 So, in the morning we sort of I motivated for you the use of general contours in quantum field theory and started giving you some of the relations between the objects that are natural on the contour and the correlators you actually want to compute. So, I give you these childish rules for computing contour objects in terms of these nested correlators let me quickly tell you how I derive them it is not that hard I will just sketch the derivation in two seconds and then move on to the next thing I wanted to say. So, these are derived by basically using the identity that you take the theta function of n arguments I will define this in a second, permute them and then sum over permutations that it will click and you will get the identity ok. So, the theta function of n arguments is just the sequential product of theta functions of two arguments and this just says that 1 is bigger than 2, 2 is bigger than 3, 3 is bigger than 4 and so on. So, basically what you do is you take your correlator that I that I wrote in terms of the average difference basis stick in this identity use the fact that you have a sum over permutation. So, you can permute the indices of the times and then take the average and differences write them in terms of left and right use the fact that there is a childish ordering. So, the left operators goes to one side the right operators goes to the other side and by the time you are done you will get commutators, anticommutators. So, it is very straightforward, but that is what there is and for the k-o-t-o contour you basically do the same thing, but sort of run the Keldish rules one and one then you run the Keldish rules two and two then you run the Keldish rules three and three and so on and then work your way inward and that is why you got the rules that I wrote down. The main difference is that these guys are time ordered. So, they have a theta function that the even index guys are anti-time ordered. So, they have a anti-time ordering function just that for the even indexed legs the left comes before the right. So, that which is the time order. So, that was that just completes the discussion for the Keldish rules. I just have one last thing to say about this and then we will simplify discussion a little bit more. The other thing we were discussing was these left right correlators or still average difference does not matter which were too many and we want to see how to map them down into our white man basis of n factorial correlators. I am going to do this in two steps. I am going to ask the following question let us say given a particular you can ask what is the minimal contour I need to represent that particular correlator. A related question is for end point functions what is the maximal number of time folds you need to insert. So, in the k-order of contour there is a maximum k beyond which you do not really care about and let me first convince you the answer to this. So, k I will convince you that for end point functions it is a basis consider call proper q or 2s with q ranging from 1 to maximum value which is the integral part of n plus 1 over 2. I am going to ignore the one odd case of q equals 0 which is the time ordered guy I am just going to call that q equals 1. The Feynman contour I am going to call I am just going to input into the Schminger Keldish contour. So, why is this? So, let us get the maximum value well the maximal out of time order basically requires you to start somewhere say and then you let us say you evolved let us take let us take an example let us take n equals 3 n equals 2 will be boring. So, let us take an equal. So, you could start with you have operators O 1 O 2 O 3 I could first come and insert the operator 2. I can say well if I if I insert 1 after that then I am going forward in time. So, that does not that does not give me anything extra let me want let me force myself to go back in time and insert 3 and then go back forward in time and insert 1. So, remember that on the contour the times were the time insertions were fixed the time locations were fixed. So, this is some fiducial time and 3 3 4 3 was less than 2 was less than 1. So, to get the maximal number of folds I have to allow myself to do the maximal number of Heisenberg forward backward evolutions which means the only permutations you are allowed are once where you have a sequence where you sort of start somewhere you go up you go down you go up you go down you go up you go down and this sequence of permutations is basically easy to classify and you can see that once you write down such a permutation it is a it is just a question of how many ups and downs you want to do and you just need to do as many as the integer part of n plus 1 over 2 ups and down to fill in all these guys ok. These permutations have a name these permutations depending on who you are you can either call them accordion permutations or tremolo permutations or what have you and they are they are related to other things in in counting problems I will get to in a second. So, the maximal q o t o this is an example, but this can be this should convince you you can just run run the inductive argument from here that q q max is integer part of n plus 1 over 2 which tells you why these have not really been analyzed that much in literature because mostly you are interested in response. Response is a two point function you disturb something you measure response, but the nice thing about two point functions is q max is one that computed by Schwinger-Keldisch contours you do not need anything else if you want to compute only two point functions and that makes sense because there are only two time orderings you can either have o 1 followed by o 2 or the anti time ordered guy o 2 o 1 that is it and both of them fit nicely into a Schwinger-Keldisch contour this guy is simply this guy, but three point functions let us just finish off the game here for three point functions there are six correlators four of them I will just call one two three two one three one two and three two one they are all easy to see are computed by one or two because this is just three so these are the four three point functions you would have gotten if all you did was Schwinger-Keldisch there were two more one of them is this one which is one three two then the other is two three one which is given by with the one here two here and then you can put the three here then you can put the two so by proper q otos all I meant was the minimal number of forward and backward legs I need to draw a particular ordering any more legs is redundant because I can use unitarity to collapse them you can always add dummy legs to the bottom so I can always make this a two or two in a very simple way by just saying that is not very exciting and you see that is what that is what is responsible for this large number because once I have the correlator here I can start drawing other pictures all of these involve dummy Heisenberg evolutions because this part on this part of the same this part on this part of the same you can keep collapsing the content so if you fix the number of OTO legs you have lots of correlators you had to have lots of relations but you have lots of relations because they just come by sliding the operators around as if this was an abacus and you just moved operators around like beads and like on an abacus you can't move the operators around through an operator you can just slide them when the unobstructed all the relations I give you a embedding you can just stare at them and eyeball them by saying how many sliding do I have so and that should account for this 2k to the power n so let me give you the answers for the sliding so this is all just a matter of doing some counting doing some bookkeeping and understanding how to sort of canonically embed this but the canonical embedding should be clear now because all you want to do is given a sequence you try to write down your correlator by isolating pieces which involve either forward evolution or backward evolution and then putting from these kind of forward descents or backward descents at different places now this has a name in permutation theory this is this is a in the language of permutation counting these are called permutations are given run structure the run basically indicates sequences where you sort of are ever increasing or ever decreasing and then you have peaks and valleys where sort of you have top of a sequence and bottom of a sequence so these have these can be done again I am not I think there is a sort of interesting numerical accidents that happen one of which I will tell you but let me just say the answers to this counting problem so I can ask given n factorial endpoint functions how many of them are encoded in a canonical proper q audio so for example in two point functions both of these we just obtained from one audio with my convention that this guy also was one audio in three point functions four of them were one audio and two were two audio so and that was basically giving you a decomposition of six a partitioning of six into four plus two there were four of them which were one audio and two were two audio in general this counting should occur as follows this n factorial should be split into a number of sets with g and q being a number of endpoint functions which are computed a proper q audio which means that they cannot be computed by anything less they could be computed by something more but that is redundant that is a and there is an answer for this I will tell you how we arrived at this answer just give you the counting is the coefficient of there is a generating function you can write write down for this it is complicated but it is it is not actually that complicated it is some coefficient in the Taylor expansion of a poly log function the second question you can ask is ok that is the problem of asking given some n how many how many what is the k you need what is the how many how many of them can be represented in different contours the second problem is suppose I gave you a contour with some k how do I decompose this 2 k to the n. So, the 2 k to the n itself splits up into the following sum which says that each correlator which is each ordering which is computed as a particular q audio correlator appear some has some degeneracy it appears so many times which is equivalent to this statement that if I take this 3 to 1 ordering and ask how many ways can I represent it in a 2 OTO there are many different ways all related by the sliding and h n q just counts that that is that degeneracy and this is given as the coefficient of e to the k in some other generating series it just says that n is bigger than to give both of these series are easy to count I want to go through the exercise but it is done in the literature the only thing I want to remember right now from this exercise is that there is a physical question what are all the correlators you can compute with a given out of time order there is a mathematical embedding of that into these OTO contours which are the things that you would use to do functional integrals but those OTO contours come with large redundancies and therefore many relations and these relations are going to be interesting for us and that is what I am going to focus on but this is sort of the precursor to doing that are there any questions about this oh mu is just some dummy variable it is a variable for a generating function you take this function and expand it out in powers of mu the power of mu to the q will tell you how many endpoint functions can be embedded in exactly q OTOs they cannot be done in anything smaller they cannot take and but that is the those many of them will fit into so for example here G 3 1 was 4 and G 3 2 is 2 and here G 2 1 is so let us try to abstract this into some set of statements that we can use as a constraint on the low energy theory so everything I said presupposes that you had a functional integral you know exactly what you are doing all the information was provided to you there was some initial state I was agnostic about what that initial state was it could be some density matrix it could be the vacuum state does not matter you could just run this exercise right because all of that information is buried in this row which showed up in my generating function and then and then we went through this exercise so let me first diagrammatically convince you that this is a useful way to understand what is going on and then tell you what the statement is at the level of generating functions so diagrammatically the following statement appears to be true let us say you have an operator here which is 1 and then you have something happening in the contour before this I am completely agnostic to what other operator insertions are and I am drawing a 1 OTO contour for second act but I can draw something else as well so the same statement would hold true but let me do this first I think since here all I have is the identity matrix nothing obstructs me from moving this operator down to the next leg nothing nothing happened these are two exactly the same two representing the same correlator in functional integrals in our so here this is a one right operator here it is a one left operator and low and we only have derived an interesting statement which says that sorry let me try to distinguish my 0s and O's I am just subtracting them I am just saying that and the f basically stands for future most if I take the difference operator and put it future most the correlator is 0 this has a name this is usually called the largest time equation although it was not technically invented to talk about correlation function it was invented by Weltman to talk about how S matrix elements know about causality because Weltman's idea of how to implement Bogolib of causality in scattering amplitudes led him to conjecture something called the largest time equation but I'm I just added for you diagrammatic no no no no magic required but in higher OTOs there's a lot more the three OTO and you have equations for every turning point so you have turning point equations which follow from sliding rules operators same same statement again using the fact that everything here all these turning points are just identity matrices so some of these here are largest time like equations whereas this will be so-called smallest time equations I'll just call them turning point equations because that's what they are so these relations are what sort of capture part of the phenomenology of what these h and q h q n's h and k q's are doing right they're telling you that some correlators are related to some others but you can get all of them by sliding and the sliding is really being represented here as correlation function identities and the reason I want to do this is that correlator identities if I capture them in a useful manner can be used to constrain the low energy theory okay so suppose this was a microscopic theory where you had these correlator identities if I package them in a nice fashion for you then I can use that packaging to say whatever my low energy theory is I need to get these what identities and therefore I can use my encoding to sort of tell me what the constraints are in my low energy theory so that's where we are going we're going to sort of re-express all this in some slightly useful language and then say we have the tools to go back and ask questions about the low energy theory because so far all have been doing is with respect to the microscopics and eventually I want to sort of unanchor myself from this knowledge of the microscopic theory and just be able to write down the low energy effective action in very general circumstances as we would do naturally in the context of the Wilsonian program so so let's let's see this these two relations directly from something from the generating function and then try to capture them in a language we found useful to sort of achievements so all this discussion about these OTO's you can find in the paper which as I said strangely enough should have been done but never was done you can find this in a paper I wrote with who is that ICTS between Iran also at ICTS and Felix Hall who that we see paper from not so long ago and what I'm about to tell you now is work I've been doing with Felix and Logan and the primary reference I'm going to give you is a series of long review like papers we wrote about finger keldish its application to low energy dynamics I'll also tell you some other pieces of work that have appeared in the last couple of years once I say say a few more things okay let's talk about finger keldish per second so that's what I primarily focus on the as you will see that already sufficiently rich and I'll tell you what the upshot is for the higher OTO's but the story is a lot more involved so we had this generating functional last time which was functional of two sources the left and right source I want to ask two questions one is where does where do those sliding rules come from and to what are the most general constraints here the first thing I'll just answer the question about the constraints and then see the sliding rules so let's consider this generating functional when the two sources are set equal I can do this sources are just external things that I can sort of turn them on at will but look what happened traces cyclic you and you dagger are unit you are unitary trace collapsed which means that if you functionally differentiate this with respect to sources you just get 0 now what do these equal sources couples to is equal sources they couple to the difference operator because you have coupling of source to operate in you is j right or right but the left operator comes to the dagger which means that source operator coupling there is j left or left but with a minus in other words in the finger keldish theory the source operator couplings has natural Lorentz signature metric if I switch to my average difference basis that's like going to light com the operator couples to different source and the difference operator couples to an average source so this guy where the two sources are equal doesn't turn on the difference source it only turns on the average source but when I turn on the average source had I been functionally differentiating this guy I would have been pulling down powers of the difference of pulling down difference operators you are just learning that pulling down difference operators gives you nothing the correlator vanishes so unitarity implies arbitrary sequence of difference operator correlators to be a zero in the theory the keldish rules know about this the keldish rules know about this and my largest time equation because keldish bracket engineer this to be zero which is why when I computed a average b difference last time the Schminger keldish ordering the a b commutator which tells me that a had better be in the future of b if b were in the future of a there is no contribution and same another application of keldish rules would have told you that this correlator is 0 because the the innermost keldish bracket will click only when the future most operator is not a difference and here both are differences both will give you 0 you get nothing I did this for two point functions you can run this exercise for any number of operator insertions you will see that all of these are true but they had to be true because they're coming from something very fundamental they're coming from unitarity and this is why encoding this unitarity constraint which is this kind of difference operator correlator vanishing or this largest time equation which also by the way the consequence of unitarity because this one is just telling me that well you can either turn on a source here or you can turn on a source here but they're just saying you evolve the system after some time and then you do the same thing from that time onwards to your later time but and if you functionally vary with respect to the same thing you'll again get back a similar relation except that there will be some residual use in you daggers left but that doesn't matter what we are going to try to do is encode these in a language that's amenable to doing RG so that's my goal but before I do that let me tell you that things are much more fun and much more interesting once we go to higher OTOs because there are a lot more relations I'll just say that I won't tell you what the encoding is because that's not something we've worked out in full detail yet but you can guess based on what I'm going to tell you about the Schminger-Keldrich theory that is a very natural generalization so let's just do two OTOs as an example I have z which is a functional now of k1 right k1 left k2 right to left four sets of sources corresponding to the four legs of this two OTO contour and look you have lots of identities coming from three different sets of what I'm going to call localizations let me explain these three things I can take this generating function I can set the sources on the two right and two left to be the same then by unitarity this collapses into a one OTO I can set the sources for two left and one right to be the same or these two two left equals one right one left equals two right two right equals two left collapse this to that's why it's partial because it doesn't go back all the way down it just becomes a one OTO contour and these encode all the relations if you saw where you could take a Schminger-Keldrich correlator and by sliding implemented into a two OTO contour total localizations happen by the for the trivial case when I set all sources the same because then everything is trivializes to trace row or I can do something seem different I can set these two to be something and those two to be something else and that's still give me a total localization this precursor localization is a bit more funky if I set the one right equals one left then I can't collapse them because they're obstructed by the by the tools but what I can do is subsume them into redefining my initial state and then it's effectively a one OTO but for redefined initial state and that's why and these are related to what people have been discussing in various contexts of precursor or post-cursor operators that in this context it's basically just means what what I just defined for you so quantum field theory with given out of time orderings has all this rich structure and all of these are built into these correlators but the nice thing is that you can abstract all of the discussion into a sequence of statements that are valid about correlation function identities key things are things and the key name of the game here is that they all follow from unitalented what we now want to do is take these statements and put them in a context where we can sort of make progress and everything I'm going to say now will be for the finger killish theory the one OTO theory because that's that that will be quite sufficient for the rest of the lectures there's an upgrade to the of all of those statements to these higher how much time do I have very good okay so that's about time to basically repackage this information set the stage for our next discussion let me ask you one thing if you were told without any of this background that you have some number of correlation functions you sort of started computing them and then you start getting zeros getting zeros for a class of correlators which don't care about where those operators are inserted because I've been completely agnostic about where these operators are inserted in space and time the statement is just true because the locations of the jays where the so the facial temporal locations of jays don't don't enter into the game at all all I care about is the right source equals left source end of story so there's some complicated microscopic theory you do some calculations you start seeing zeros I mean usually their identities that depend on what the theory is what your Hamiltonian is which enters into the evolution operator etc etc but I'm making a statement about a wide class of correlators being completely blind to these statements so they must be an underlying reason why this happens because to my knowledge only one nice set of examples of quantum field theories where this is guaranteed by symmetry and those are quantum field theories where there's a topological symmetry topological symmetry has the power to guarantee you that correlation functions of certain operators which are in some suitable sense in the cohomology are trivial so we're going to just take that intuition and run with it and ask can we invoke some topological symmetry and guarantee that the relations you're seeing in the Schwinger-Keldrich theory are made manifest so now on to something completely different as the saying goes and now I'm going to sort of reformulate everything that we've seen in a language which is very much tuned to topological BRST symmetries requiring ODIF to be BRST exact would guarantee that and I'll just say here in parentheses similar statements will hold for turning point equations so the post so that right now this is just a postulate I'll sort of argue for this in a second and argue that that exists two supercharges a nil potent such that difference operator is both q of something and q bar of something call these SK supercharges because they're coming from the Schwinger-Keldrich construction note that I'm requiring existence of two supercharges there's a parallel work of Hong Liu and his collaborators who seem to want to do do away with one of these and work with one I think this structure is nicer it it manifests CPT if you want to think about and in the example I'm going to write down in a second you can see that the two charges of both CPT conjugates of each other they have a conserved ghost number and so on and so forth so I think this is more natural way of doing things before I sort of say more about these operators let me just give you an example so that you'll see what's going on take a free scalar theory whose action in d dimensions for the Schwinger-Keldrich contour would involve the right field with the usual free master scalar action but the left field will have also the same action with the relative sign the relative sign again comes because of the difference between the u and the u dagger this theory actually has a symmetry but among it there's one which is interesting for my purposes which is you can usually you can feel actions when you're computing correlators you can freely define actions because when you do the path integral you integrate over all all field configurations so actually what the field is doesn't really matter change of variables but in here if you do a field redefinition which sort of does the same thing to the right field as it does to the left field the action doesn't change there's an invariance of the action so in some sense there's a gauge symmetry here that you can gauge fix you can gauge fix this side to 0 and then run the usual Fadir-Popov trick to implement that gauge fixing we do that then you can write down an action which is the gauge fixed action but now it'll have the BRST goes which come from this gauge fixing and it's not so hard to convince yourself that the action is going to look like this which I will rewrite in terms of the average difference fields plus these ghost terms that came from this gauge fixing one for each real component of the scalar the scalars are complex size size let's say coordinate dependent not feel different not not feel dependent you could do something more but it doesn't it'll end up being the same thing it'll just give something more complicated here the gauge fixing condition but I took it to be just coordinate dependent so when the gauge fixing condition is just delta of psi and then the Jacobian factor gives me this piece so you see that in addition to my regular fields which were light right and left or averages and differences I have a pair of ghosts and these ghosts and this theory has this BRST symmetry in this case you may have to use equations of motion it's off-shell it's sorry it's on-shell yes so I'm going to write something which is off-shell but at an operator level so inspired by that construction I'm just going to demand that for every what we what we did in a Schwinger-Keldisch construction when we took an operator O or O hat in our single copy theory and we encoded it into an O right and an O left for being able to put them on the Schwinger-Keldisch contour now in addition to that to sort of keep track of uniterity I'm going to introduce two fields here which are the analog of the C and C C bar which are going to carry opposite grassman statistics to the O right and O left they're going to be the ghost partners of O right and O left so introduce o g and o g bar it goes numbers and minus one respectively such that 2 sk on o g minus o d and bar sk on o g o d or in other words you can just write down a nice sequence of relations which is the BRST complex which starts out with the average operator descends along a ghost descends along a q sk to a ghost operator and then descends along a q bar sk to a difference operator and then there's another descent direction and these arrows indicate these kind of commutator structures these operators you can compute the commutators the nil potent they are specified them for you acting on the sequence of operations they're actually nicer nicely represented as a superspace is useful for doing calculations so let me just introduce that for now and then stop I want some superspace with two coordinates theta and theta bar so that those and q bars act by derivations in the super directions put all of this structure into one single super operator and I'll put my super operator with a little circle on top to denote them to be separate from my from an operator the average operator the bottom component the ghost and its conjugate are the intermediate components the difference operator which is the top component the theta theta bar component of the super this looks it's a very simple superspace I just have two two grassman odd objects they have no fermion labels they just grassman odd objects they just square to zero so it's it's very easy to just run with this and the statement is that pretty much all the identities one has one can then use the superspace technology to recover all the keldish rules and so on and so forth with some suitable dressing so you can rewrite the lesson the model of this discussion is simply that you can just take the finger keldish construction is given to you and all the constraints that come from the microscopics can be taken care of by working in this natural Schwinger keldish superspace that's the first lesson next time we will try to do this in the context of thermal field theories and then see what it implies hydrodynamics so I'll stop here