 Čakaj, zato, kako se počutite, počutite vlastnje lektu z Mukund. Čakaj, počutite, kako smo, in tukaj, počutite s tem, počutite, počutite vlastnih lektu, kantečnju, in nekaj jeenz, da je prijeljno, način ono zelo, ki so pričeli, da se pričali, da so prišeljno, in vse jeznaj, da se pričeli, začali, da ni se pričali, ki ampak vse jeznaj, in začeli, da je zelo, in da se pričeli, pa postavljamo, da je se pričeli, in da se pričaljno, v teplogi semetriji. Zelo sem tukaj, da se pravamo o algebrih, kaj je to kombinacija kaldiša, brsti semetriji in kms kondition, kaj je inerent in termofil teorič. Tukaj, zelo sem tukaj, je tukaj, da sem tukaj tukaj, način, ko je pričo v kvantam-filtrih, ko je mikroskopik odličenosti vizuču, in tudi svoj svoj, da motivite in vznosil nečeljih energijistih teori. Tako, ki smo spetili v Natti vših strani, vsevni, način, da se je zelo vzor in vsevni vzor, njegi neč o kaj se priključnih atom, tudi se lahko vzorni način v začanje, kurs grene v systemu in tudi o vsej mikroskopičnih vredajov, primaričnih termodanimičnih, in tudi je presešljena v systemu, ki je tudi tukaj, ker je tudi izgleda taj formalizm v nekaj način. Tako, moj objevčen je, da je vse več nekaj način izgleda hodinamikov, ki je nekaj nekaj način teori. Znamenim, da je vse nekaj nekaj način o hodinamikov v toj kontekstu, in tudi tudi, kako je tudi formalizm izgleda, imam imač nekaj so, da boš prišlič, da bomo na nečem izgledati, da sem da pokazala, da ne bomo na kraju dobro nekaj, kako smo pošličila. Prejsem, da vam pošličila, da je ovo Cardiča. Zato, da so prišličili, da vzopseli hodnje. Zelo sem pošlič, da ga se različila, da je tako, da je, da je dino, da je vzopsela, da je tudi, da je tudi, da je tudi, So in near-equilibrium situations. They're interested in long wavelength, you can say this, long wavelength physics, so these have frequencies and momenta, which are much smaller than the thermal scale set by the system, okay. So in this regime you can sort of take a quantum system and And you can focus not on the microscopic details, but on a few macroscopic variables. Basically, roughly speaking, in thermal equilibrium, you tend to monitor simple quantities like macroscopic variables like energy densities and charge densities and so on. In this hydrodynamic regime, these guys are local functions and they vary from place to place. And the dynamics of how they vary from place to place is the theory of hydrodynamics. So there are a few macroscopic variables. The dynamical data of our system, which we keep track of in this limit, is simply the conserved currents. Let's say the energy momentum tensor as a function of space and time. Or if you have charges, charge currents as a function of space and time. And the reason these are the only things you keep track of is you have a system, it's almost in equilibrium, it's not quite. It's deviations away from equilibrium or on macroscopic scales. But what's happening is that all the disturbances that have driven the system away from equilibrium have mostly died out. And the things that survive to long distances and late times are things that are constrained by conservation to decay back slowly. So you could imagine that you take an equilibrated system. It could be an air in this room or a glass of water, which has been sitting there for a while. And you come and hit it. There will be a lot of transient effects initially where lots of things will be happening. But if you wait long enough, the only sort of modes in the system that will survive are the modes that sort of talk about the conserved quantities. The local energy densities, the local charge densities, the local stresses and strains, et cetera, buried in this object. And so the dynamics of the system is actually very simple. If you think of these as quantum operators, then the dynamics is basically their conservation. If you have gauge charges, then if you have charge currents and your background gauge fields for these charge currents, then you have a covariant conservation of these charges. You see, a lot of the details of your microscopic data has disappeared by the time you've gone down to this. Because usually these things are constraints on the system. They don't determine the dynamics. In the hydrodynamic limit, they're the only things that survive because everything else has sort of quieted down. So it's a rather special dynamical system where conservation dictates how the system evolves. And so one needs to understand how to write down these structures in terms of some basic variables. So you can think of, basically what you have is you can think of there being some kind of fluid, which has some flow pattern. And in local domains, you can have local, your little domains of local equilibrium. And here there's some local temperature, T of x. Here there's some other local temperature. And you have exchange of charges, energy, momentum, so on, across these domains. So usually what people tend to do is tend to talk about a flow velocity, which is some flux vector. It's usually normalized. And I'm going to work in the space times. I'm going to talk about relativistic fluids. So my normalization is that the velocity is normalized to be unit time like. And there's a local temperature effects. And maybe you have charges, you have local chemical potentials, et cetera. And these are the data of the system as the system evolves. And what you need to do is to establish what is the energy momentum tensor as a functional of these fluid data for convenience. I'll also allow myself the freedom to turn on background sources. So these are the dynamical variables. These are the things which will satisfy equations of motion. But in addition, I'm going to allow myself the freedom to have, probe my system by putting my fluid on either curved space or, as Nati was doing, turn on background electromagnetic fields to sort of make the charges move. So there will be some metric data and some gauge field data. But these are, I emphasize, backgrounds. And they are classical. We already saw the metric show up last time when we were trying to talk about equilibrium on arbitrary curved space with varying temperature. This is a variant of that, where now I have the temperature not just varying in space, but also in time. And therefore this velocity field is also changing in the function of space and time. So once you have this, this gives you the dynamics, the conservation equations for which constraints what these guys, what the u's and the t's and the mu's do. So, and you can convince yourself that these are the right number of equations because this guy is a conservation equation in d dimensions. So this has d equations. This is one extra equation for a scalar equation for charge conservation. In u, I have d minus 1 variables. In t, I have one variable. And mu, I have one variable. So the system is deterministic. It has exactly the right number of degrees of freedom as there are equations. Now, life would be very simple if this was all there was for hydrodynamics. And usually what we know phenomenologically is that you not only have this for the conserved currents, but the dynamics is also constrained by the second law of thermodynamics. And since everything is local, everything happens locally, it's not happening globally. I don't have a statement which is... I'll make a local version of the usual statement which says that entropy should increase by declaring that there exists some current, local current, which you can think of as a local flow of entropy whose divergence satisfies the second law. So the entropy is produced in this form. Now, usually what this does for you is that it constraints the terms that can show up in the energy momentum tensor. I'll write down the energy momentum tensor in a second, but that's the general structure of the phenomenological theory. What you also tend to usually do is you tend to ask that this statement hold configuration by configuration, that it holds once you solve the fluid equations, once you have an expression for T mu nu and you found what U mu and T are, you then ask that the entropy on that configuration have positive definite gradient. It's a bit sort of awkward to sort of have to solve this on shell because it's a constrained system, you have to solve equations and then ask what happens on top of that. So it turns out to be actually useful to give an official statement and this turned out to be quite powerful in basically giving a complete understanding of what this phenomenological theory of hydrodynamics is. So let me explain that in a slightly useful form. So a couple of years ago, inspired by various developments in the last decade, a few of us gave an official formalism for tackling phenomenological features of hydrodynamics and for simplicity, I'll drop this chemical potential, I'll just write equations in terms of just local temperatures, you can add charges if you want, it won't change the discussion over much. So the natural variable I'm going to pick is the unnormalized time like vector beta mu, it's basically the flow vector normalized by the temperature and I'll note here that this vector is a killing field we introduced last time when the system is in equilibrium. This is part of the reason to introduce this this way, the time like vector field in equilibrium that this vector field is killing. Now in terms of this guy, I can say that the official entropy statement is basically d mu j mu s plus basically some kind of Legendre transform, so this starts to look like some kind of first law, this is like ds, 1 over t de equals something. So this is required to be positive and you can actually do some simple tricks to this, let me just write out one more equation and then you'll see why this is very useful and why this statement is very helpful. So let's define, let's Legendre transform the entropy current into a free energy current by basically putting together pieces involving the energy momentum tensor. So again this is like s, the time like component is just s and here it's basically energy density divided by temperature. Just think of the time like component because this guy is time like, so in some inertial frame this picks out t00 and it's normalized by 1 over t, this picks out the entropy in the time like component. So this is really a free energy current up to sign. How you define the free energy. This guy turns out to satisfy a very simple equation. So the analog of this, analog in this definition is this. I have sort of just plugging in and rearrange this but this guy is just the lead derivative of the background metric with respect to the field beta. So this is by definition the lead drag of a metric along a vector field is basically the symmetrized covariant derivative on the vector field, sorry on the one form. So this is very helpful because you see that this piece will vanish if beta is killing because this is really the piece, this is twice the killings equation. So you can basically start by asking questions in the following hierarchical manner. You can start by saying, okay let me look at equilibrium. In equilibrium nothing should happen, no entropy should be produced. So delta must be zero. So let's classify all possible free energy currents that can show up which are conserved currents. So you sort of simplify the problem from the point of view of trying to classify energy momentum tensors which are two tensors to the point problem of classifying conserved currents and you can argue that all such conserved currents must come from a master generating partition which is like a partition function. So I'll write this as z equilibrium of beta mu equilibrium and g mu nu. It's just some scalar function built out of these guys and that scalar function if you are doing usual statistical mechanics of quantum field theory on Minkowski space with constant temperature everywhere, homogeneous constant temperature everywhere is a familiar partition function. I'm just allowing myself the freedom to upgrade from that picture to the configuration I was drawing last time where my thermal circle was fibred over this base manifold sigma and the fibrration structure dictates what the structure of this partition function is. So we can write down expressions for this but you can write this down so the z equilibrium basically starts out its life as the homogeneous piece which is an integral over sigma the base manifold spatial manifold times so let's call what we had yesterday gamma the spatial metric plus because this is spatial metric and then there's a piece here which you can think of as the free energy of the function of temperature plus gradient corrections. So this is the piece you would have if temperature was uniform everywhere but things are changing on this manifold so I can sort of and I'm interested only in physics in this long wavelength regime so I'm allowing myself the freedom that moment are small compared to temperature so I can expand everything in terms of derivatives. So if you want we can write down the language we had last time you would have terms like so this is remember Akal is the client ansatz the geometry that's the metric and then you can have derivatives of terms involving derivatives of sigma terms involving derivatives of A so all the gradient terms and you can assemble them and study this order by order in perturbation theory in this gradient perturbation theory very good so indeed so right now I'm just writing a general expansion and I would just have to figure out the coefficients by actually evaluating the partition sum for the theory because I mean already the free energy cares about what the spectrum of the theory was but whatever it is it is some scalar function of the temperature so everything I'm going to say I'm going to be reasonably abstract I'm not going to specify to a particular theory towards the end I'll give you one example just as an interesting statement that one needs to understand but for most of the discussion I'm going to assume that you have a theory in mind so it could be QCD, it could be some other theory and you have to do some level of calculation to understand what this is but if you do it you will understand the coefficients but our logic is going to be like you work with effective field theories so let me remind you the model I have one second model I have in mind so think of QCD and think of chiral symmetry breaking when you have pions Numbu-Goldstone bosons so of course you could start by the theory in terms of quarks and so on but we are not interested in physics on the UV scale you are only interested in physics around the pion scale so you have an effective theory of pions which you can write down in terms of the pion effective action but in the pion effective action that pion couplings that are undetermined so the analog of the pion couplings are now functions of temperature the scalar functions because we are sort of we are not reducing this to quantum mechanics but we are reducing this to a field theory in d-1 dimensions let a question you expect the existence of a partition function in this case because there is a sense in which there is global equilibrium although this circle this time the way the circle is it is fiber non-privilega over the base all that is telling you is that locally but different observers measure temperature is just spatially varying but there is no explicit time dependence and that is all that determines the equilibrium absence of explicit time dependence a better way to say is what is the analogy I gave you last time which is you know that you can have spatial gradients and have equilibrium by just looking at the atmosphere the atmosphere is not a system where the temperature is homogeneous all the way through the temperature is changing in fact exponentially going down as a function of the altitude and that is a sort of baby version of this we are just allowing ourselves arbitrary dependence to talk about hydrodynamics I would have to allow myself even the curvatures to vary slowly but in principle you could take any curved manifold with a time like killing field and put your field theory on it and it will be in equilibrium it will be in equilibrium with respect to the killing field so any observer who follows orbits of the killing field by definition does not see anything because along the killing field nothing changes so this person cannot see anything and that is what equilibrium intuitively means and that is the intuition we are exploiting so that was something that was done 45 years ago by a bunch of people from TIFR and another group in combination of Germany and Vancouver but you can ask can you go beyond equilibrium can you classify everything can you basically tell me what is the so I have this equation which is supposed to be valid offshore it is supposed to be valid before imposing the equation of motion of the energy momentum tensor conservation can you just tell me what are all the structures that can show up in this equation to do that all I need to tell you is what is the set of vectors that can show up in an entropy current what is the set of tensors that can show up in energy momentum tensor such that this combination is positive definite so phrase very abstract leads so I am asking I have vectors and symmetric two tensors built out of some basic objects in this case just beta mu and some background metric can you just do some algebra and say what combinations are allowed for positive definiteness it is almost the question is almost like asking I have some quadratic form how do I constrain coefficients so this quadratic form is positive definite it is just bit more complicated because it is a question in tensor space not in terms of scalar quantities which is what you would use for quadratic forms but basically all the intuition you have for doing quadratic forms you can employ here if you suitably use variables so I will tell you the answer of this analysis and refer you to the literature because that is going to give me a benchmark for what I want next you sort of ask this question and then you find that allowed classes fits into eight distinct classes I mean let me also jmu s because it does determine jmu s directly and we gave them names I will just tell you what the names are and two of them I will describe for you in some detail the rest I am going to skip so this is the class that we would all be familiar with so this is the class of dissipative data let me write in energy momentum tensor that you would usually write for hydrodynamics you write them energy density you write some pressure both of which you would get from this free energy energy density just transform of the free energy the pressure is basically the free energy itself and then go to the next order this is the shear tensor is basically the symmetrize traceless derivative of the velocity with some coefficient which is a function of temperature called the shear viscosity and this guy belongs here so the first simplest thing you do in hydrodynamics is you see that the system has friction the friction is encoded in the viscosity and this viscosity is in this dissipative class this is the thing that everybody familiar with conductivity for example we have charge current belongs here but then I have this plethora of things which by definition are not dissipative what do they do so this dissipative class is a class that you get by solving this in homogeneous equation with a nonzero delta on the right hand side it's the only class that contributes to delta and the claim is by playing around with the intuition about quadratic forms you can assemble this dissipative stress tensor so there is a t mu nu if you want you can put a d here to say that it is a part of the stress tensor that comes from dissipation and this guy can be assembled along with j mu s dissipative into a piece that gives you a positive definite object explicitly this is the easy one you can show that this guy has to be of the form again the intuition is that this guy is some kind of basic building block sorry only one of them there because when you take this guy and put it in here then you start seeing that it starts looking like a perfect square because it's a perfect square in a space where you take two tensors and two tensors which are symmetric and then you combine them with a four tensor and as long as this four tensor respects the symmetries it's actually even symmetric under the exchange of mu nu and rho sigma and this positive definite it's a quadratic form in the space of two tensors then you're in business so t mu nu can be decomposed into eight distinct pieces in the dissipative class you can argue that t mu nu has to take the form of a four tensor contracted with the lead derivative of the matrix with respect to beta this guy fits in that class the second term fits in this class so this guy I claim can be written like this that was equilibrium sorry this is equilibrium this is not equilibrium because this is dissipation and this guy belongs here yes but at this point as I was saying before it's an arbitrary four tensor which is a function of temperature it's only a requirement either it's symmetric in mu nu and rho sigma and symmetric under the exchange of mu nu with rho sigma if you want I can put a second bracket here let me write it a bit better claim all hydrodynamic dissipative strength pieces can be assembled in this form this was something that came out of a very impressive analysis of shantani butacharia that we managed to explore to show that this is true so I'm not going to show you how to write this in terms of this but it can be done I'll refer you to the literature but then there's this plethora of things and in fact Atisha alluded to something because this piece is in equilibrium this does not belong to t mu nu d it actually belongs here in equilibrium and this comes from z equilibrium the stress tensor in hs comes by taking this guy and varying it with respect to the metric you get the stress tensor by putting a system on curved space and varying with respect to the metric and that's what you get from here and clearly if you vary the free energy piece with respect to the metric which is basically the piece in this root gamma then you would get both the epsilon piece and the p piece I'll leave that as an exercise to show that that's true and the relation between p and the free energy is basically p equals minus the free energy and epsilon is basically p minus tp prime or something like that which is the usual edge under transform statement now this is whole other 5 classes 6 classes that don't seem to know anything about dissipation so first of all claim that all of dissipation is contained in this 4 tensor everything else satisfies this homogeneous equation with the right hand side set to 0 which means there are many solutions to the homogeneous equation question is how do you classify them one class I already gave you but there are many others which are not obtained from that homogeneous which are other solutions to the homogeneous equation so this guy so here for example this is another easy thing to write can take the same structure but now take it antisymmetric in mu nu and rho sigma exchange now you take this and plug this in here you have 2 L beta g mu nu copies of L beta g but the indices are antisymmetric so this term vanishes and therefore it's a consistent solution with delta equals 0 so you also simultaneously solve for g sigma I'm not writing out what g sigma looks like or the entropy current looks like in all these cases I'm just writing out what the energy momentum tensor is because that's a physical object there can be arbitrary ordering derivatives so Landau-Liftschitz has this piece to 0th order Landau-Liftschitz has this piece to 1 derivative which is explicit here so basically eta is some constant tensor that gives you that eta Landau-Liftschitz doesn't have this because this does not show up at first order in derivatives in any system except parity violating systems in 2 plus 1 dimensions and Landau-Liftschitz didn't consider parity violating systems telling coming I haven't told you what they are so here there is no energy momentum transport you actually have a conserved entropy current this is relevant in circumstances where you have topological degeneracy in the ground state because they have non-zero ground state entropy but no energy momentum transport this guy is the first case where Landau-Liftschitz could have figured out had they gone 1 higher order but they didn't and so here there is there's basically something like vorticity squared vorticity is basically the anti-symmetraizd derivative of the velocity sorry I misspeak, this is not here this is here there's a part of sigma squared which is here let me say it that way I'll write down what this is in a second the other 3 someone asked Nati the other day a very interesting question what about anomalies toft anomalies in thermal physics again Landau-Liftschitz didn't know about toft anomalies because it was before toft and there's a very interesting statement that one can make so to do that I have to actually turn back the charges because anomalies don't show up unless I have charges so for neutral fluid what I said is complete but let's say you have a charged fluid and let's say this fluid is charged under some background global symmetry which has a toft anomaly now what you would say is the following statement that if you don't turn on and this is a statement Nati made that if you don't turn on background fields then the current is conserved if you turn on background fields then there's a term here which I will call some kind of Hall current which is a functional of the background f mu nu that's how anomalies work in 4 dimensions the conservation of J would be violated by an E dot B term on the right hand side but E dot B is a background electric and magnetic field it's not a gauge current so this is coming from the triangle diagram Nati was drawing with 3 vertices so this is all coupling to external currents now when you turn on temperature and you can ask the following question you can take the following order of limits you can turn on background fields turn on temperature solve keep this equation intact you can ask how much the current is not conserved by the background and then having work dot word J is turn off the background fields the statement is that J has pieces that nobody anomaly even if you don't turn on background fields so in other words J would be the charge density times velocity minus conductivity times let me call the sigma c sigma comes up in 100 different places and there is a vector here basically the gradient of the chemical potential gives you conductivity I put a quotes here because this is not quite the full gradient it is just the spatial part of the gradient and but then there is another piece which comes from this contribution there is a piece here which comes proportional to the vorticity contract with velocity and this piece basically tells you that there is a current flowing in a direction normal to the velocity this is roughly speaking in the direction of velocity this current is in the direction of velocity but here there is a transverse current and the coefficient here is set by the Toft anomaly it is precisely fixed and you can get it directly from some basic data of the theory without doing any of this magic that guy is here this is a big surprise I think various people have thought about this in the past but the first smoking gun signal that such things exist came out of an ADS CFT analysis that was done by Logan Aragum and various students at TIFR about 10 years ago it caused various people to question landau lift sheets and when you sort of did all the analysis again taking into account things that they had missed parity violating terms and so on it works precisely in this general formalism now there is one more wrinkle which is that in certain number of dimensions you can have Lorentz anomalies if you have a flavor current that is anomalous it can also give you a frame anomaly for the Lorentz rotations those guys also contribute and there is some time called mixed gauge gravitational anomalies and they show up here and once you have this you also have terms here which come from again write down a stress tensor piece it involves some 5 tensor and some other contractions of G with L beta I am not writing a full tensor structure but there is some other tensor structure you can work out which belongs in this class so what I want to emphasize is that much of this could have been done many years ago but it was helpful to have a lot of intuition that came primarily from holographic analysis and classes of stress tensor that were derived there to sort of complete this statement so what we have now is a statement about what hydrodynamics ought to be as a low energy theory we have to have a theory which gives you all these 8 classes these 8 classes are no more than these 8 classes so you have a benchmark for the low energy theory so you can ask take your favorite microscopic theory it is in thermal state it is not quite in equilibrium start with a microscopic theory integrate things out and motivate an effective action that captures these pieces of data so you can try this following naive thing which is inspired by the equilibrium partition function and you can see how far you get and that quickly makes you realize that the story is lot more richer and you have to do lot more which then in the last 10 minutes time you back to where I began in the last couple of lectures so you could say construct an effective action for hydro whose job is to give you those 8 classes and that was an official classification so if you have an action because that action by definition is official objection every one of you would give me is that dissipation is tricky because it seems like you are losing information so how can you write down an action so let's come back to that let's postpone that question for a second let's just ask can we write down an action that at least gives us the remaining 7 classes that don't know about dissipation because they are conservative the energy momentum is conserved they are not producing any entropy so how can you not write an action so here we have something that looks like an action function it's on the spatial manifold but you see because it's on the spatial manifold and the time direction is just a killing direction nothing prevents you from upgrading this to a full integral over space time because that just basically homogeniously removing a factor of beta so why don't I write an action which says some action which is given in terms of Lagrangian density which is the functional of g mu nu and beta mu scalar function built out of two pieces of data anti-done I've just allowed myself to forget the fact that these things were supposed to be killing fields just write down this action of this kind two problems one and the most obvious one if you write down an action with a variable which is a vector field you'll get some vectorial equations of motion alright but no guarantees that you get those equations of motion of conservation equations because energy momentum conservation implies Euler Lagrange equations but the opposite arrow is not always true it just takes simple examples and you'll see because these follows from background difiumorphism invariance these follows by varying the physical physical fields they're not the same thing so clearly what has to be true is that not all of beta can be dynamical data so there's a way around this and there's a way around this that almost everybody who's done, who's read the first chapter of Polchinski knows the number go to action has gravity on the world sheet and the equation of the motion of the world sheet gravity tell you that the energy momentum is conserved and basically you can use that trick go about this so you basically don't keep all of beta mu independent but only a part of beta mu which is like a target space coordinates pulled back onto some kind of world sheet so here what you do is basically think of some world volume I'll use the word d-brand it's a world volume but this is space filling world volume and here is my physical space time and here live beta mu and g mu nu and the world volume has some intrinsic coordinate sigma oh god, sigma again sigma a and I have some map which map me from the world volume to space time which gives me space time coordinates as a function of world volume coordinates and on this world volume I have the pullback of beta to be beta a and the pullback of g mu nu to be g ab the dynamical variables are x mu and you can convince yourself very quickly that if you do this embedding equations of motion of x mu are precisely energy momentum conservations pushed forward into space time so there is a sigma model interpretation of hydrodynamics where you can think of x mu at the dynamical variables you can write down the theory living on some auxiliary space in fact you can think of my this beta as some kind of flag on this reference on this world volume so you really have a d-brand like theory the novelty from the d-brand is that this d-brand carries a flag it carries a local knowledge of what the local temperature and velocities are so you can think of this space time as having the vibration structure and this is pushed forward space time now that sounds great that solves this problem that fixes here t mu t mu nu equals zero follows from implied by variation of Lagrangian with respect to x equals zero but it doesn't solve this problem what does it solve it gets right this structure because it's almost guaranteed to get this structure right from the point of view of the equilibrium analysis and it gets this structure right so it gets you the energy momentum tensor which belongs to these two classes it just doesn't do the job for the rest of them so the question remains what do you do to get the data for the remaining sectors and there are two hints that something must give two hints that something must give at the fact that the microscopic theory was on a shingar keldish contour what we wanted to do was start here integrate things out and write down some kind of classical effective field theory for these x mu variables so where did these x mu variables come from is the first question we should ask ourselves because the microscopic theory had some maybe quarks and gluons you didn't have x mu's x mu's are some emergent variables they are like pions in QCD and the answer is provided by the contour because on this contour just as I have two sets of fields I also had two sets of sources I had right and left sources which I was calling j right and j left that means that I have g mu in your right and g mu in your left in the microscopic theory not only that in structural shingar keldish theory there are interactions between the left and the right contour as I start in the microscopic theory I have all this structure if I start integrating out some degrees of freedom on the right and left I will start non-trivially coupling the two bits by definition because if you just write propagators the propagators talk across the contours I should keep track of everything and the first question you should have asked yourself is why do you have only one classical degree of freedom where is the other guy because there was a right and a left so they should have been something like a right plus left or a right minus left but only one of them is visible in the low energy theory and the answer to this again comes from something that one tends to ignore in hydrodynamics one tends to do the average description of how the fluid flows but as in any statistical description there are fluctuations on top of this average because if you are doing statistical mechanics what you usually ask in a canonical ensemble is what is the average value of energy that gives you the answer that comes from the micro canonical ensemble but then you ask yourself oh what is the fluctuations around the canonical value of an average value of energy those are buried in the difference fields the average field is really the right plus left in some heuristic sense in the fieldish average fluctuations are constrained by the right minus left and so you should write down a theory which has both of these things and moreover you can think of these guys as the goldstone modes breaking diffio right times diffio left down to the common diffio but in the low energy theory you are only sensitive to the averages everything that is not in the averages is buried in the fluctuations so long story short to write down the hydrodynamic effective field theory you cannot just stop here you have to take insight from this contour and then write down a theory which has all these pieces of data the trouble is if one just goes about it lively one is not going to match the fluctuations correctly because we saw there were microscopic constraints on precisely this sector this was the piece that was constrained by the topological invariance and so to go from the macroscopic theory you want to write down so the macroscopic theory should be constrained by our SK KMS algebra one can do this can I have two minutes and what I am going to just write down is the answer to this problem without telling you where it comes from and refer you to the literature of saying that where was this action write it here basically saying that you can write down a gauge sigma model this X field upgraded to the super field X we were talking about yesterday and the symmetry we are gauging is thermal difiumorphisms and one advantage of doing it this way is that not only do you get pretty much everything in this classification but you can also write down the dissipative action because in some rough sense in the same sense as Hiroshi was talking about in the previous lecture there is a sense in which the two copies roughly speaking take care of how one of them that is a very heuristic statement can explain that bit better but you can write down a Lagrangian theory which captures this structure so I will put up some notes and refer you to the literature on where this all is constructed but I hope I have given you a flavor for where what the constraints are in the microscopic theory and what is the at least what is the target goal for the macroscopic theory that you want to achieve and the link between I have sort of elated over but hopefully you have enough information at your disposal to sort of go ahead and learn that for yourself ok, let me stop here