to five t u u etc just everything you can make out of t u u's would also contribute and uh these are also exact all the all the currents that show up in the uh verisora algebra so the existence of the verisora algebra in two dimensions uh is is existence of all these extra twist zero operators and those would also show okay so so far to summarize these words in equations what we said is that as v goes to zero the o p e psi u v psi minus u minus v i'm going to normalize by the vacuum two point function of those operators just to get rid of some factors is approximately one okay i didn't i didn't mention the one yet the one is there because the identity you can also just have the identity operator showing up in the o p e that just always has twist zero so that just always shows up as a one okay so that's always the leading term and then these terms we just talked about so there's um just plugging in all our factors from dimensional analysis we have a minus u v to the d minus two over two u squared some m equals zero to infinity b sub m u to the m du to the m t u u of zero where the b m are some constants to be determined this is just putting this that's just this this sentence here with all the right factors stuck in for dimensional analysis so the leading terms are the t u u's and their u derivatives i'm going to set d equals four just so i don't have to carry those d's around but the discussion is completely the same in any dimension higher than two the answer is just the c's i'm going to calculate the b's in just a second so this is this is the vacuum expectation value on the top by writing since i haven't written any brackets this is anything yeah good yeah okay so to find the b sub m's and this is always how you find op is this is always how you figure out what op's are is you plug them into a three-point function okay so that's kind of the whole purpose of op's is to take uh complicated things like four point functions and higher point functions and turn everything into a statement about three point functions and uh that's how you figure out what the op is in the first place uh the the coefficient this b sub m coefficient is going to show up like if we were to plug this into a three-point function with a t and then take the expectation value then all these terms would show up because of the t t two-point function and in a conformal field theory three-point functions are known so i'll write the formula psi u v psi minus u minus v t u u of u three v three is just a known function which you can look up in various papers and it's c psi psi t or you can just figure it out from conformal invariance c psi psi t to u squared minus two u v to the one minus delta psi over u squared minus u three cubed v squared minus v three squared this is just a known function and uh we're only interested in the light cone limit v goes to zero so we can drop that term you need is the t t two-point function so the formula for that is t u u of u comma zero t u u of u three v three is equal to c sub t over u minus u three to the six v three cubed again this is fixed by conformal invariance c t is just some coefficient is actually it's a very important coefficient uh because c t is very roughly some number of fields in the theory or number of degrees of freedom like if this were a scalar then this would be about one if it were 100 scalars then it would be about 100 up to some factors of pi and stuff okay so i'm not going to do the algebra now but you can imagine what to do next right you take this you take this formula you put a t here and then you just evaluate it and make sure it gives the right three-point functions right i've given you enough information that you can do this exercise and uh you'll find b sub m is equal to 240 over c t times m plus three m plus five m plus one factorial okay so just some number it's completely fixed by conformal invariance independent of the theory in principle we're done understanding the light cone ope this is the answer you take this this and you plug it into that sum and uh you you're you've succeeded so that's the dominant contribution in the light cone limit but it's going to be very informative to resum this into an integral so let me write the answer and then i'll tell you comes from so in integral form the ope formula as v goes to zero is one plus delta psi over c t i forgot to mention that uh c psi psi t is negative delta psi the reason for that is the word identity so the the ope of the stress tensor is you can the stress tensor is related to the generator of conformal transformations so you can go through a standard exercise and derive this this is one plus delta psi over c t v u squared times the integral from minus u to u u tilde one minus u tilde squared over u squared all squared t u u of u and everything else zero it's not it's not totally obvious you could resum this thing into an integral but it is easy to check and that's kind of the best i can that's kind of the best i can do to explain this formula is that you can so you just take this formula and you expand t in derivatives and you just tailor expand t around zero then the the uh integral then order by order you do the integral it's totally trivial just this kernel times powers of u tilde uh and you could just check that that gives the right that that gives the right thing okay so this is just a statement that this integral against u tilde to the m gives you the right gives you the right coefficients okay so that's exercise two check that this gives v sub m to actually find the formula just sort of requires some guesswork you can just try plugging in an arbitrary function there expand both of them in tailor series work out the first free orders look for a pattern and then try to cook up one that works that's kind of how to how to come up with that okay so we're almost at the main conclusion for today uh but we're going to take one last step to reach our main formula which is that now we want to consider a double limit which is the limit v goes to zero followed by u goes to infinity this this limit is not done together it's done consecutively so first you take v to zero then you take u to infinity or uh in other words you're working on a limit where v is much less than u inverse is much less than one in spacetime that just means that first so we had our two points first we took the light cone limit so they're sitting practically on on the light cone here but then we're also going to take them to be far apart in in the u direction so we're sort of in a limit where they're sitting on the light cone but they're sitting far apart you can kind of see what this has to do with the experiment i i outlined in the beginning where you want to you want to try to throw a probe through some stuff this is kind of the kind of the limit you're talking about right you're you're throwing it very fast so it's a no limit and you're going to look at it from far away so that's widely separate so in this limit we can just look at that formula here and what happens is that the kernel just drops out right so the as we take u to infinity the limits of the integral just go to infinity the kernel just drops out and we see our friend the null energy operator so the formula is now psi u v psi minus u minus v i won't write the pre-factor again is one plus delta psi over ct v u squared times curly e where curly e is the integral du for minus infinity to infinity t u u of u v equals zero transverse direction set to zero so that was in purpose of this discussion and now you can start to see where the why we might hope to be able to derive the average null energy condition by looking at correlators in the light like limit it's because it really is just the first opera it really is just the most important operator the null energy operator is the most important operator in the light cone limit in this double limit which is limit relevant to doing these this causality test so we'll do the causality test tomorrow and derive the statement that this first correction must be positive and that'll give us the anac so we'll continue with that in the next lecture okay thanks