Lorentzian Conformal Field Theory - Part 3 - Simon Caron-Huot

[00:00:15]So yeah, so so last time we uh derived this uh all important bootstrap equation and and discuss uh some of the input into it, some of its properties. So, so there was for example reflection positivity that we saw imply that this operator product expansion that represents the fourpoint correlation function converge absolutely in any conver in any kinematics or any kinematic any four-point configurations at least we talk about Loren today And also it ensures that uh wick rotation uh exists and defines a unitary theory. So reflection positive is this property that looks a bit weird in the space that some things and this reflection is average to positive but it actually is the right thing to uh it to get these properties on the right and define a unitary Lorenzian theory in the usual sense and and we also saw that uh uh there was a when you do this wick rotation there's this concept that uh you can order the imaginary part of time that you eventually take to epsilon at the end. So ordering of imaginary part of time and that map very simply to the operator ordering on the minkowski picture.

[00:02:00]Okay.

[00:02:04]So in that way the correlator and continue gives you all the possible realtime correlators you could hope to get.

[00:02:14]Right. So, so today we'll uh talk more about the loren geometry and and so so we'll cover a bit more fun foundational material fundamental stuff if you want also on generalized free fields and then I want to switch mode and uh tell you a bit more about applications for example how do how do we apply this equation to learn something about uh the 3D CTF that we talked about on Monday and uh another theories.

[00:02:48]So, right. So, so at the end of last lecture I mentioned okay that there's a sort of a slightly nonrival way of doing the wick rotation that that is actually quite natural in this uh in this uh radial quantization picture right when we add x and and y one thing we can do is to wick rotate if you think of this radial direction as time you can wick rotate that time action, right? So, so, so that would give us a picture where we have a Lorenzian cylinder.

[00:03:35]And so so in terms of coordinate somehow the nx mu become e to the i t time some unit vector and but it's better to just draw it geometrically and I would try to draw something that looks like a cylinder that's not too skewed because then everything else will look funny.

[00:04:04]Okay. And and that is a cylinder.

[00:04:10]That's time. And this direction is a sphere.

[00:04:16]And d minus one dimensional sphere. It's like space of this direction.

[00:04:25]And a basic question one could ask is this is two different ways of this is a Minkowski signature theory. We have two different ways of defining a Minkowski space theory. Are they related? So how does this relate to the the R D minus one, one, right?

[00:04:56]It's got to be related because it would be weird to get two different CFTs and Lauren and singers from the same starting point. So there's got to be some relations and okay a lot of the calculation that I'm going to uh to I will not do calculations here. I would just somehow draw the picture so that you get the the mental image of what's going on. If you actually want to do calculation in this, I suggest you read on the embedding formalism.

[00:05:25]And a good reference also for the geometrical aspects not necessarily on formalism is a paper by uh Craftchuk and Simon Stuffin on light ray operators 0098.

[00:05:42]So a lot of this geometry stuff is in that paper and so so so I will try to explain things at a more uh geometrical level uh today. So so so the first step of of understanding the relation is well we could we can start from a time slice.

[00:06:02]If you take a time slice, it's so so if you take for example the t square zero slice then then on a cylinder we just have a sphere s minus one and and it's sort of natural to guess that this sphere will be a will be very very simply related to a time slice of of of minkowski case according to this theoreographic projection that we discussed earlier would be r minus one union a point at infinity now I will put a little subscript zero on this point okay so I just need you to accept that hopefully that reasonable like what else could this time slice map to okay so that that but that's what comes out of calculations but just need you to accept it Okay.

[00:07:02]So, so there's going to be some point on this cylinder that corresponds to this point infinity zero.

[00:07:14]And I will call this point this point is conventionally called space like infinity because well we went to infinity on a space slice. Okay.

[00:07:29]Right. So, so there's a special point on the cylinder which is like the north pole on the sphere on the So, if you do ST like the north pole on the sphere and if you were to stereographically project it, it would map to the point at infinity on the plane.

[00:07:46]Great.

[00:07:49]That's first step.

[00:07:51]And actually, sorry, not number them because I have other numbers later.

[00:07:59]And and actually we only need one more observation in order to figure out the geometry.

[00:08:04]The other observation we need is that from our perspective, well infinity is very far and and and we're mortals. So like in a finite time, we're never going to be able to cross the like cone from infinity, right? Because there's a light pulse that's emitted from very very far.

[00:08:25]We're never going to cross it in our lifetime.

[00:08:28]So, so mortals can never cross the lyon from infinity. Yeah.

[00:08:41]And I explain a bit more. Okay. I don't want to define what mortal mean, but okay, that's us. We live in Minkowski space.

[00:08:50]That's what we explore and never cross.

[00:08:53]And the like cone from this point omega 0 we would call null infinity which often call plus minus.

[00:09:04]So so starting from this point which we declare with space like infinity we draw the future like cone from it. It's going to go somewhere here and in the back of the cylinder goes here. And you can also draw the past like cone in the front and the back. And and what I'm saying is that if we live here, we're never going to be able to cross this like cone.

[00:09:37]So this region of the cylinder is the Minkowski space we have access to.

[00:09:47]when we map uh under this uh trans confirmal transformation that maps the cylinder to the plane to to the Minkowski space. So, so, so yeah, so Minkaski space that's a claim. When you confirmly map Minkaski space into the cylinder, it embeds as a set of points that are space-like from infinity.

[00:10:21]the space of points that are space-like from space like infinity is Mikowski space that's what the confirmable embedding of Mikowski space in the cylinder does okay so in some sense in this confir transformation has a weird rescaling of the clocks so so so this infinite time from a perspective uh going to this uh not infinite And this is called a point carry patch of the cylinder of the cylinder. Okay.

[00:11:19]There are other ways to draw this picture that are maybe helpful.

[00:11:25]So, so for example, well, if if D is equal to two, the sphere is just a circle and you can visual visualize it easy more easily by drawing an interval and periodically identifying the left and the right boundary.

[00:11:43]And then this point at infinity as well this is the same point are identified and now this looks like a diamond.

[00:11:53]So I've done the pointar patch you will see it in papers drawn as a diamond like this.

[00:11:59]So which is a technically correct picture in in uh in 2D but it's you can always draw the pictures you want and and also another way this is drawn that will resonate with with other lecturers is that well in higher dimension something else you could say is you just draw the pen diagram where each point represents a sphere D minus2 sphere and then that you could also draw the penros diagram like So, so but in 2D you don't have to in 2D the sphere would be like a two points and you don't really have to identify those and see. So it just gives a folded triangle pen diagram in 2D is just a folded version of this of this of this diamond.

[00:12:52]So these are all all ways of drawing the same point patch right? So you should be able to see that these how these pictures relate like if you start from infinity zero and you unroll along the cylinder it grows and then it contracts again. So this picture is already like this.

[00:13:22]>> Uh no. So this uh yeah so so if you normalize a cylinder to have radius one >> then this time is exactly pi.

[00:13:37]Yeah it takes a time pi to get across.

[00:13:39]And it's kind of an amazing thing about the cylinder is that so here you can shoot light in many directions and like initially it looks like the light rays are expanding out but then they all at the time pi they all refocus on the antipodal point at the same time.

[00:13:55]So the the the ly cone really closes again at that point.

[00:14:01]So >> comes back.

[00:14:03]>> Yeah. Then it could keep going. So you could draw the future like of this point and it will be another >> another thing like this.

[00:14:17]Yeah.

[00:14:19]Okay. So so so let me make a number of comments about about this picture. So okay zero for comment is that many of you may have seen this cylinder in the context of ADS CFT. So and of and and actually this geometry is is very well def discussed in many ads uh uh lecture notes and and and what I want to say is that yeah so in ADSCft the interior of the cylinder gets filled by a gravity theory.

[00:14:56]Interior of cylinder gets filled with dynamic gravity.

[00:15:09]That's the ads space which antiditor space which fills in the uh the bulk and and air. We're not really using that the CTF we're talking about. It's just about the boundary and we're just talking about the boundary or the surface of that cylinder. Okay.

[00:15:27]So if we only use a boundary of cylinder okay the surface of it okay the interior plays no role if you're only talking about CTFs you interior never plays any role okay and the claim of ads CFT is that the theory at the boundary can be equivalently described in terms of the theory at the interior you can use one or the other description rather both at the I'm done.

[00:16:03]All right.

[00:16:08]So, yeah, the other comment I want to make is that the null infinity is where massless radiation collects.

[00:16:19]So, so like if this person here and I don't know this person here plays with a hammer and emits light and radiation this will go this will move at the speed of light and it will hit some point A on null infinity plus I draw it here it's I plus and radiation goes to Here I plus in the back, I plus in the front.

[00:16:50]The interior plays no role. Radiation up there. So null infinity is there.

[00:16:59]Is where massless radiation goes.

[00:17:13]And but what you see actually on this picture is that there's nothing particularly special about null infinity on the cylinder. I just took a random point. I called it infinity and I draw it's n cone but I could have done the null cone from any point and it's the same and and in fact if you take a n cone from a different uh uh take a null cone from for example a point here n infinity and you ask what's the null cone from this point it would be something that cuts through the cylinder and looks like a null plane.

[00:17:52]So, so null infinity in the CFT null infinity is equivalent to any null plane.

[00:18:11]So a null plane is just a solution to the constraint say x0 - x1 is equal to zero. Okay, null sheet also called a light sheet.

[00:18:24]So, so in CTF, so that is a sense in which in also in Loren signature there's nothing special about infinity.

[00:18:35]But you should think about infinity in in in signature as not just a point but more like this as a n sheet.

[00:18:45]Okay.

[00:18:48]And there are special points where the null sheets intersect, but there really is a is a uh the null sheet. Okay.

[00:18:56]And and that that is probably one of the deeper connections between the CTF concept, CT CFT business and and the celestial graphic program because the CTF the celestial graphic program to a large extent is concerned with understanding what operators we can insert and what states we can we insert at null infinity.

[00:19:18]And what we're saying here is that in CTF the measurements you can do at null infinity they're exactly the measurements you can do on any null sheet. So in CTF you can answer this question more uh you have more constraints to answer this question.

[00:19:36]So, so this is relation between detectors that uh Yan is discussing in his lectures.

[00:19:48]Infinity map to operators. You can insert on the like sheet and very often people focus on like three operators.

[00:20:01]For example, things like integral over time, dx+, d++ sort of objects that appear that that appear will appear in nan's lecture. This sort of object in a CTF you can insert this object on any like array and when you put this object when you happen to put this object at infinity then they measure the energy flux in the direction sphere.

[00:20:27]But in CTF you can insert it anywhere because there's nothing special about infinity.

[00:20:42]>> Yeah.

[00:20:46]>> Yeah.

[00:20:49]>> It is a point the maximal co dimension and null infinity is co- dimension one.

[00:20:58]infinity.

[00:21:04]>> Yeah. So that that's maximal co dimension because you have angle fixed and time fixed.

[00:21:10]>> Yeah.

[00:21:12]And uh Yeah.

[00:21:19]Yeah. And r is infinity. So r is also fixed.

[00:21:27]Yeah, other questions.

[00:21:32]Okay, I have a long list of comments.

[00:21:35]Basically, what I would say about about about this geometry.

[00:21:39]Uh, another uh feature that is nice about the cylinder is that uh the confirmal group acts smoothly on the cylinder but not in the pankar patch. So, let me explain. So the confirmal group acts smoothly on on cylinder but not on on on point patch.

[00:22:12]Okay. And the confirmed group is a bit uh actually it's a cover it's a cover of so the Lawren confirmed group is a cover soda 2 or not described it in detail but it's discussed in many SCST references it's an infinite cover of of that thing and it attacks smoothly and you can see easily that it does not preserve a point patch because clearly some uh well some of the symmetries of this thing is like a just a rotation of a cylinder is a symmetry, right? And if you rotate the cylinder, it can take a point here that's is close to boundary and make it cross the boundary, right?

[00:22:54]And and if you were to literally look at the confirmal transformation of Minkowski coordinates, you take this coordinate which is in the far future and you apply this little confirmal transformation to it that's infantismal, very small and and the coordinate will look like it it maps to a point in the far past.

[00:23:18]So, so in terms of the coordinates of Minkowski space confirmation look very singular and it looks like the past is close the future and it's just weird and and and and really the weird thing about it is that we don't really time time is not periodic. We don't live in this Pac-Man world where past is the future.

[00:23:43]What really happens is that the point just leaks out of the point patch and that's it.

[00:23:50]Okay, so confirmal transformations in general take you outside the pankar patch but they all very smooth and nice on the cylinder.

[00:24:00]>> Yeah, we should think about one. Ah well well that takes us to the next question.

[00:24:13]Now that the oper now that suppose we have the measurement here we do this control transformation now it's outside the point car patch uh how can we interpret this can we interpret this still in the viewpoint of point of mikowski space uh that that is the next question I want to address. Yeah, it is. Uh, but yeah, in terms of patch, it looks like we just exit the patch. Although we don't want to say that now we're no longer in spaceime like it sounds weird. We're still somewhere.

[00:24:45]So, so uh yeah, so that that's going to be a little bit more of a technical comment, but uh and this is something that is not so well known. So a lot of the things I'm I'm saying here are not so well known and and they're like because they've not been applied so much but they but they are somehow implicit and in this paper that I quoted and I also tried to make them a little bit more explicit in my paper on autoraphic cameras but it's it's not been used so much but uh I think there's a nice fact that is implied by the calculation that these people which is that uh so I will I will state it as a fact because I really believe that that that that it is a fact and and and and focus on the calculation is is that can I ask the following question suppose I take a okay I need okay I will just draw it on the this picture here All right.

[00:25:55]So let's call it the a patch.

[00:25:58]Okay.

[00:26:00]And this point at infinity you understand now why I call it omega 0.

[00:26:05]I call the other point omega 1. So in relativity this point is often called future time like infinity because it's it's at infinity of our point patch.

[00:26:20]However, we can also think of it as a space like infinity of the next patch.

[00:26:28]And then you could keep going and have some infinity two here. That's a space like infinity of another patch. It's called a B patch and a C patch.

[00:26:39]Right? So, so the cylinder, so you can actually tile the cylinder by a succession of patches, right? The whole cylinder is naturally tiled by a succession of patches.

[00:26:54]Okay, so let's ask the question. Suppose I start on the cylinder and I take a correlation function of a bunch of operators and C, sorry, C is poorly drawn. C should be here.

[00:27:12]So, and I take a correlation function of things that don't fit in one patch. What the hell does it compute?

[00:27:24]It's got to compute something because I could compute it starting from the UK CTF and doing some analytic continuation. So it's got to give something that has meaning in in in Yikoski space and and the claim is the following that if you compute now I put the because I'm signing I put the vacuum so the vacuum expectation value of a bunch of operator say in the A patch and bunch of operator on the B patch on the cylinder So the claim is that if you take this correlation time order correlation function on the cylinder with all these things in different patches.

[00:28:13]The claim is that this map mapped to something very well defined in Minkowski space which is going to be the time order product of all the things on the A patch multiply on the left by the anti-time order product of the things on the B patch and and and so somehow the way that the past the the the paths alternate that you go to the future of of one patch and then you backward evolve back to the past and then you forward evolve again and so on. So the things that are on patch patch A are time order the thing that have been patch b are anti-time order and and so on.

[00:29:02]Okay.

[00:29:05]If you compute something, >> yeah, then you will have time order product of A separated from time order product in C, but like a white nonman sum over all states in between.

[00:29:22]It's not not a single overall time order product, but two separated time order.

[00:29:32]>> Yeah, this is a normal time slice.

[00:29:35]and and yeah >> in terms of the time could be later.

[00:29:49]>> Yeah. Yeah. So if you do maybe that will answer your question not or if it doesn't let you know. So for example when you take this little little confirm transformation which take point from a to b then this point a here will be on the left because it's time order and to reappear on the right of the b ones.

[00:30:09]So that's in that sense this this this this this recipe is smooth under uh confirm transformation of the further >> ah uh then it doesn't matter because on the cylinder they clearly commute they have they are space like from each other. Yeah.

[00:30:39]Yeah. So these ambiguities never appear >> in this case on the left hand side a patch of >> uh yeah but it it again it doesn't matter because everything on the A patch that whose time ordering could be ambiguous is space like from something in the B patch.

[00:31:02]Yeah. But so you can assume that anything in A is is before B. But this T here on the left, what I really mean by this T is literally this time slice ordering >> and that gives the same.

[00:31:16]>> Yeah.

[00:31:27]>> Well, commentators may vanish but not the corators.

[00:31:34]Okay. Yeah. So that was a bit of a technical but it just wanted to explain how how to think about correlation function on in general they map to the cylinder and actually here this equal sign has some confirmable factor from each going to each other and some of these confirmal factors have funny phases that they explain in that paper but uh yeah this is a personality sign with a predictable proportionality constant.

[00:32:07]Uh yeah. Okay. So now let let's let's talk about a bit more uh uh going back to uh the uh the uh fourpoint function.

[00:32:20]So, so when take a fourpoint function uh remember that in the wick rotation we say that uh the row for example in the symmetrical frame row from last time row minus row one - one in the symmetrical frame. Now after the wick rotation row becomes I e x i t with a little uh one minus i epsilon and and times say plus minus i theta. So basically row becomes something of magnitude one but the magnitude is slightly less than one because of the i epsilon.

[00:33:07]So, so for t less than zero, so this guy inside then the row is e to the minus epsilon which is less than one.

[00:33:24]Okay. And the reason I'm mentioning this is that this is a very important result.

[00:33:30]It tells us that we remember that the op converge when row is less than one.

[00:33:36]So it tells us that in Mikasi signature no matter where the points are the Ipsilon makes the op convergent.

[00:33:47]So it means that the op they have operator one and two it converge but now it no longer converge absolutely like when when a sum converge only because of a epsilon just to not not go into technical detail at the physics level it means that it converges a distribution the sense of a distribution So the limit epsilon goes to zero defines a distribution merge as a distribution for any configuration of four of Mowski points for any Minkowski configuration.

[00:34:45]But what is important is that the epsilons of one and two be basically order such that one and two are the rightmost two operators in the product. So what makes the epsilon work out here is that 01 and O2 are the right are the operators that attacked on the vacuum. So, so the general statement is that OP of 01 O2 on the vacuum turn into okay and the one 2K a vacuum. The OP converge whenever two operators act on the vacuum.

[00:35:29]whenever.

[00:35:33]Oops.

[00:35:35]App.

[00:35:51]Okay, that's a very formal statement.

[00:35:54]Let me just do an example to squeeze something out of it.

[00:36:04]So, so the claim is the following. So, that that claim is going to be related to a microcosality.

[00:36:10]We're going to use some commutator, right? So suppose I have four points that are in configuration and I compute the expectation value of wait a second 03 04 O2 sorry 04 01 02 and we're going to take a commutator between four and one and that window is going to vanish when 01 and 04 for our space like right so so that statement in Mikasi signature that computators vanish spaceelike is what people call microoality it implies you cannot send information faster than light and and and so on so what I'm saying with this convergence statement is that we can unpack this comator as a difference of two terms Doesn't matter where the or other orbiters are 2 three and so on. So we have 0 is equal to 3 4 1 2 minus 3 I did the wrong choice. I think it was.

[00:37:32]Oh, sorry. I wanted comator of 042.

[00:37:35]Sorry.

[00:37:37]3421 minus 3 2 0 4 0 4 0 4 0 4 0 4 0 4 0 4 0 4 0 4 0 41.

[00:37:47]So we have this thing and what I'm saying with this over product convergence thing is that I can no matter what the configuration is I'm always allowed to insert complete basis here.

[00:38:01]You can see the sum over states here right and same here.

[00:38:15]So when you insert a complete basis of states of the CTF between 1 2 and 3 4 that is the definition of the OP in the one two channel. So that sum here is computed by the picture where you compute this one two three four and the other term here I have minus and it's computed by the sum but now I have 2 three and 4 one that is equal to zero So that's a familiar equation and and so what I'm what I'm saying here is that this equation is actually true everywhere in Mikowski signature provided x1 - x4 is time is space like it's all we need to assume we can cross many many like cones and do crazy stuff.

[00:39:19]This equation is still valid as long as one and four are space like >> now it should be two and four to fix my typo. Thank you.

[00:39:33]Thank you.

[00:39:38]>> You will kiss. So this relation will kiss the branch cut but the two sides will somehow be evaluated on the same side of the branch cut and they will still match on the branch cut.

[00:39:50]Yeah. So basically uh crossing this crossing equation really in Minkowski signature is a statement of microcosality.

[00:40:04]So it has interpretation directly in in Minkowski space. Yeah, >> just just from looking at the time that you got there.

[00:40:11]>> Yeah.

[00:40:12]>> Like is it like two and four? Does that mean two and three or two?

[00:40:18]>> No, two and three could be this could be time like uh Yeah, the kinematics could be Yeah, you could have very Yeah, you could have sort of kinematics.

[00:40:29]>> Nope. So we could have Yeah, that's the weird thing because I could also have derived the same conclusion by assuming one and three are space like and two and four can be anything and it's still true. Yeah, >> it's a very weak assumption.

[00:40:42]>> So but that doesn't mean that each two and four are like one and three are also like >> no no exactly. So so but let's just draw a picture right. So if two and four are space like this is all we're assuming and that's me time goes up.

[00:40:59]I can put one and three anywhere and three could be in the future of both and it's still going to >> Yeah. So search condition. Yes.

[00:41:24]>> Yeah. Yeah. Yeah. Exactly. What what what is important here is that I was not writing a time order product was writing literally this whiteman product. So 01 O2 act on the vacuum and then I can insert complet of state.

[00:41:40]Yeah.

[00:41:45]said, "Why is this order like?"

[00:42:05]>> Yeah, I mean, the thing that's slightly fancier here is that uh uh So, so nuclear and space you can indeed uh uh when two two and four are separated like you could insert complete basis of of state on a slice that look like that on a slice that looks like that and look like they ordered differently but uh what is different here is that I have all these spectators that are timelike separated from these guys. So you cannot really apply the argument. So what you're getting here is stronger in that sense. But if you just focus on the two guys here in nuclear and signature the commutativity is indeed clear because we have all this time like spectators here.

[00:42:45]>> Yeah.

[00:42:54]>> They act as >> anchors. Uhhuh.

[00:42:59]Yeah. One and four. Yeah. Yeah, they add some references to that uh that that that that picture.

[00:43:08]>> On the other hand, if you put like two and four not space like separate >> Yeah.

[00:43:13]>> This is not vanishing.

[00:43:14]>> This is not vanishing. Yeah. Well, unless one and three happen to be spaceike. But if neither are spaceike, then then it it would not vanish.

[00:43:23]Yeah.

[00:43:26]Yeah.

[00:43:31]All right.

[00:43:45]>> Uh yeah yeah yeah. So so so then you approach a branch cut of your uh of your corator and if you try to cross the branch cut then it will no longer converge.

[00:43:59]both channels.

[00:44:00]>> Uh it could be that's only a single one of them that's up converging.

[00:44:09]It really depends on your kinematic. It could also be that they continue to converge, but now it converts to a non-zero answer. That can also happen.

[00:44:19]I take that. Yeah. Sorry. If the only thing you're doing is you cross a like from between two and four, this will continue to converge, but it will converge to a non-zero answer.

[00:44:31]>> So you have >> uh no this argument goes through you you get it from uh whenever two acts on the vacuum. So so so so in this argument I could have added uh many many spectators on the left and it still works true. Uh but the implications of that equation are much less studied. So this is what we want to go next. What are the implications of these equations? Yeah. Question >> any four points? Yeah.

[00:45:18]Yeah, this this this this what I said.

[00:45:20]Now it's still old, but as long as the two key guys that we're crossing are space like from each other.

[00:45:30]>> Sorry.

[00:45:34]>> Yeah, if everything's time like then uh then then yeah, there's no uh There's no analog of crossing.

[00:45:44]All right.

[00:45:47]So, so to get a sense, I I feel that at this point it's good to to get a bit more concrete about what this OP looks like.

[00:46:00]And I thought I would talk about at least one example where we can solve it analytically and and have formulas and uh and so so so this situation of generalized free field.

[00:46:18]So so so it's a simple model. The idea is you define a twopoint function to be say 1 / x1 2 to the 2 delta or some parameter delta and we define IR points by w contraction.

[00:46:45]So we take like a gshian measure with that two point function. So for example, we have the fourpoint function.

[00:46:54]You have something like O there' be one contraction looks like this. The other one that looks like this. The other one that looks like this. You have sum of these three contractions.

[00:47:06]And that gives one / x12 delta x3 4 2 delta. Let me call it delta O plus right so that's a that's a correlation function of a generalized free field it satisfies permutation environments it satisfies all the satisfy confirmal symmetry obviously so so these are controllable uh yeah this you can think of as a conflator so so this uh passes all a CTF except you don't have a local stress tensor or general delta.

[00:47:45]So define the CTF that if CT CFT correlators without a stressed answer.

[00:47:56]Okay, so that's the general field. as far as defines C since it defines CFT corators and it admits and it it pass if delta is some is is large enough it satisfies reflection positivity and so on. So we can apply the OP. All of this goes through.

[00:48:22]So so in order to study it OP one I'm going to study it OP in a in a one two channel the the basic way to do this is we uh first make the correlator confirm invariant. We're going to define a function G. We want to define a function that depends only on the cross ratio that we talked about yesterday. These X's are annoying. There's too many of them and they're very annoying. So if I take the corer and multiply it by this uh this this this term 2 delta O X34 2 delta O time O that gives a spe function we can just write down explicitly. It's going to be one and the other term this is mostly an exercise in recalling the definition. So I will just write down the answer. It's a power of ZZ bar 1 - Z 1 - Z bar the delta O and there's a term ZZ bar delta O. Okay. So three terms become that. So that's the as a function of just the two cross ratios. This is the answer for that call.

[00:49:37]Okay, by the way, uh posted some exercise where you can practice on uh on this uh on this concepts. So there is exactly one exercise on this uh this results for the uh exact OP de composition of the general field. So I encourage you to try it and and because of that I'm not going to write down exact formulas, but I want to give you a flavor of what the formula looks like. Okay. The formula itself is just like a dozen a product of a dozen gamma functions. So it's elementary in a sense but it's long.

[00:50:11]>> Yeah.

[00:50:22]the sign. Yes. So, yeah. So, in everything's positive and this is unambiguous and this is also something some complex conjugate. Everything's positive and it's clear. uh what happens when you go to Minkowski is that uh this x square will become negative when it's time like and then this becomes really one over minus something minus its absolute value say plus i for example and the i is determined by which type of the cut and that basically becomes one over something times x you can see one is this plus i the delta this become x to the minus i pi delta So the epsilon tell you what's the face.

[00:51:06]Nice. So that way you can make sense of the this this object in Minkowski.

[00:51:18]So so yeah let's so what's the op decomposition of that thing? So we want to write this G Z bar as a sum of delta and J of this confirmal block some unknown coefficient squared we call it C square sorry C square G FF or of delta and J times some confirm block G delta J zero.

[00:51:53]So we would like to write the OP de composition of this thing.

[00:51:59]And and in order to have a mental picture of the OP de composition, it's really useful to to think of this thing as basically it's not quite that, but it's basically it's basically a power of delta.

[00:52:15]It's basically it's basically a series expansion. I discussed a series expansion in a row last time. You can also think of it as a series expansion in z. So this is basically uh magnitude of z to the delta times i j ar z. Okay. So it's basically the op is basically a tailor expansion in in in zed. Okay. And you could you could write it like this. It's basically one plus order zed. Okay. in zar.

[00:52:51]So, so the leading term of a block is just a simple power law and the subleading terms is a bunch of extra stuff.

[00:52:59]So, so just by thinking of just by thinking about the series expansion of that at small zed, we basically understand the anatomy of the OP.

[00:53:09]So, what we see is that at large at small zed, the thing that dominates is the one.

[00:53:15]Okay. So the dominant the dominant term is one which is correspond to delta and and j equ= 0 0.

[00:53:32]So there's an operator of dimension zero that's being exchanged.

[00:53:39]There's a fancy name for the state the state of the CTF that has these quantum numbers identity. Yeah, also called just a vacuum. So yeah, so identity exchange just called identity. So exchange of identity it's called identity because it's the identity operator which by the state operator correspondence defines a vacuum on the sphere and and the picture of this term that's always a dominant term because like if you ask anyone uh uh say this is one and two and this is three and four right if you ask anyone what the correlations between these four measurements may look like well they would say well one is probably correlated with two very much but not very much correlated with three and four so the correlator should be just one two * 3 4 the independent averages right so so this contribution is just the independent statistical average of one and two and then three and four independent okay so that's density exchange it's always there and it dominates or just one block.

[00:55:05]>> Yeah. Each block Yeah. So each each block delta J looks like this plus subleing terms, >> right?

[00:55:15]>> Yeah. So each block has an infinite series expansion.

[00:55:19]For example, in two dimension is some hypergeic functions that is written in the uh in the problem set.

[00:55:24]>> I was told that they have exact formula for every >> exactly.

[00:55:33]Now if we look at the next term, well the next term go like z bar to the delta delta o.

[00:55:43]So well all the other terms all the other terms go like zb bar to the delta o plus integer powers of of z and then z bar integer right and and basically we can read off what the quantum numbers of this thing are because z is unit of length.

[00:56:08]So, so the operator that they multiply must have delta is equal to 2 delta O plus integer and the integer is the number of extra Z let me call it Z to the A Z bar to the B it's 2 delta O plus A plus B that's a scaling dimension of that thing and you also read off the angular momentum of that It's just a minus b. So if you have more z's than z bar then you have angular momentum because you have you pick a phase under rotation.

[00:57:08]So notice that the minimal amount the minimum dimension for a given spin is is attained when you just uh set one of these to zero. Maybe I should put absolute value here is when you set one of them to zero. So, so in general delta is 2 delta o plus 2 n + j where n is non- negative and j is is non negative.

[00:57:37]So these are the possible dimensions which appear in this format.

[00:58:03]So yeah, so what have we done? So just by staring at the series expansion of this thing and asking what can appear, we figure out what the J and what the possible J and delta are in a sum.

[00:58:20]And then if we work harder and actually look at the coefficients, we could figure out what these C squares are.

[00:58:28]And then to compute this the term with higher and higher dimension, you have to series expand to I and I order. Okay?

[00:58:35]And if you compute the first 10 orders and ask Mathematica to do fine sequence function, you'll get an analytic answer and that's the answer that's recorded in this homework. There are nicer way to derive it but but basically you can derive it like that. So let me just give an interpretation for what these operators are.

[00:58:59]So, so what's the idea of of this op that it's basically a tailaylor series expansion and so so maybe I will move now to the the this uh board.

[00:59:14]So, so if you think of it as a as a series expansion, what we're trying to do is we're trying to expand the product of O time O where they are slightly different location.

[00:59:34]So we're trying what we're trying to do is you're trying to expand this as a sum of stuff and and we saw that there's always a first term that's one and the coefficient of that first term is just the we contraction between the two O's that's easy the other terms will not involve this but you could simply say that well let's just expand in X and if you natively expand in X you'll get O times derivatives of O and let let me look prettier by putting the zero on the left and the X on the right you get something like this O plus X mu O mu o plus sum of all mu1 mu n of call it muk mu1 mu k Right? If you were to just naively do a tailor expansion of this thing, you'll write that down. And then what is the scaling dimension of this? Well, it's two delta O plus an integer. So this is just what we're seeing.

[01:00:50]So the OP of the generalized free field is just a naive taylor expansion reorganized in terms of so-called primaries and descendants.

[01:00:59]and and we can be a little bit more precise because so all these derivatives here they commute with each other. So this is a symmetrical tensors in K indices but a tensor can have a trace part or a part that carries spin. So if you split this part into a traceless part and a and and and and and a part with with with trace with with no trace, you can write this as a power of d² to the n time d mu1 d muj o where these are like minus traces.

[01:01:38]So you take a trans traceless combination of derivative in order to ensure that is angular momentum J and you and and the other derivative you just contract into llassian that are rotational invariant and this guy has dimension.

[01:01:53]So this operator is called a double twist. some reason also fit double trace operator with n boxes and spin j and it scaling dimension what I wrote before is 2 delta o plus 2 n plus j as you can see just by adding the dimensions of the parts.

[01:02:25]So in general free field that is the answer and generary field is like a gshian theory. So in some sense it doesn't interact. When we add interactions all of these operators will move around and they will pick some anomalous dimension and so on. But at the moment they're not they're not yet. So you could imagine doing some kind of persporary field and seeing this scaling dimension get corrected and at the moment in the general field it just stops here. This is the exact result.

[01:03:13]>> Yeah.

[01:03:17]coefficient.

[01:03:22]>> Well, they're sort of correlated uh and but the the main new thing that appears also is that you get new operators in the sum.

[01:03:31]So, so, so, so in general field, this somehow exhausts the OP. But when you go to a generic interacting theory, in addition to the guys that maybe you can identify a small perturbation of this with slightly modified dimension and slightly modified op coefficient, you also have new guys.

[01:03:53]This at the basic level, you could write things like O with a bunch of derivative and so on. Like there's a bunch of simple things you can write down, but they may also be more complicated like bound states that are harder to interpret this language.

[01:04:08]Okay. Okay. That gives you an idea of the anatomy of a of a CTF.

[01:04:16]Yeah. Yeah. So I won't have time to go to 3Dizing today. So I will I will spend more time on this uh on on on this story. And one thing I I wanted to see is that uh uh the generalized free field is nice not just because it is an exactly solvable answer.

[01:04:40]But it turns out I claim or justify that these GFF coefficients that as I mentioned can be computed exactly give you the correct asotics in any CFT.

[01:04:57]So, so any CFT.

[01:05:01]So, the OP data of any CFT at this limit of large delta but fixed spin approach generalized free field.

[01:05:26]So, so what does that mean? That means that last time we argued that on general ground that the OP was absolutely convergent in nuclear and singer. And now we can be much more precise and and say exactly how fast it converge because we have a estimate of the large order behavior at large delta of the op.

[01:05:52]So we can say something much stronger and and the reason we can say something like this is that okay I would I would just take a simple version of the OP maybe a simple model mental model you can have the the OP is that the correlator says a function this nice row symmetrical variables we introduced last time is a sum delta and I'm going to put row bar equal to row is basically just tailor expansion in row. Okay, let's just just use this variables are slightly more clever than zed and just do a tailor expansion this variable or the scaling blocks from last time and I'm going to ignore the spin by the C C delta roughly speaking here is a sum of all spins but let's just study this model it's a good model for you can do it spin by spin eventually but let's just study this model and and and and the point is that this object We know that there's an op like this that we can apply in this channel. This was the one two channel that we're using here. But we can also apply the OP in the other channel say as uh as two and three approach each other and that will give us a series expansion around row goes to one.

[01:07:24]So the other thing we know is that it's a sum of delta of the same data times uh 1 minus row to some powers.

[01:07:37]So we know exactly and in fact there's some minus 2 delta O here like this there's some extra uh power from the crossing but basically you have this uh you can expand it you know it's series around zero but you also know it's series around one.

[01:07:58]So what we're going to do is that we will just think about such a series and I'm just going to keep the leading the most dominant term at one and and all the terms near zero.

[01:08:16]So, so we need to find we need to solve the following sum of the delta of C delta row delta is equal to 1 / 1 - row to the 2 delta O. You need to solve this.

[01:08:35]And you see that if you had a finite sum on the left, you could never do it. Pronomial cannot have a singularity at equal to one.

[01:08:46]You need an infinite sum on the left and you need the tail of the sum to just have the correct density of C delta to give you the correct power behavior.

[01:08:56]Right? So, so you need you need basically integral up to delta of D delta C delta zero to this to behave like delta to the 2 delta O and yeah so this the integrated spectral density needs to have this asic behavior in order to give you the correct singularity in that in that s the simple case that's familiar maybe if you had a pole like a simple example of this if you had a You have 1 + row + row square plus dot dot. That gives a geometric series of 1 minus 1 / 1 - row.

[01:09:52]And you see that the power one here is related to having the integral of all power rule linearly with row with with with x.

[01:10:03]So you need this this behavior.

[01:10:08]In fact, one can really quantify this sort of error in these things. There are these so-called tubarian theorems that people have written about that allow to bound the errors and what you get is that this integral has to go to like one plus order one delta. Okay.

[01:10:29]So you can really get precise statements about the asintotics at large delta of the spectral density.

[01:10:45]Maybe um finish with with with the parenthesis.

[01:10:57]Right? So this doesn't really tell you where the operator are.

[01:11:02]It just tells you roughly speaking the the total number of operators weighted by their coefficient as to approaches parallel.

[01:11:16]So, so if you were to plot as a function of delta this uh integral of d delta over delta to this power so I'm integrating maybe I should put primes in the integral to make it clear what I'm integrating.

[01:11:39]So and let me put data minus one here just to make it nice and linear.

[01:11:46]So this is the integral well this integral is really like a sum of okay so I think I I think I must have confused you because I should have written it as a sum should have written it as a sum over the prime that this what I really meant here sorry about that sum over the op data divided by this this power law to the delta.

[01:12:23]What and let me put a minus one here.

[01:12:26]What what happens is that it has to asmtote to a straight line.

[01:12:34]But in general this sum jump as discrete jumps every time you hit an operator.

[01:12:40]What it does is that it actually looks like a staircase like this.

[01:12:50]Right.

[01:12:52]So this is the the straight line is the asytoic and and the staircase is the actual.

[01:13:03]Okay. And what the statement says is that the jump the jump cannot be more than order one.

[01:13:12]Okay, the largest jump that's our estimate. I I think it's probably possible to actually put a number here, not just an order, but a number, but I don't think it's been done.

[01:13:25]But that is the that's the picture of what we know about this synthetics and and okay, here's a little side comment. You can also consider so this is what happened in GFF. In GFF you have these stair case and they basically jump by one each K each step. You can also imagine a different scenario.

[01:13:47]For example, if you confirm field theory is dual to a theory of gravity inside the bulk at very large delta. Delta is the energy on the sphere. So you expect to have a large energy in the bulk and you expect to have large you expect to produce black holes for example.

[01:14:06]So and and a black hole should have a high density of state and a very random spectrum. So in a black hole regime instead you will expect this to be very very tiny tiny tiny steps where the distance between each guy is order e to the minus s. So the the energy spacing is one over the entropy one over well one over the number of states which is exponential in the entropy. Right? So in the black hole scenario if you go very far along this curve you expect instead to have a very very smooth curve that is uh well each step is exponentially small and and and you're approximating this smooth curve. Okay. So this gives you a different extreme limit of this. So GFF is in some generaliz field in some sense is as discrete as it could be and and the black hole is as continuous as it could be. Yeah.

[01:15:02]Uh, okay. I guess I'm over time. I'd like to make a little Can I make a little two minute comment? It's sort of interesting to ask uh uh when you compute the op when you compute a sum like this we compute this op we sum this op coefficient times block and and an answer question is uh suppose you have a discretization here the question is what's the error that is induced how different is the result compared to just an integral so So dense sum versus integral like what's the error between that.

[01:15:50]So right, you might say, well, if you're doing a dense sum and all the jump is delta E, you might say that maybe the error is delta E.

[01:16:04]Error O delta E, that's a naive guess, right? If you're doing a discrete sum, you just say well the function is you can approximate it by its value plus its derivative and the error should be proportional to the to the decreasing like the discrete the error in agreement sum.

[01:16:25]This is actually not this is way too pessimistic.

[01:16:29]There's a famous theorem by Erler McLaren which says that the error the linear part of the error here actually goes like the sum of the derivative of the function you're integrating which is basically the function at the end point.

[01:16:54]So actually the error cancels and and the only part where you get really care about the error is the end points. So the error when you do an integral is equal to error from end points and and the error from the bulk region in fact all the derivative cancel out the error is nonproductively small plus order e to the minus some number over delta e which is e to the minus e to the s This result was known by everyone 200 years ago because if you're going to compute integrals numerically by doing reman sums, you you really care about where your errors are coming from.

[01:17:52]But uh that tells you that the crossing equation tells you general things about the spectrum but it doesn't tell you much about the uh tells you general thing about the density of op coefficients but it doesn't it's not very good at picking up the spectrum at high energies. However, the endpoint region at low energy is going to be very important for this crossing equations. And this is what we can actually constrain using bootstrap is this uh is this low energy endpoint of the spectrum where the fact that it's discrete is really a big deal. Okay, so that's what we're going to discuss uh next time.