 Yes, thank you very much. So I continue in some sense, also because Nicos talked in a way about viscosity, which can be understood as a dissipative phenomenon, but now I want to somehow make a bridge between different formalisms and try to explore how dissipation can actually be described in terms of the 1PI effective action and analytically continued 1PI effective action, which is the object that we study oftentimes with the Wetterich equation, for example. And so now, if you think about how dissipation, well, this one does not work. But I can switch probably like that, yes. So first of all, in quantum field theory, dissipation is a little bit of a tricky phenomenon because dissipation, by definition, is the generation of entropy. But on the other side, unitary time evolution, as you would expect it from most quantum field theories of interest, conserves entropy. So on the other side, that's just a little bit of a formal question in practice. Oftentimes, one is interested in a description that has only an incomplete amount of information. For example, one would like to describe a physics system in terms of certain expectation values of fundamental quantum fields, maybe. Maybe also of composite operators or some order parameters. And then in addition to that, there might be also a way to describe information that is not contained in that. For example, in terms of a thermal bars or so. And so that is a kind of description that cannot possibly capture all the information about a very detailed quantum state of the system. And thereby, it's rather clear that it does not capture the full amount of information. And thereby, also, some entropy generation can be present effectively. So now, as we have some examples in mind, how dissipation can come about, if you integrate out certain modes, maybe it's useful to look at these two particular examples. The first is if there's a situation with a decay. So let's say we want to formulate a field three for muons. And so we know that muons can decay into electrons and electron neutrinos and muon neutrinos. And of course, the full electrovec theory is a unitary quantum field theory. But on the other side, you may wish to integrate out the electron fields and the neutrinos for some reason and then only consider the effective field three that thereby arises for the muons. And now on that level, because of this decay that's possible, there is actually already a dissipative term in this effective field three for the muons. And that's something we would then have to deal with in such a formalism that I will now describe in more detail. Then another example is just electromagnetic field. And there you know that for very large field strengths above the swinger threshold, there's electron-positron pair production. And if you now integrate out again the electron-positron field, which you can do, then you have a field three for the electromagnetic field, which also has dissipative effects because of the possibility of electron-positron pair production. Now, one possibility to describe such situations is the so-called double-time pass formalism. So that's also called Swinger-Keldish formalism. That can be used in general out of equilibrium situations. And there's a lot of literature, of course, on that. It can be formulated. OK, thank you. Thank you. So in principle, one can treat arbitrary initial state density matrices in this formalism. In Brussels, it's mainly Gaussian initial states. And then it can be formulated in terms of two fields. One is the so-called quantum field. The other is the average field. And this formalism allows to treat also dissipation. And it's useful for many situations, also for classical field theories. As it was explained, for example, by Nikos, that formalism was essentially the classical limit of this Swinger-Keldish formalism, or also by Leoni. And many more people use such formalisms. Now, a little bit of a problem is to recover thermal quantum field 3 in this setup. In particular, if one is interested in non-perturbative aspects. And then another thing is that one has to double the number of fields. So the formalism is a little bit algebraically involved. So there is some interest to try to see whether one can also extend the conventional thermal quantum field 3 formalism to describe also dissipation. So now what I want to explore is now a kind of close to equilibrium formalism. Close to equilibrium meaning that I do not necessarily want to describe arbitrary far from equilibrium situations, but rather situations where one deviates from equilibrium, but in a sense that one can still access in a certain way by extending the equilibrium formalism. That will become more clear in due course. One example is viscous fluid dynamics, which is a local equilibrium approximation. And then there might be additional order parameter fields. So it's not only the fluid degrees of freedom. Now, usually this is discussed not in terms of effective actions, but in terms of phenomenological constitutive relations, or as a limit of kinetic theory, or also in the ADS-CFT cospons. So there are already some theoretical formalisms. But I will be interested in deriving a formalism directly based on quantum field theory concepts. And then use this approach of analytic continuation as an alternative to Schminger-Keldisch. And as I said, it's a direct generalization of the equilibrium formalism. Now, the first thing that I need is a setup that can describe local equilibrium states. Because the reason is that dissipation is somehow the transfer of energy and momentum from certain macroscopic degrees of freedom, say from an order parameter, to the thermal degrees of freedom, so to the fluid degrees of freedom. So even if one starts initially with a zero temperature situation after some time, due to this dissipation and the local thermalization, the temperature will rise. So it's then useful to put this setup immediately in such a context where the temperature depends on space and time, and also the fluid velocity. And it's actually, in particular, this combination of the fluid velocity and the temperature that will enter the formalism. It's also convenient to use immediately general coordinates and the metric of this form. In particular, if you think about the formalism of general relativity with an arbitrary coordinate system, then already to describe thermal equilibrium situations one needs a temperature that depends on space in general. Now global thermal equilibrium in this formalism corresponds then to this beta mu vector being a killing vector. So that means that this relation here is fulfilled. But then if this beta has another form, which is not necessarily a killing vector, then this describes close to equilibrium situations but not necessarily equilibrium situations. Now, you can directly compare the local density matrix with the translation operator, and then essentially go the standard steps that one does to introduce the Matsubara formalism, but now in a way that this beta depends on space and time. And in particular, for the time being, this configuration space is some hypersurface of space. You can consider a hypersurface of fixed time. And then on this hypersurface, there is a certain temperature and velocity field. And then one can work with fields that are periodic in an imaginary space direction just as a generalization of the Matsubara formalism in such a way. They have this periodicity where the plus sign is now for bosons and the minus sign is for fermions. And then one can define a partition function on such a hypersurface of space, which is then of this form. And it has an Euclidean quantum field theory in now this configuration in general time and space dependent temperature and velocity field. Now, if one assumes that such an approximate local equilibrium description is valid at all times, then one can actually shift this hypersurface in time. So it is actually such a description can be used at any point in space time. And here's now a kind of comparison between the global thermal equilibrium situation and now this local thermal equilibrium situation where now the Matsubara torus is space and time dependent. Now one can now go on and define an effective action as usual by taking the Legendre transform of this Schwinger functional Uw. And then it's a function of the expectation values corresponding to the derivatives of the Schwinger function. And then there's an Euclidean field equation of this form here, which resembles in some way the classical equation of motion already. But it's not quite yet what one can use. We still have to discuss the analytic continuation from this Euclidean configuration to a Minkowski space time. And that will be one of the main ingredients of this formalism today. So first of all, to understand a little bit more about analytic structure, let's go back to global thermal equilibrium so that this beta is of this particular form here. And then one can directly get propagators and inverse propagators in the Matsubara formalism in the standard way. And from the definition of effective action, well, the inverse propagator is inverse to the propagator. So that's clear. Now the propagator has a Kellen-Lehmann spectral representation. And from that one can read off all the interesting properties of the analytic structure. In particular, now here I've already shown the continuation to real frequency. So omega is now a frequency in the Minkowski direction. But you can actually take omega to be anywhere in the complex frequency plane. And now there's this spectral representation in terms of an integral over some variable set. And then the denominator is of this particular form here. And there's a spectral density which has certain properties, in particular for unitary, going to field threes and so on, it's positive. And but it can also, for example, in QCD for gluons, it can also be negative. But it's rather general that one has such a spectral representation. Now, as I said, one can analytically continue now from the Matsubara axis. So this axis here of imaginary frequencies and actually discrete imaginary frequencies to the whole complex plane. And in general, there is now a branch cut along the real frequency axis from this integral here. That can also be isolated poles, of course. But everything that is of interest in some sense, all the non-analyticities are on the real frequency axis. And one can obtain different propagators by evaluating now this object G at different points in the complex plane. In particular, the Matsubara propagator one obtains on this axis here. But then one can also get the retarded and the advanced propagators by going slightly above the real frequency axis or slightly below. And one can get the finite propagator by going below the real frequency axis for negative frequencies and above the real frequencies, frequency axis for positive frequencies. Now, this was now in global equilibrium. One can also now go to another coordinate system where the fluid velocity may not point in time direction and then instead of discussing everything in terms of the frequency, one would look at this particular combination which corresponds to the frequency in the fluid vest frame. That's just a simple step. But now the next step is a little bit more interesting and that is now to decompose the inverse propagator into two parts. So we said that the propagator can have only poles and branch cuts along the real frequency axis and that means that the inverse propagator or more precisely the eigenvalues of the inverse propagator can only have zero crossings or branch cuts along the real frequency axis which now in this general coordinate system corresponds to the real axis of this variable omega. And that means that one can decompose the inverse propagator in these two terms. So one is actually, both of this P1 and P2 are regular. When one crosses the real frequency axis but there's now this sine function here which mediates a discontinuity. So this SI is the signal of the imaginary part of omega so that precisely jumps from minus one to one if one crosses the real frequency axis and if one then chooses this P2 and P1 conveniently one can take them to be regular and then this way describe this inverse propagator function. Now that's one of the crucial steps here that this sign of the frequency appears here. Now that's in frequency space one can also extend this to position space and in particular what is the sign of the frequency becomes then or the sign of the imaginary part of the frequency becomes then the sign of the real part of a derivative in this direction of the fluid velocity. So that's a little bit maybe an abstract step so what has been a sign of a frequency becomes now an operator. Actually the sign of an operator. And that's a formal thing but one can actually work with this object here as I will try to convince you. And then this can also be represent you can see that it's now a derivative in the direction of the fluid velocity and that can also be written as a literative and that has some advantages. Now in the direction of the fluid velocity or in the direction of beta which is just parallel to that. And the sign operator that I can already say now will actually also appear in the analytically continued quantum effective action gamma. Now that is now a particular form that one can expect. In particular the quadratic part of this gamma can now be of such a form. So there's the field expectation value twice and then in between the inverse propagator which has now these two parts and one of them is multiplied by the sign operator. And now you can also do certain partial integrations but okay so that's in general the structure that somehow at a different point at one point or the other design operator will appear. Now that was for the quadratic part of the effective action. You can also extend this a little bit to non-linear terms or to higher order correlation functions. It's not that easy because there's no spectral representation but what one can do, one can do an inverse Haber-Zarhtonovich trick in some sense and do the discussion first in terms of auxiliary fields. There one understands the analytic structure and then integrate these auxiliary fields out again and thereby also understand the analytic structure of higher order correlation functions to some extent. I'm not sure whether everything is understood yet but to some extent it's possible and then also design function will appear there in different combinations. Now the question is whether one can get the equations of motion from that. In particular, and that's a non-trivial step. So in particular if one would start from the time-ordered effective action or this Feynman description with the I epsilon then one can also get a well-defined effective action but this now taking the functional derivative with respect to the expectation value you also get such an equation here but that is actually not a valid field equation. You could not use such an action and simply obtain an equation of motion that way. It would not be causal in general and it would also not be real meaning that if you have a real scalar field then this equation will tell you that it does not remain real in the time evolution. So it's not in agreement with the reality constraint. So that's nicely discussed in this book here so in general that's taken as an argument to go to the Schwinger-Keldisch formalism but I will now argue that there's an alternative within the analytically continued effective action which also gives a causal and real field equation and that is the following. So I take this analytically continued action I define something I call a retarded functional derivative and I will explain what that is and then there's indeed a field equation that is actually real and also causal. And so what I do is now I have this effective action that has the sine operator. One can take as usual the variational derivative with respect to all the fields and then still the sine operator appears and the sine operator appears either to the left or to the right of this variational derivative of this field variation and then one can choose the sines. If the sine operator is to the left of this field variation you have one has to choose a negative sine if the time evolution is in forward direction if it appears to the right then one has to choose a positive sine and then after that one can just proceed as usual and one obtains thereby a field equation that is actually causal. Now that's if the time evolves forward so that the entropy is produced if the time runs in forward direction in a situation where the time would be reversed one would have to choose the opposite sign. Now I will make this a little bit more concrete and I will also discuss examples but maybe before that I can already say that this indeed leads to a causal equation of motion that one can see by taking another functional derivative of this first so by this retarded functional derivative and then that gives simply a derivative of the source with respect to the expectation value. Now this can be inverted and the inverse of this right hand side is just the retarded correlation function and so then one obtains an equation that says that the field expectation value at a certain point in space time can only depend or can only be manipulated by changing the source at a point that is actually in the past of this point X or maybe on the past light cone of this point X. So one can then really see that this leads to a causal time evolution. Okay so now maybe it's time to do the first example. In particular the simplest example probably is the Demt harmonic oscillator. So that's harmonic oscillators are of course something that is rather simple and then but taking account of a damping term in an effective action is already non-trivial. So this is the equation of motion that's a standard harmonic oscillator these two terms and then the C mediates a damping term. Now what is the action for that? So of course you know the action of the harmonic oscillator but how can one take care of a damping term? Well naively you would think that maybe in frequency space you could write something like that here where this is now the so-called damping ratio but that doesn't work. In particular this combination of fields here is even with respect to omega and this would be an odd term so it would just simply cancel out or in the time representation this would be a total derivative so that is not something one can do but what one can do is one can now consider the inverse propagator to be of this form where there's now the sine function appearing so that there's actually now a branch cut in this inverse propagator along the real frequency axis and what have been the poles before at the two frequencies or plus and minus omega this has now been bordered into a branch cut and so now one can take the variational derivative either in frequency space or in time space with respect to the field which is in this case X and then one can somehow after taking this derivative one can now choose the sine in this way and that defines now this retarded function derivative and indeed by choosing the sines in this way one then obtains the equation of motion for the damped harmonic oscillator so that works in this sense. Now there's another example that's related and that is now for scalar field and here now the action is already in field theoretic language also I've put it into general coordinate system and now it depends also on this beta mu because there's in general now a terminal pass and so one possibility for such an action is to write it down in terms of derivative expansion and so there's an effective potential which depends now on this or in symmetric combination of fields and also on the temperature and then there are kinetic coefficients which also depends now on rho and T in general and then there is this term here and this term is now the one that mediates this discontinuity in the inverse propagator or that is responsible for dissipation and that so that's the term that is actually now odd here it has a first derivative acting on the field and it involves the sine operator SR and now one can take the variational derivative and work out all the different terms and then one can again choose the sine according to such a prescription here and one obtains then a field equation of this form which is what one is used to so that are the two terms from the kinetic term this is from the effective potential here and now this is this additional term that describes dissipation and so it's a kind of generalized Langgarden equation with an additional damping term now the one advantage of this of having an action instead of writing down immediately such an equation of motion is that the action has more information to it and now I wanted to explore this a little bit in particular there are now two important questions that come up so one has now a modified variational principle that also can take care of dissipation but the question is what happens to the dissipated energy and momentum from that somehow is lost by this field expectation values and what about other conserved quantum numbers if there's any additional symmetries and then also what about entropy production can we maybe say how much entropy is being produced so now first we have to look at the energy momentum tensor and the energy momentum tensor can also be obtained from this analytically continued effective action by a similar prescription so one takes now variational derivative with respect to the metric and so then this gives the expectation value of T menu and if you would now look at a situation where there are also gravitational terms then this would precisely give the Einstein's Einstein's field equation now one can now decompose actually this gamma into two parts one is a gamma R that's a reduced part or the regular part and then there's a dissipative term which has all the terms that have these discontinuities or which are written in terms of the sine operator and then also the energy momentum tensor has two parts one is the regular part and the other is the dissipative part now one can ask what is actually with general coordinate transformations and that's interesting because usually one would say that energy momentum conservation is a consequence of general coordinate invariance and now this is what I want to investigate now here in this formalism so general coordinate transformations infinitesimal ones can be written as a kind of gauge transformations of the metric and also of all the other fields in particular the metric transforms in this way and then the temperature is a vector so it transforms under general coordinate transformations according to the vector representation and then all the other fields can be transformed as well now in this case I look at the scalar field and the scalar field would transform in this way but you could also do this now for vector or also Spina fields and now this reduced action is actually invariant under general coordinate transformations if there are no anomalies which I assume here now from this one can obtain a relation and in particular now I first want to discuss this situation without any dissipation and then this reduced action does not depend on beta we can consider maybe just a conventional vacuum situation and then it depends only on the field expectation values and gminu and now it satisfies a field equation of this form so that it's already stationary with respect to the field variations and then you can also ask now for the variation of the metric and then this leads to the gauge variation leads then to this relation here and because epsilon is arbitrary one can read off that the energy momentum tensor has to be co-variantly conserved so it's a consequence of general coordinate invariance now in particular one has then really shown that the expectation value is conserved now let's go to a situation with dissipation now three points change actually one can still take the variational derivative of this reduced action and vary the fields vary the metric and vary the beta but now first of all there is additional field beta which you have to take care of that's not a dynamical field so it does not satisfy a field equation and also there is now, yeah, the variation with respect to phi is also a little bit different because the variation of the reduced action does not, it's not stationary there is an equation of motion satisfied by phi but for that one has to take also the dissipative terms into account so one can write this variation of the reduced action as some field m can also be written as minus the variation of the dissipative terms and so that defines a new field m and then also the energy momentum tensor is now not conserved because that's only the reduced part there's an addition also in the dissipative term so one can replace this covariant derivative by the covariant derivative of the with you of the dissipative term and then also beta does not drop out anymore so that defines again a new field k and so one obtains now from this relation here actually a non-trivial equation and in particular four additional equations because epsilon has four parameters where there's no epsilon on this slide but there are four different ways to do for independent ways to do a general coordinate transformation so this are four additional differential equation that actually can be written in this form and they can be understood to determine beta mu so the temperature and velocity field this formalism in principle can also be used in a situation where the fluid velocity and temperature are somehow given from some external constraints so maybe there's an external heat bars but if one wants to use it in a situation where this is not the case kind of autonomous situation then one can use this additional four equation to determine the time evolution of the fluid velocity and of the temperature now another interesting thing is end-to-be production so one can actually take but one can project these four equations with one can contract them with beta and then this is one one is left with only one equation and now I claim that this equation here actually describes the production of entropy so from the four equations we had before I now contract it with beta and then one can actually see that what is here on the right hand side is something like an entropy current in particular if the reduced action has only the effective potential in it then one can take the derivative with respect to beta mu keeping the fields or keeping actually the metric fixed and then this gives precisely s times fluid velocity and that's precisely the entropy current so the right hand side here is the divergence of the entropy current and I believe this can actually be extended to many situations sometimes one may have to change a little bit the definition of the entropy current but I think that such a relation can actually be written in many situations and so what one obtains is now that the divergence of the entropy current is given by this left hand side and so that means that also the left hand side has to be positive by a local form of the second law of thermodynamics entropy generation must be positive and so the left hand side here must actually also be positive and so it is actually the entropy generation by the processes concerning the energy momentum tensor that's in general shear and bulk dissipation but then also the field equation for phi if there's a dissipative term can actually produce entropy and I will describe examples for that as well so here's now actually an example so the action that we had previously for the scalar field and now one can obtain the energy momentum tensor it's now of this form here and it generalizes directly the energy momentum tensor of a scalar field as you are used to it so these are these terms here and also Jimmy Newell times these terms but at the same time it's also a generalization of the energy momentum tensor of an ideal fluid namely if you would drop this set term here then everything that would be left is just a fluid velocity times certain derivatives of the fluid of the effective potential as a function of temperature and that can actually be expressed in terms of the energy momentum tensor of an ideal fluid due to this relation here so the time derivative of u gives just the temperature derivative of u gives the entropy density and then this is multiplied by t and that is just epsilon plus p and so then this is also the energy momentum tensor of an ideal fluid now the general covariance now leads to these four differential equations and they can be understood as differential equations that determine the time evolution of the fluid velocity and the temperature field now what is now the entropy current so one can now do this manipulations as before and one can get an expression for the entropy current by taking a functional derivative with respect to the this temperature vector and that's what one obtains for the entropy current and so the entropy density would now be the time the temperature derivative of the effective potential and also of this kinetic term if it depends on temperature and then the fact that entropy generation has to be positive semi-definite gives a relation for C so in this effective action the C has to be positive or zero in order for entropy to be entropy production to be positive and then one can for example consider a situation where this field phi is spatially homogeneous and only varies in time and then so it has certain oscillations and then this can by this term C can produce entropy and for example you can consider the phase of free heating after inflation in the universe there's also a scalar field that starts to oscillate and then by such a term C as it can appear in this analytically continued effective action the entropy can actually be produced okay so now that was one example the other example is actually even simpler that's just an ideal fluid and so there the effective action would be of this form it does not depend on any fields it's just also for an ideal fluid there are no dissipative terms so it's just an integral over the effective action as a function of temperature where temperature is now related to the norm of this beta vector and then the variation with respect to the metric at fixed beta mu actually leads to this form of T mu nu and that's precisely the form of an ideal fluid and so then the general covariance leads to two equations which are just the Euler equation and the energy conservation equation now this can now be extended and one can also take now dissipative terms into account and in particular there are these two dissipative terms that can be present in such an effective action which now involves again this sign operator here now acting here on the metric on the terms that are here on the right hand side and then one can go through the formalism again and obtain by this variational derivative with this certain sign choices one can now obtain the energy momentum tensor of a dissipative fluid which has now also the shear and bulk viscosity terms in it and then this indeed then also produces entropy one can see that the entropy production is given by this relation here so that shear and bulk viscosity have to be positive in order to fulfill the second law okay that's where I wanted to go so there's now this variational principle for theories with dissipation based on analytic continuation and it one needs a local equilibrium setup which somehow describes a kind of generalized chips ensemble with a temperature and fluid velocity that depends on space and time and in addition to that they are also then the order parameter field so it's not really a complete equilibrium situation but a local approximate equilibrium and it works for close to equilibrium situation for example fluid dynamics coupled to additional fields and but I believe it's also rather general formalism although not as general as the full Schwinger-Kelch formalism so general covariance and energy momentum conservation lead to this equation for the fluid velocity and the temperature and also one can investigate entropy production and then the local form of the second law can be implemented or can be let's say investigated at the level of the effective action and then of course an interesting very interesting question that I did not cover is how one can derive or how one can obtain this effective action from starting from a microscopic action and yeah that's again another issue that one can also use flow equation to some extent but I believe there's more to come on that okay thank you very much