Lectures on open quantum systems given by Sebastian Lille that you've got to be introduced to already yesterday. So please Okay, thank you very much. Yeah, welcome to the second lecture. I mean as yesterday I remind you that I mean asking questions is really really very much desired especially in view of of slides that tend to go too fast. So please always interrupt me specifically today's lecture will start with a few I mean technical wrap-ups for this Keldish path integral and then we'll I want to put forward some applications that illustrate how the microscopic breaking of equilibrium conditions morph to large-scale observable phenomena and a nice point about this is I mean I can stop essentially wherever in the middle and we still have some at least if I don't mess up totally we still have some result and I can cut parts of the second lecture and in the third lecture we'll then apply this Keldish formalism to this measurement induced phase transition problem. So please ask questions. So just as a reminder for yesterday we started introducing the Lindblad equation which you can write compactly in this form you have a the system and density matrix time evolution is generated by Heisenberg von Neumann commutator and then some dissipator and if you write out all these terms you recognize there is a both a left and right action onto the density matrix and that is called this this whole Lindblad operator which is a total thing here is called a super operator which still acts linearly onto the density matrix so you can find in formally an Exponentiated solution so that starts from some density matrix That's again here. Yeah, and then evolves this density matrix on two times strings Simply because we need to transform every of these indices of the matrix over time So as in unitary evolution of a density matrix this structure of two times I mean sorting out of a density matrix generalizes to the open system evolution as you can see here left and right action and what we then did was this Keldish path integral construction where the result is Shown again here just as a reminder and in particular I want to remind you of this fact that the action the left or right action on the density matrix is Remembered in the Keldish path integral by an additional index Yeah, which labels the contour which labels the side on which the original operator was acting on to the density matrix And you see here really nicely this structure that is written up here in the operatorial formalism really pops up again in this Lindblad action if you like That's written down here. Yeah, precisely with the right pre-factors that guarantee for example the probability conservation Okay, and then we started with a few structural developments first of all we discussed the issue of probability conservation in this field theory and Although this sounds like very technical. I mean of course probability conservation or unitarity of quantum mechanics Is really a very important symmetry of any physical evolution? And we use that here in the context later to motivate the operation of the Keldish rotation Which? Starts from the observation that the Keldish action Unlike many other actions that you encounter for example for equilibrium problems has the special property that it vanishes When we evaluate it on a configuration where we pin the configuration on the plus and on the minus contour to be the same so then this action vanishes and that motivated actually to Introduce other coordinates to parameterize this action not in this plus and minus contour basis but to switch to a basis of center of mass and Relative coordinates called Keldish rotation this operation and the fields that are coming out of this Rotation here are called classical and quantum fields They have the property that then the probability conservation property takes this need form here that if we set one field to zero namely the difference between plus and minus contour then the action vanishes and The interpretation of this field was that the classical field here can Condense it can acquire a finite expectation value a field expectation value So it can describe a spontaneous surgery breaking phenomenon while by construction since the average of the since the average of the Field inserted on the plus contour and on the minus contour yields identical averages the quantum field can never acquire a finite expectation value and that Motivates this denomination quantum field a field that cannot condense that can just fluctuate Good So this is also a good preparation here to now step by step Include or try to understand or get a feeling for this Keldish action by by step by step including fluctuations So one first thing is if you want to include fluctuations Maybe a good idea is to look at the limit where fluctuations are not very prominent Yeah, so and then we know what we are actually expanding about later on and this is what I term here Deterministic limit of the Keldish functional integral. So we notice that Probability conservation is the property that if we take the field actually the field pi q exactly to zero Yeah, so then this is probability conservation and now so the zero order Expansion in the quantum field is just probability conservation Now I want to make the point that the first order expansion of the Keldish action in terms of the quantum field here and Leads to the deterministic limit Yeah, so if we expand our action to first order in this quantum field This will describe the physics of a single field configuration in other words and that the deterministic limit of this problem Now let's think about which is the circumstance under which this could occur So what could be an ordering principle that really allows us to neglect? Everything which is not linear in this quantum field And this is precisely the Situation that you encounter when there is a spontaneous symmetry breaking in the problem in this case Spontaneous symmetry breaking means that one momentum mode There's a winner takes it all phenomenon one momentum mode gets Microscopically occupied so it scales with the square root of the number of system constituents so that's the phenomenon of Spontaneous symmetry breaking one mode really gets macroscopically occupied and in real space This means that the field actually because of Fourier transformation property it scales like n to the zero Conversely the quantum field here it can never acquire any Macroscopic expectation value it can never scale with system size It will generically in terms of a scaling argument scale with the zero power of n and therefore if you go to real space Now you see here a difference in scaling of the quantum Versus the classical field if we assume there's a condensation phenomenon going on Now therefore we can then expand we have now found a criterion or a principle that really allows us to Expand to leading order in the quantum field because it scales Subleadingly compared to this classical field and the result of this is then an action Yeah, so I just take this action written in these coordinates classical and quantum field and I expand it to first order in this quantum field This will give me the leading contribution to the action Yeah, and then we have actually an integral of the structure that you recognize a Fourier Expansion structure we have something like that. I mean Fourier would be this x equals a representation of a Delta function and that's precisely the structure that we have up here this P corresponds to phi quantum. This corresponds to d s d phi classical And therefore You can see when I really drop every but this linear term in the quantum field I'm representing here essentially a delta function for the classical equation of motion The equation of motion is the or the classical action principle so the classical action here is minimized and that Minimize means that it takes value 0 and that's the only point and that's the only field configuration that will contribute to this whole path integral here So this is really how Summation over all possible configuration that is prescribed by this full functional integral here how this collapses to the only contribution coming from the deterministic path From the one configuration that minimizes the classical action the classical Health action so so no fluctuation included. Yes, so that classical in the sense of no fluctuation Contribute so that is that is Yeah, I'm not assuming any specific form of the action here Yeah, so we are looking at a very general theory and I'm essentially Deriving the classical action principle if you like so and it's only valid So when we have an argument why we could truncate the full action to linear order in this quantum field And that the intuition is really when we have a Condensation phenomenon So this is really then one precisely this condensed field configuration gives the dominant contribution to the path integral It's it's it's less than the subtle point approximate subtle point usually means you look at quadratic fluctuations This is even less. Yeah, so I'm of course, we get to that But I think it's always good yet to have a kind of intuition for a starting point and then So now we are looking at this black path and then we'll explore also situations where other contributions Contribute so we'll build around that limit. So there were two questions. Yeah, and so Physically, it's really this this scenario Yeah, so so we need we to in order to justify the operation of just expanding to linear order in the quantum field We need an ordering principle So imagine the classical field would also like without condensation phenomenon It would also scale like and and to the zero Sorry like n to the minus one half if there's no winner takes it all effect No condensation around then both these fields scale the same and then you simply do not have the situation Where one special field configuration dominates the whole summation prescribed by the functional integral? That's the point. Yeah, so really the condensation phenomenon Physically means winner takes it all one field configuration is Dominating the whole integral And that is really so the physics is a macroscopic occupation of single modes Oh the condensation Both a condensation what we started yesterday with this workhorse model and will great detail come back to that That is a both a condensation a magnetization Transition from a para magnet into a ferro magnet. That's typically when you have extensive order parameters That's the situation of One configuration dominates Any ordering? Yeah, right. So classical. Yeah fluctuation less Yeah, so there's just one path one one configuration dominating the story. There was another one the forgetting about the initial condition is something that Sorry in the initial density matrix Technically it comes from taking the time of the initial time to minus infinity physically The the point is that I mean dissipation is when dissipation is in the game Yeah, indeed. I mean you expect the system go to go to a stationary states And we want to describe here the physics in stationary state, which is overriding any initial condition Right, so that's a and so to say that the dissipative Tendent or couplings in the problem determine the the form of the stationary state. That's one thing Technically it's actually not so trivial But what you can do if you want to work hard and one should do this of course once you can take a damp harmonic oscillator and formulate really the initial value As in this Keldish formalism and then you can see really that if you take this time to minus infinity limit the Information on this initial condition dies out exponentially fast But the physics is really this a damp problem finds a stationary state that is independent of the initial condition I mean that that would be the as a the situation Very interesting situations. Yeah of MBL for example I think when when you add openness So I don't know of good examples. So that would I mean except you have something that is known as Zero modes of this lint blood operator If you have a zero mode meaning that there is a state in the system that is really not touched by this by this dissipation that does not decay and Imagine that there's not only a single zero mode But the whole subspace of zero modes then this space will subspace will remember the initial condition And that's actually interesting effects like you can stabilize by dissipation You can stabilize topological states of matter and they they typically are characterized by edge mode subspaces Which are higher dimensional higher than one and which remember initial conditions other questions good, okay, so We have now we appreciate now this deterministic limit Yeah, and we have really shown that I mean there is a technical way to Convince ourselves that kind of the there is a classical action principle Yeah, that that gives us a dominant configuration if we assume physically that there is this condensation phenomenon Okay, and that's actually really that that comes back to your question We have it basically from a completely different perspective yesterday gotten the exact same result for a specific Theory, namely for this workhorse lint blood 5-4 theory where we've seen really that I mean where we've done a lot of Heavy approximations and we've seen in this way we can also model this condensation phenomenon. Yeah, so this total factorization of the density matrix in space and then That mean field theory essentially, okay, so let me summarize that I said it's many times there, but here is still a crisp summary Deterministic limit of this Keldish functional integral is dominated by a single field configuration Which minimizes and maybe the better technical term is not the classical action, but the bear Keldish action So the thing that's really up there in the exponent Sorry for the notation mismatch here This should be just s yeah, and it the applications of this scenario the physical Applications are both a condensation as we just said or also I mean there's an interesting connection to this field of non-permission physics, yeah, which basically looks at Fluck at Disappear at non-hermition Hamiltonian So Hamiltonians which have a Hermitian and an anti Hermitian part both of them occur Yeah, so that's just and and and this finds a systematic Explanation in this framework in the sense of you could regard this as the Deterministic limit of a Keldish action problem. Yeah, and Then you can ask so why could this be a good idea and the answer is here on the on the blackboard Yeah, so this is a good idea. For example, if you deal with classical wave optics, yeah, so then you will Be able to describe the system and of systematically in such a deterministic limit of of the Keldish path integral And maybe I give you another piece of intuition. Yeah, so I coming back to this statement. I made yesterday and Another way of looking at that is the system has a collective degree of freedom So and this is the circumstance when there's a collective degree of freedom you can neglect Fluctuations and examples come from our classical world like like the damned harmonic oscillator Yeah, so although formally this is not probability conserving this dynamics. So you could add some noise to to this damn harmonic oscillator. It's just not necessary because this Coordinate that oscillates is a highly collective coordinate made of many Microscopic particles and the energy scale on which this oscillates. It's much much bigger than kb times the bolt Boltzmann constant times temperature. Yeah, so that fluctuations in this problem are really negligible So that is really applications if you like of the deterministic limit of this Keldish functional integral. Yeah Yes, so if you want yeah I can give you the recipe Yeah, so if if I give you a lindblad equation and you want to see what is the non-Hermitian Hamiltonian and when first and One when is it justified? So these these two points are important to answer.  I do this Keldish rotation. I compute this Well, first of all, I would have to come I compute this this Keldish action I do this Keldish rotation and then I ask myself Well in this problem, is there actually a collective variable in place? Yeah, for example, is there a Bose a condensation going on or I mean is there a photon condensation going on classical Optics is photon condensation Yeah, and then you have a good reason to neglect the noise Yeah, and you could and then your non-Hermitian Hamiltonian is really basically Standing here. Yeah, so it is you get the time evolution for the field expectation value for the field Yeah, and on the right-hand side of this equation stands your non-Hermitian Hamilton. I mean here We actually didn't set it to zero. We actually integrated it out exactly So it was not a I mean the approximation was I can linearize the action in the quantum field And then I recognize. Oh, there's an integral prescribed From the Keldish path integral over the quantum field But since it only occurs linearly. I can just do this integration by using the identity of I mean the functional Delta representation for a representation of a delta function and this delta function pins all the Configurations of s phi c to zero, which is the classical equation of motion Actually really an exact treatment of phi c if the approximation of phi q in Linear is justified and there we need some physics to justify that. Okay more good so Then we come a bit to fluctuation Yeah, so classical Field theories so you could can look generally of a field theories in this way It doesn't matter if Keldish or some other field theory, but classical Physics classical field theories. They are really a single field configuration Determines everything so that's really a deterministic theory classical theories are deterministic But quantum and statistical field theories So they have this property Keldish or not there so that you really sum over all accessible all possible Field configurations beyond this classical path. So they are in this sense probabilistic. Yeah, so there's a Kind of probability weight associated to every of these other than field configurations Now specifically in the Keldish we've now seen probability conservation zero order and quantum field Deterministic limit first order and quantum field and now what about when we want to quantify deviations away from this deterministic limit? Yeah, and the kind of Correlations that measure that the departure from the deterministic limit. They come in two flavors One of them because now we have these two field variables if we want to look at it purely mathematically We have what is called correlation functions They measure the strength the picture is they measure the strengths of fluctuations away from the deterministic Configuration and they are sort of encoded in averages of this type here in the deterministic limit Yeah, as we've seen yeah, so every information is just in the field expectation value So then in the deterministic limit this factorizes and whenever you find a deviation from this factorization You can know you you know that must be the effect of fluctuations away from this deterministic limit the other class of correlation functions is What is in the context of the Keldish path integral known as a response? Function yeah, so this Describes the impact. That's really a structurally different question. You're asking to the system here. You're asking How strong is the impact of an external perturbation? Yeah, so this is a system intrinsic property here measures how strong are fluctuations away from the deterministic limit This thing here will give you an answer. I put an external probe field and then I want to know how the system reacts That's the response function Okay, and now I give you the prescription Yeah, how you can compute these two objects and hopefully along the way still get a little bit more intuition about them So to this end what you do you introduce source terms as Maybe you're familiar from that with in statistical mechanics. You introduce source terms With the idea that you can use these source terms here to generate Correlation functions you if you take the Variational derivative of this object here or maybe written in this Keldish basis here You can see I get the phi c down downstairs in the expectation value by just taking a derivative With respect to J quantum star in this case so this would give me generate me and the Field expectation value and then we can go do more and we can take more derivatives For example, we get what I introduced as the single particle response function by taking another derivative however now with respect to this JC field In the in this basis and you can see uh-huh indeed This is the structure. I was announcing on the previous slide and you also have hopefully a little bit of intuition now Yeah, what what this correlation this abstract a correlation function here means well It just means I have this expectation value and now I wiggle a little bit with an external field JC and To linear order. Yeah, I get this correlation function So this is linear response theory in this Keldish formalism. You can think about this extremely physically Yeah, I Think of a cavity model that we looked at yesterday. Yeah, so just a damned harmonic oscillator And so the Hamiltonian is omega naught a dagger a and now I add something like a source term J times m a plus a dagger or a Something like that Yeah Then I do a Keldish is then I transfer this into Keldish action Yeah, and I get in terms of the Keldish action for this term here. I get integral over t j plus a That's mission conjugate minus j minus a minus And now I say, okay, this should be really this J should be an external the physical source Yeah, so it should not carry a Keldish index here and now you see so this is let me This is the classical JC and now you see. Okay, this is indeed JC a Plus minus a minus Yeah, so in classical external source a real field that I impose onto the system It couples precisely to this a Quantum field in the Keldish path integral. So that is the intuition a real physical Classical external source it couples right away to this quantum field and that that's exactly what stands up here Without having any physical picture. We can go to this, but that's the physical meaning of this of this responses Okay, and then I mean as a as one more piece of information Yeah, you can in a once and for all a little bit tedious and technical way Which I don't do in the lectures here You can also relate these correlators here of the Keldish path integral to actually really Correlators in the operator formalism. Yeah, so that's often times useful and practical calculations here to know Okay, what does this object actually stand for in the operator formalism? There must be a one-to-one translation if you do everything in the correct way and then as a piece of information that I give you Yes, so it's precisely the commutator of fields for in the bosonic field case Now let's come to this other class of correlators that that I was Specifying namely those with a phi C phi C star And so here I just go kind of by by the formalism here how to generate this correlation function here Well, I have to take the twice the derivative with respect to these quantum fields and as a piece of information Yeah, so the best intuition I can give you for that is really that this measures deviations from the deterministic limit And so we just try to find a way how to generate that in this formulation here And then relating it to the operator formalism It is not the commutator that it represents It's the anti commutator that it represents and if you evaluate the anti commutator at equal times Then this describes precisely how strongly states are occupied. So that's another interpretation of this correlation function of the equal time correlation function It is how states are occupied two times the occupation number plus one which comes from I mean using commutation relations Yes, J is an external source here. Yeah Yeah, that's what I tried here. Yeah, so maybe maybe I'm in between doing it too long and What I tried here, yes, think of this is a term. Yeah, so you can imagine I really start wiggling I have a driving of this harmonic oscillator with an external source that I call J Often it's also called omega if this is really a physical classical field Yeah, it will not undergo this construction on the plus and minus contour, of course Because it's a parameter for this theory and not a fluctuating and not an operator the operators They get an index because I have to represent them in terms of current state But external fields are parameters. They don't get an index and I mean if you then say, okay I essentially I'm only going this term if I pull this through in the Keldish construction Then there is never an index Because it's just an external parameter But and but and this physical field here it couples to the difference of operators then and that gives us then an interpretation of now I think too close that gives us an interpretation of This more more formal construction here where I just To be able to to get all fields down by taking derivatives gives us an interpretation of the classical component to this And this is really a physical field that I can wiggle and the j quantum is an object that I just need in order for it To be able say to to represent this correlator here as a functional variation of the relative. Okay, that's the point It's a bit technical this part of but it's also an important Important stuff and I think if you miss it out, then you can't really understand the sequel. Yeah Maybe wait a bit we come to the structure that that emerges in the end Yeah, so then then we can get back to this question Okay, right here we go. Yeah, so you assemble then yeah So we want to in in field theory We want then to build a green's function and because of this to indices C and Q Yeah, we should get a two by two a green's function structure Yeah, and the structure that we find for this problem is okay here is this G Keldish defined in this way then we have the retarded greens function that we just put put it here in the upper corner and Then there is the Hermitian conjugate to that Yeah, which comes when you exchange jc to jc star and jq to star to jq So then you get what is known as the advanced greens function if you translate back to the operator language It has support on the other sign of the theta function So these are relations that you can then pretty neatly see I mean, I don't want to prove them all but just give us a piece of information So we have all details if you like So let me now come to this structure. Yeah, you've noticed some people have already noticed In advance. Yeah, so that there's a zero down there. Yeah, so which relates to probability conservation So let's see again. Yeah, so let's see how this comes about and To this end. Yeah, I don't develop again the full theory that would be stretching it But we can go here via an example Yeah, we get just go via a simple example and you take my word that this generalizes to full full interacting theories in a special Continuum, but I don't prove this here. Let let's just see with an example It gives you the right picture. So here what I did is I take this damn Harmonic oscillator master equation that we that we really carefully translated to Kelly's yesterday and then I Rided in these classical and quantum fields. I write it out You see here. Oh, there is this zero and this zero is Precisely necessary, you know as a consequence of probability consequence. I mean comes out of from the from the calculation But it also must be like that because I have this probability conservation is the property that the action vanishes when I set the quantum field to zero and that's why there has to be a zero in this corner because this zero pre-multiplied or multiplies phi quantum of t times zero times if you stretch out this bilinear it multiply it comes it is this Contraction of the matrix. Yeah, so there must be a zero because otherwise if there were anything non-zero Yeah, so then I would immediately violate this property Okay That's how it is. Yeah, and this is much more general than in the Then in the this simple problem because of this general property and then we can go to Fourier space And the claim is now that this object that stands here these There's a lot of notation now the claim is that this thing that stands here is actually the inverse of the grains function Who is familiar with such a statement? Then I know how quickly I can go there. Who's familiar? Okay, I go mid-speed Then so I think yeah, I write down a formula Yeah, so that reminds you of this here is the inverse of a matrix and this inverse is the grains function of the problem, so Let me do this in a kind of matrix notation. Yeah, so s I write it now in general some a b Phi a G minus 1 phi b The start for a complex action doesn't matter a kelv dish or not. Yeah, and then we have the Gaussian integral maybe minus i times all the i's are now Important sum a So this is I mean a matrix notation for the action in the presence of Sources, so this a index is time continuous time index and kelv dish index Quantum and classical then I'm representing here and the result of this integration normalization times e to the minus G a b so the inverse of this matrix here with n The normalization given by the determinant of g minus 1 Okay, so that is the that is how we can convince ourselves That indeed I mean this object here is the greens function because we can I mean for the simple Harmonic theory here. We just do this integral here. That's precisely what I just did what I just did so this average here This quadratic action standing here, and if I do this integral Then I get here the inverse of this g minus 1 matrix and then I can argue Okay, this is our definition of the greens function in terms of variations of the partition function with respect to this external sources And on the other hand I so I can act with these derivatives on this formula Or I can act with these derivatives on this formula after I've done the integration and then I get Then this g function comes down. So this tells me indeed this relation this thing here is Precisely the inverse is the greens function and the inverse of the greens function occurs here as the action kernel So summary of this We write the inverse greens function. We write it We write this kernel like this With this it's totally random Doubleized again, we write the inverse greens function in this parameterization there with this P APRPK and the zero representing probability conservation and then we invert this object Yeah, just do it mathematically. We see how the zero that pops up here Shows up down here. Yeah, and this precisely tells us now the interpretation of this zero so the second variation with respect to the Classical field here That is probability conservation again that implies this and again. I say this is I mean now specifically done explicitly for a quadratic problem, but it's it generalizes to interacting theories this structure Probability conservation implies here is a zero Okay, good. So let's go quickly with an example and then we are through with the with the technical and developments Namely, I mean if you want for example, what is really a concrete example for a response function in the real-time domain? well, then I mean we can compute the the Time Fourier transform of the retarded greens function in the frequency domain and we find this structure here as we expect Yeah for a damp harmonic oscillator there's a decay with rate kappa and on top of it some Oscillations with frequency omega naught that shows up here in the oscillator the imaginary part of the frequency Domain a greens function is known as the spectral density for the problem and for this damp problem The spectral density is just a Lorentzian function There's also something that you might recognize from other formulations of physical problems And when we come now to an example for the correlation functions Yeah, and we specifically look at the correlation function that equal time I was saying this gives us the occupation of this mode. So let's do it for This damp cavity. What's your expectation? What should be the mode occupation for this dump cavity? How many particles are there in the cavity in the infinite time limit? Particle density there the occupation density expectations. I guess there should not be much left. Yeah, if I just have a decaying cavity We expect certainly that there's no Density left and indeed this comes out of this calculation Yeah, so here we understand that this equal time Keldrich greens function is two times the occupation number plus one And if you go through the calculation and just do this integral here, you find this is indeed one So this means that the end the physical expectation value is really zero totally in line with expectations. So summary The correlation functions encoded in G. Keldrich here They give us statistical properties how modes are occupied and how strong are deviations from the deterministic Configuration and the response here gives us what is also known as spectral properties It tells us how so to say a system response when we hit it So and then you will see some oscillations and these oscillation damp out. So that's interpretation of this quantity Okay Good. So now here you see how slow we are, but that's totally fine. We can smoothly go To the to the next to the next topic and now look at in this framework That we just acquired now at a real many-body problem where we do again this work horse Lindbladian that I already explained yesterday So it's a many-body system with a Hamiltonian with kinetic energy and with Interactions and we have then in the dissipative sector single particle pumping single particle losses and two-body losses Okay, let's now look at the Keldrich action for this precise problem doing all the steps Somehow it's really tedious, but this is the result. It's also very straightforward tedious, but really straightforward This is the structure of the Keldrich action in the after Keldrich rotation that we get out here You can recognize indeed the structure that I was advertising for the dam harmonic oscillator With the difference so that in the single particle sector here now We have spatial degrees of freedom not only a temporal degree of freedom But we also have spatial degrees of freedom which show up then as so the kinetic energy for example as this momentum square over to We also see yeah, so that There is decay parts in the single particle sector. So this is the single particle loss rate This is the single particle pump single body pumping rate And then on this a Keldrich component here We have also these two scales showing up with the interesting point So that in this PR Keldrich Green's function the difference of loss and pumping occurs and in this pk sector just as an observation the sum of them occurs and So let's get a bit into what this physically means I want to do two things now I want to use actually this asymmetry that occurs here to do simplifications in the limit yeah that this Lost minus pumping parameter here becomes very small So this means that we pump the system as strongly almost as strongly as we lose particles And this intuitively gives you the gives you an instability of the system When you pump harder than you lose then in the end you would pump in as many particles as you like But physically remember we have the two-body loss and the problem So this will give us a stable stable Situation where the the pumping this over pumping of the system is actually compensated by two-body losses and gives us a Stable state of matter, but upshot there whenever I make this parameter small I'm tuning myself close to a phase transition close to an instability to a to this Condensation that we already started yesterday So and and this mathematically this will allow us to take a limit of the Keldish path integral Which is more intelligent than the deterministic limit It's more than that but still doesn't have the complexity of the full Lindblad equation and it's a Very close to a critical point. This is very often a very good idea in generic situations is a good idea Yeah, there's a massive meaning Yeah, so if the action has to vanish When fight quantum is here, I can never write a term which is purely so it's this relation that stands up there Very good observation and it precisely this there cannot be a term which only has classical fields Then we would violate probability conservation. Yes, so it's it's a very deep property that that Surfaces everywhere in this theory, you're totally right more questions Very good that you're following so on this level indeed for God totally forgot Good that you're reminding me and totally forgot to mention. Yeah, so that our non linearities Yeah, so they are also now processed in this Keldish rotation. For example, this parameter This was in the microscopic model the elastic collisions And the kappa was the two particle losses and you can see here this lumped up parameter So the elastic collisions they come up in this combinations and the Two-body losses they come with even and with odd instances of the quantum But everything is linear at least linear in the quantum field. Otherwise, we probably violate probability conservation So this is a fantastic sanity check To verify s of phi quantum equals zero Okay, let's take this limit Yeah, and actually we'll even be able to connect it back to this deterministic limit But as an excursion or to motivate that I would like to introduce a physical problem Which is borrowed from condense matter physics so called exit on polarity on condensates people describe these in terms of What is known as launch of my equation just a second to go and then we'll connect our Semiclassical limit of the Keldish path integral to precisely this launch of my equation that gives us a picture on how this Relates and also gives us a prescription Yeah, if you if I give you a Keldish integral problem and you want to connect it to launch of one then this is the path That we are now developing Okay exit on polarity tons. These are semi conductor hetero structures So you have here three layers of semi conductor and these green ones. They are extremely highly reflective So they form like a cavity where light can then Bounce back and forth and this light can actually be coupled to the exit on degrees of freedom These are particle hole excitations in a semi conductor They're just called exit tons here and let me look here at the uncoupled scenario We're just look at the dispersion relation for the photons that are confined into this yellow plane This will be a quadratic dispersion. Of course photons are relativistic linear dispersion But due to the confinement in the z direction up here. They they become non relativistic. They get a huge gap So they behave like non relativistic particles in this plane if you like and the exit tons on the scale of the curvature of the foot Photons they they are absolutely flat Now as I said, you can couple these guys and when you couple them Yeah, you get actually the the effective so it's really diagonalizing a quadratic form We have coupling on the off diagonal and then you get actually this gray lines here as the real Eigen modes of this system here and then you can pump One of the Eigen modes You just come in with a laser and make it put it in resonance with this upper branch here upper so called polaritons You know this hybrid particles of exit tons and and photons polaritons you pump the upper branch This is unstable. It's a dirty piece of condensed matter and these modes will decay and in particular They will feed the lower polariton branch in an incoherent way upshot We cannot go into all details It will give us an in incoherent a single particle pumping of the lower polariton degrees of freedom here So this is this process Of course, these lower polaritons are also not totally stable Exitations of matter because these mirrors of the cavity here. They are again a bit lossy So you will lose exit tons and you will not lose them a single exit. Sorry polaritons. You will also lose them in pairs So this is then what these people describe this problem typically in terms of this stochastic Driven-dissipative cross-pedevsky equation So here they study the amplitude evolution of these lower polariton degrees of freedom And it is there is some propagation of these modes So that's this curvature of this lower polariton branch here gives you an effective mass here Then there is pump and loss processes as I was motivating they come up in this fashion here There's elastic collisions between this exit this polaritons and there's also these two body losses that being motivated phenomenologically If you want to structure and there's of course this noise If you want to structure this problem a little bit you can recognize here a structure that we also found in the limblad equation There is some coherent evolution Of this field amplitude here. There's some dissipative evolution of this field amplitude And now the element. There's also this element of noise a random force that acts on the system and This has precisely this idea that the system is now allowed to explore configurations beyond a single one Imagine I put off this noise So then I have here really a deterministic equation Which has both Reversible dynamics and irreversible dissipative dynamics then we are in this limit of non emission Systems if you like But now we have a random force that attacks the system all the time from the outside And this kicks the configuration around in possible configuration space So there must be some really clear relation Between this path integral way of thinking about configurations summing over many configurations and this Stochastic concept where we have introduced something explicit that kicks around the field configuration And in this way allows to explore configurations of this deterministic path So now I want to sharpen this precise connection as a special limit of the keldish path integral this equation should come out Let's see how this works None of them so the the the Polaritons, so so I mean So it's pretty indirect. So so the polaritons you can formulate a Hamiltonian for excitons a Hamiltonian for photons You diagonalize this Hamiltonian you get the polaritons and you get also polariton elastic interactions And then there is dirt in this conence matter piece That allows the that or I mean really very concretely So these mirrors here are lossy. So the photon has not only a Hamiltonian. It also has a decay So you would have to start from a limb blood Where you have a Hamiltonian describing the photons and the decay of the photons and that is all I mean in this phenomenological way priced in Conence matter way, you know, okay, so again, I mean the physics of these systems here I mean bozer condensation has been observed in such systems Yeah, so and they obey and and you can see this bozer condensation by by dropping everything that is complicated in this problem Yeah, so no spatial dependence. Let's look for a stationary state and let's forget about the noise And then again, we come to this intuition. I want to emphasize it again. It's a little repetitive Yeah, but you see when That's basically the equation we wrote already yesterday as this mean field limit of the lint blood equation Yeah, so it's all very strongly related And you see here or the solution of this simple Algebraic equation gives you when the pump exceeds the loss. Well, then you get a condensation going on Okay, so how does this relate to lint blood? That was we want to explore now And so now we go back to this more technical discussion And try to simplify this action again, but in a little bit more Cautious way than we did that in the deterministic limit There is one important observation And that is this When I tune the system to the critical point of Of the to the instability to the condensation instability, then this parameter here goes to zero So this means that this retarded and advanced inverse keldish-greens function here they scale with say momentum square Because the the constant term is removed precisely by this fine tuning to the critical point Conversely This piece here entering the inverse keldish-greens function Well, this hasn't sees nothing special. You get okay gamma loss equals gamma pump But this is a function that scales with q to the zero power. So there's no scaling And now you can go back to your quantum field theory course And do a what is called three level randomization group analysis or a canonical power counting for your action You require that an action it appears in the exponent In action appears in the exponent of a functional integral So it must be a dimension less quantity And you require that the total action is is dimensionless and in and we know also the scaling Of all the ingredients of the greens function here And in this way by requiring the total action be dimensionless We can figure out the scaling dimensions of the classical and the quantum field And because of this mismatch in scaling of retarded advanced and keldish component of the greens function One finds that the scaling dimensions of these fields are different. They split off So there is a strong asymmetry again between scaling of quantum and classical field here And the observation is in particular when you apply this counting here for the fields to the quadratic terms You will see that the more quantum fields there are in a non-linear term The more irrelevant this is in the sense of the randomization group and the statement that you can see is that in dimensions larger than two couplings For as a non-linear couplings here with more than two quantum fields are total irrelevant in the sense of the randomization group So we drastically simplify the structure of the action by immediately skipping these terms So this really is a massive simplification of the problem and in particular I will now show you how this simplification Brings us precisely to this longevity equation in very clean terms Okay, so that's the agenda So first of all I write down The action that we get when we do this simplification when we drop these these non-linear terms that that are rg irrelevant Even if you did not maybe follow all the fully The argument Let's let's keep in mind. Yeah, so that we can massively Simplify by rg irrelevance arguments the action in a way that the only terms that remain are linear in the quantum field But now also something it is quadratic in the quantum field But nothing more all these non-linear terms except this one which is linear in the quantum field all they they are gone Okay, so what does this buy us? It's just a formula to begin with But now we can show that this actually brings an equivalence to these longevity equations so And this is done here So here I recall the structure of this action Something that is linear in the quantum field something that is quadratic in the quantum field And this here is the action that we have to vary in order to produce this term So and it has precisely a reversible contribution and a decay contribution which comes with an eye in front Okay, so now we want to manipulate This pass integral further to see the connection to longevity equations And a nice trick that you do here is we want to get rid of this quadratic term in the quantum field and we want to Make it actually linear This idea of decoupling field powers is in some Context known as havat satonovich decoupling but even without this word I just give you the simple identity where I have to watch The signs that that shows you how I can get rid of or how I can decouple this quadratic term Is this identity say for a single complex Integration and of course we have here Functional integrals, but the identity that we really need is this So this is again a nice Gaussian integral with when I do this integration. I get the result This is the result of this integration Yeah, so or in other words when I start from this term here as it's the case up there I can read the equation this way To what is known as decouple this term decouple in the sense that now I produce something That is really linear in this quantum field And when we have a situation that is linear in the quantum field, we know what to do because that's exactly what we did previously We just integrate over The quantum field and produce Here, so so that is the action after this decoupling So with this xi field This here is the variation of the bear action And I can use now again the representation of the delta function in terms of a Fourier Transformation to produce now the delta constraint that we had previously So this is just a variation of the bear action this thing But now there is this shift with xi This shift with xi came from decoupling the quadratic field quadratic term in quantum fields and What it means this xi Comes out when we just read this formula what it means So there is a shift in xi And this xi is a random variable with gaussian distribution And this is precisely what is described the lingerie equation So it is essentially the classical equation of motion that we got in the deterministic limit augmented by a random force Whose statistics is prescribed by this havat-sortanovic decoupling xi term This is the this is the the idea. So we can get this Driven dissipative lingerie equation here As a semi classical limit where we dropped higher order terms in the quantum field than quadratic By good arguments, yeah, namely being close to a critical point. So Uh, we could simplify the problem to to this and we show down that this Simplified semi classical limit of the kelvish path integral is equivalent to this lingerie equation. Okay. Are there questions for this? good So now this is the picture that that you a physical picture that you can get out of this So there is a kind of really, um Um Weak universality, yeah, that comes just from doing this semi classical limit So in some sense, you could look at this linblad 5 4 theory that we looked at Close to a critical point or this exciton polariton micro physics They we can all describe this in terms of very complicated microscopic linblad equations But when we coarse grain In the vicinity of a critical point close to the onset of the condensation phenomenon Yeah, so then this problem simplifies actually to a to a lingerie problem And from there, yeah from such a mesoscopic starting point one could then do actually more sophisticated calculation to really extract the the macroscopic physics out of that and this is really A reflection of this concept of universality that so to say if you coarse grain if you zoom out of the problem Then often simplifications occur that um That make the problem a little bit more um easy to tackle. Yeah Right. I mean so so the the question can be now. What did we drop? Yeah, so is this actually So, um, is this Maybe you can formulate it. Is it justified to to drop these quantum couplings? Yeah, and you can I mean these arguments They hold I mean on very general grounds close to a critical point. Yeah Except for fine-tuned situations. Yeah, that that this noise level also Also, this peak held ish has a zero But let's assume this is not the case and then you can ask, okay If I if I keep all these terms, yeah, for example in an rg flow calculation I I don't know this argument. I haven't heard of it. So I keep all these other terms What will they do? Yeah, so that's maybe a way of reformulating your question And then you can do that and you will see that these terms have lower or weaker infrared divergencies Yeah, so they will drop out of the randomization group flow Even if you take them into account and that is so does it telling you that this argument this high level argument really works But you can also keep all these terms if you have Very much time You do the heavy calculation and you will see there is a fine difference in in leading Infrared divergencies that are coming from these fluctuations while these quantum fluctuations They are really overshadowed by by those. So there is an interesting fine structure in Quantum versus classical fluctuations Or yeah, so that that are under these conditions. You can really see the quantum fluctuations are suppressed It's a little bit like a finite temperature phase transitions versus zero temperature phase transitions So finite temperature means finite noise level in this more general language here And of course, it's interesting to look at also situation where this is Manifestively violated. This is the analog of quantum phase transitions that also exist in such systems But I mean that's now I can't discuss everything further questions Yeah, that's also a very good question. Yeah, so you can formulate it quite generally. So what is um, which concepts Of physics translate to this out of equilibrium realm and which don't And if you think about a spontaneous symmetry breaking It is a concept that works in and out of equilibrium. No matter So there is a laser A laser lazing mode is a spontaneous symmetry breaking of phase rotation symmetry So this concept has nothing to do with in or out of equilibrium And and therefore yes, of course, it's just a matter of how can we see that in this formalism That there is a spontaneous symmetry breaking phenomenon, but this is not I mean spontaneous symmetry breaking It's one mode gets macroscopically occupied. That's exactly what happens in a laser So there's it's not conditioned on equilibrium But it's actually a good thing to to always think is this effect is this question I'm asking very specific to equilibrium or not Shutter good. So this is the wrap up of this section here So so what we what we did essentially so we went along this line here We started from the lindblad equation and we mapped it into this lindblad keldish functional integral So then we did in the as the last step Yeah, we we more we took a limit The semi classical limit and we mapped Then this complex keldish functional integral under certain circumstances We did that are not always present We could map it into a simpler object Something that is for example relevant to this exciton polariton concrete systems And from this here from this functional integral formulation We translated it to a longevity equation formulation of the problem So so the upper ones are always equivalences and then there's a there's a limiting case that you go when when we walk from up Up down and actually I would like to embed this a little bit more into Formulations of of of quantum statistical problems quite in general Namely There is always a formulation in terms of deterministic evolution of probabilistic objects Very strange formulation, but a density matrix Or is kind of a probability distribution So and under this lindblad equation it evolves deterministically I don't see any noise in the lindblad equation But I'm evolving a probabilistic object. So there's always one formulation of this type A deterministic evolution of probabilistic objects And then there's always another alternative way of formulating the problem in terms of a stochastic process So here we build in explicitly some noise And you've seen how this noise comes out of for example This construction transiting from a functional integral from a probabilistic to this longevity formulation So this is a quite a general principle In in in theories and that that you can formulate deterministic probabilistic theories or stochastic versions of it And for the quantum problem, this would actually be the Heisenberg Langevin equation here in the quantum case and Also this stochastic Schrödinger equation that we get to know in the in the afternoon And here these two lines here So these these two lines are I like to refer to them as essentially or all of these here Sorry, all of these they are differential formulations. So we are evolving forward something one time step by time step While I mean these integral formulations here. So this is like the Integrated integral version of the Schrödinger equation. Yeah, where one has one deals with Exponentiated Exponentiated evolution operators and essentially how they to propagate over whole time string Yeah, so this is kind of the structure of theories that that we are dealing with here Okay, so now comes this question Yeah, that I mean goes in the direction of what you were saying or asking so When we have now this Lindblad equation, how do we see even that we are out of equilibrium? So now I would like to discuss a bit this question So what is non-equilibrium about this Lindblad equations is pretty intuitive You're pumping you're hammering here on this density matrix and So how do we quantify that really more precisely? If I give you kind of a childish action, can you tell me if this is in or out of equilibrium? So that's the question Okay, and and a little bit of I mean again embedding also in in the series of lectures here So there is very many notions of what out of equilibrium actually means and a very prominent Way of of saying oh this system is out of of equilibrium is of course when it really still time evolves So time translation invariance if this is broken explicitly for example I quench the system and then I let the evolution go Then I can study phenomena like thermalization Or if I even time independently drive my my system So then I will reach this realm of flocay systems and all of this you've you've been seeing here Yeah Now what we are the notion of non-equilibrium we are discussing in these lectures is a little different It is really the The the the notion of non-equilibrium in the sense of stationary flux equilibrium states So and Point that I would like to stress because it also often leads to confusion Detecting such flux equilibria it is not enough to look at the density matrix It's really not so Often appreciated But the you will not see if I give you a density matrix I will not be able to tell you whether the system is in or out of equilibrium Because essentially very simply every Density operator can be written as the exponential of something else of a emission operator The real question that you have to answer is Does this age Up there coincide With the generator of dynamics Is this age what stands in a heisenberg equation of motion? And is there nothing else that really that would actually be Generators of dynamics like the linbladian that push you away from this limit And in order to assess this question practically You really need to look at correlation functions That indeed see that there is a gen which generator of dynamics is acting there So it requires you to look at dynamical correlation functions Any static correlation function that you can generate from the density matrix alone? So x x correlations x x prime correlations They will not tell you whether the system is in or out of equilibrium But in the moment you look at dynamical correlation functions You will see what the generator of dynamics is and there you have a chance of Detecting in or out of equilibrium in stationary state Okay good So Bit more systematically. Yeah, so another yeah, that's a very good point. Yeah, but again or that's not clear So although No, you can have long range Hamiltonians Not really. I mean no, I think I mean locality of the Hamiltonian is a natural property and That that's an argument that that is often often said I can just give you an opinion I mean so I mean that it can be that out of equilibrium. This age is highly non-local So that that the density matrix I can represent it always as I said Yeah, but maybe then out of equilibrium this age is always non-local I just say this is not a good watershed Because first there is long range Hamiltonians in equilibrium and second I don't know many good examples that this Hamiltonian really is non-local Instead, I know many examples Yeah, where even this density matrix in simple Gaussian problems Yeah, so very often you observe that the the density matrix is written as a e to the and then some locally quadratic form Even in the out of equilibrium, so I'm always I'm not saying it's it's impossible that it's long range I'm just saying it's not a good criterion because there's always counter examples of Okay, so speaking about this watershed question. Yeah another thing that that is often I mean Sad is that ah the open system character is what makes the system out of equilibrium, but this is also not a good distinction criterion because Or so to say the the the point that we are making here is mainly irreversibility of the dynamics But also this is not a good Quantifier of out of equilibrium conditions. There's plenty of examples So including I mean a phonon bath in a piece of condensed matter that you leave alone So this is an open system for the say fermionic degrees of freedom to which this Phonon modes are coupled. Yeah, so and it's a perfectly equilibrium situation There's a detailed balance between the phonons and the fermions So and the fermions they look very irreversible because they are damped by the photons phonons So what is really the important point of distinction? Yeah, is that the system is actually driven and open at the same time Yeah, and that is what I want to emphasize and a very good physical picture a very simple But good physical picture is that we will encounter non-equilibrium conditions Whenever we confuse the system in such a way by coupling it to different baths That it actually doesn't know which baths to thermalize to So this is really the important physical ingredient and we'll make this now mathematically sharp But this is the picture you should have in mind So and this is precisely the situation we were depicting And so I have a many body system and you may look at this single particle pump The single particle losses and the two body losses as just coupling it to different baths These baths have nothing to do with each other that you can tune their parameters as you like And this is the circumstance that confuses the system and doesn't allow it to thermalize to one of these baths And that that is I mean how I would qualitatively see Aha, this is really a driven open system. Yeah, we have we have different different Dynamical resources acting on the system, which don't allow it to thermalize But now let's try to make this sharp. Yeah, so this this intuition Was there a question? Um, right. So, um We start again from this, um, kelvish functional integral And um, we look at the situation now where we switch off this dissipator completely just for for the sake of of understanding something So then we reduce the problem to a heisenberg Evolution we can still path integral quantize that And then I make the statement So that the action that results when we put the dissipation to zero that the lindland lindland dissipation to zero Then you can see that the action that remains this s h That in case the hamiltonian operator is time independent. It enjoys a funny Symmetry that is written down here. So this symmetry sends time to Reverses the direction of time it also shifts time into the complex plane No matter what this and it reverses The sign of i and it has the property that it squares to one. So it's a discreet symmetry and um No matter, yeah, so I have no good feeling what this symmetry means But it's just an interesting observation that the action discarding the the dissipation for a time independent hamiltonian has this symmetry and the important question is not Why is there this symmetry? But the important question is what is the consequence of this symmetry? And this is The presence there is sort of the word identities associated to this symmetry so Which are the fluctuation dissipation theorems of thermodynamic equilibrium So just to remind you how the argument goes here So we start say from an observable You know, which we can write as a functional of these fields say Celtic fields and we write this as a functional integral This expectation value weighted with the action yes Then I mean I do a I just do a transformation of the integration variables this I can always do Let's take a unitary transformation of the integration variables And we speak of a symmetry under the following circumstance first The action is left invariant under this transformation T meaning S of t phi is the same as s of phi And also we require for a symmetry in a quantum field theory that the functional measure is invariant under this transformation And then We can actually go back to this formulation to see That the consequence of a symmetry of the action is an invariance Of This observable o Oh Symmetry transformed averaged with respect to the action that is symmetric gives the same As the action as the observable without this transformation applied And if you do this idea if you implement this idea Just for this funny symmetry transformation here you find this relation here between correlation functions Correlation and response functions And that is really so this o would be the phi c phi c and the phi c phi q So so this o is just some polynomial of of the fields that are in this in this problem and This idea of having this symmetry implies a strict relation Between the correlation functions that we discussed Yeah, so the measure of how how far do we go away from a deterministic limit and the response functions of this problem And universally If this symmetry is present here There is a relative factor between the keldish and the between the correlations and the response function Which is the bozer distribution function And this kind of emergent out of this symmetry consideration We discover essentially the bozer distribution as the function that governs the particle occupation of a system Of a bosonic system if it would do the same for fermions would have not the bozer distribution But the fermi distribution function So systems at equilibrium have this symmetry and that implies a very strict relation between correlation and response functions Okay, and the point is so whenever I take now a lindblad operator a check for the symmetry It's not there. So it's just massive massively violated and that tells us so I can just you give me your action and I will test for the presence of the symmetry I don't know where this action comes from like I just have to test for the symmetry to be sure that The system is not at equilibrium. Yeah, or it is at equilibrium Because I mean maybe this this piece here. Yeah, so this correlation and response function relation Yeah, so this holds beyond the single particle correlation where it holds for any order And so it is equivalent to I mean All subsystems being in equilibrium with each other. So the system really is in global thermal equilibrium So this this is really the symmetry behind equilibrium. Okay questions for this. Yeah, sorry Now any order means that I mean I can give you the I write correlation functions Of say third order in the field And response function of third order in the field and there will always be a kind of Relation for the whole hierarchy of correlation functions And so it's really a property of total theory and not just of low order correlation functions. Yeah Sorry the functional differential equation for this symmetry I mean, I I think you are thinking of this word identities or yeah I'm word identity. I put it in quote and quote because this is something that usually is working for for continuous symmetries where you have a meaning of of Close connection and you can I mean something that is close to something else and you can work with differential operator This is a discrete symmetry Nevertheless, I mean that's I put it quote and quote But you can extract a useful constraint from the presence of this symmetry So it's not exactly a word identity in the sense of this prst or what maybe I have this in mind It's not in this sense, but it is I mean way something that you can useful constraints derive from that. Yeah No, no, no, I mean we we do this idea that I that I go give you down there. We apply exactly this idea and yeah Just there is no no infinitesimal representation for this. There's no infinitesimal generator for this symmetry because it squares to one um I mean if you like you can look at this single particle Grinch function as a linear response. It's linear in this quantum fields Uh, and in that sense it is precisely near response I was saying, I mean there's also higher order correlation generalizations of that So it also holds in the non-linear response. You also get constraints Good. So last thing I would like to do is the um, the semi classical limit Of this Symmetry, yeah And here we can go again by by this idea. Okay, there is a shift in the complex time plane Which I can always write as an exponential as a kind of a generator that does this shift for us And in the semi classical limit, yeah, so intuitively this is for an equilibrium system. This is large temperatures So I can expand this exponential function here, and then I apply the symmetry transformation to this Keldish and classical to these Keldish fields here I do the power counting and I realize, okay, this occurrence of the quantum field is again something that is irrelevant in the sense of the renormalization group So this transformation here is the semi classical limit of this Equilibrium symmetry transformation and I pointing this out because it has a beautiful and very simple geometric representation for this equilibrium symmetry So and and this comes here So that's the point I still would like to make So we have now this action with only linear and quadratic terms in the in the in the quantum fields and this action here consists of reversible dynamics and irreversible dynamics coming with a relative I Yeah And both of these reversible and irreversible Generators here. I can write them in terms of a landau theory The landau expansion where the most important term is quadratic So and it comes as a mass term constant term then there's gradient terms and then there's interaction terms And but they come with this relative factor of I And so what I'm doing here. I'm plotting in a complex plane of couplings So all the couplings now have real and imaginary parts according to this here I'm just plotting real and imaginary parts of these couplings in the complex plane of these couplings So for example, if we have now two-body processes, the real part of lambda of the two-body process Is just really elastic two-body collisions And the imaginary part of it is inelastic two-body losses And so that what we are describing and so I make for the real part I make a sketch down here and for the imaginary part This axis and now I go through all these parameters In this landau-ginzburg functional to get here a plot in of the complex couplings here in the couplings in the complex plane Where the real part is reversible and the imaginary Irreversible and now comes the interesting point If we have a system that is in thermodynamic equilibrium So it obeys the special symmetry that we were talking including in the equilibrium in the In the semi classical limit Then the implication of this symmetry is also a geometric one In the sense that all the couplings they have to be aligned on a single ray in the complex plane While in a non-equilibrium situation, so I take a lindblad equation take the semi classical limit These couplings will just spread out somewhere in this complex plane And this has also a very good physical reason So imagine I have a system that is at equilibrium And I have at the microscopic level then only a time-independent Hamiltonian As what generates the dynamics So then the picture would be this in this complex plane of couplings On the microscopic level all the couplings They are purely real because I only have a Hamiltonian generator of dynamics So they must all be aligned on the real axis Now I start to coarse-grain this problem. I zoom out in an rg spirit And it is clear that I mean even a system at equilibrium Can show dissipation the system acts as its own bath. It wants to thermalize And what this does in the rg language thermalization is kind of this phenomenon So that these couplings acquire finite imaginary parts But the way they inquire these imaginary parts is not without rules You will never generate a situation And that's more a statement that follows from this analysis than that it's obvious But if the system microscopically has only a Hamiltonian generator of dynamics Then you will see that at all scales the couplings will remain aligned And in particular at a critical point There you have even this phenomenon that it's totally overdamped And all the real parts are going away So on the way from the microscopic to the microscopic scale at a critical point You lose completely the information of the microscopic reversible dynamics And the system becomes totally overdamped And the most important point is really that in an equilibrium situation So this equilibrium symmetry really tells you that all these couplings remain aligned while out of equilibrium And so this is not the case Right, so that would bring me I think I have to stop I don't want to overdo it, but it takes some questions Yeah, I mean right. So one of these you can take it as really the the imaginary contribution to the self-energy So this is part of it. So in the single particle sector is the self-energy in the two particle sector or something else Okay, yeah So it's not really clear what is the mathematical justification of this consequence of equilibrium What what the mathematical Interpretation of the coupling all which are all aligned Well, I mean you you can look at I mean you apply The the symmetry transformation to an action In the semi classical limit and then you ask when is the action left invariant Yeah And you find that it's only left invariant if all the couplings Yeah, so essentially it is this relation if all the couplings are on one ray or in other words When this hc is proportional hd and hc and hd are really this complicated I mean are these These objects that make together make up the action So you can see that this action it has a contribution from Reversible and from irreversible dynamics and each of them I can expand them in this landau spirit in low powers of the fields And a statement that when everyone is aligned is the statement that these two Functionals are really the same up to one common proportional constant And this is the circumstance only if this is realized Then the action is symmetric under this transformation. So that is this logic of the argument It's clear now. Let's say the last one because I think it's a bit stupid, but if there is a symmetry, I guess there is a charge Which is conserved Which not not no only if you have a continuous, so the nerda charge. So there's no no nerda charge for that Not that I would know As in not we don't expect it I think We deserve a lunch right. We sorry for going a bit over time and and one thing I mean Yeah, I'm not yet through with the physics application. So maybe I Give a start with a little bit of that In the afternoon and then we go to these measurements or I can also go on with this fear for the rest so what would you prefer to see the measurement stuff or Slowly through this Yeah So other votes for measurements say So what are we voting for like say? I mean, so should I slowly go on with this or should I Briefly summarize some consequences of this and then move to the measurements Okay more on this Raise your hands I shouldn't have asked I guess whether I should go slowly Yeah on this line here Or I should Summarize this and then we go to do this measurement business so more like measurement induced phase transitions in Maybe as an alternative point of view as to what soon one was explaining exactly, yeah, right? I mean so more a childish formalism Let's say a point of view on this phase transition. Let's vote again More on the let's say just open systems and application raise your hand Well, okay and more on the measurement induced phase transition The second one won't so So I will briefly summarize this and then okay. Thank you for your attention


So, this talk is motivated by a use case we had in scikit-learn. We needed to do something, and we were like, how do we do that? And it turned out to be a very cool thing we did. And then I thought, why not talk about it? You can find the talk on the GitHub repo, so the slides are there already. I did, a bit about me, I did my PhD in cancer diagnostics and did like algorithms and machine learning there. Then there's some machine learning consulting, and these days I mostly work on open source stuff like scikit-learn and scops and fail-in. There are some prerequisites to understanding the talk. It would make it easier, like knowing that functions can take a variable number of position logs and keyword arguments, set attribute, get attribute, those kind of things. The fact that in Python, if you have very specific and dunder methods in certain places, under certain conditions, they're called. With their output, something is done, or they have a side effect. Inheritance, type annotations, not necessarily, but we use them. And then that when you do help of an object, it tries to read certain information about that object and show you. So, what is the motivation for us to do this thing that we're gonna talk about? In scikit-learn, we have estimators, and then we have meta estimators. You can put estimators inside meta estimators, and these meta estimators will route certain things to your estimator. Like, you put a transformer and a classifier in a pipeline, and then you fit it, and then it passes x and y to your transformer, then it passes it to your predictor. But you can also have certain metadata, like sample weight would be a metadata, or in this case, my custom transformer would have a sample weight and another metadata. And what I wanna do is to be able to say, hey, my fit is requesting sample weight, it's not requesting this other metadata, don't use that. And then put that in a pipeline, and then call fit on a pipeline, and the pipeline should know how to route things. Right now, you can't do that, you almost can if you use the nightly build. But the idea is for us to have that, and we want it for this method. So, there are certain requirements on this method. One is that we don't want to change our estimators. We want that method to be dynamically generated, specifically because we also wanted this thing to work on third-party estimators. If you inherit, if you're doing your own estimator, you inherit from the base estimator, these methods should just exist. And we didn't want it to have a generic signature like we accept everything. No, if your fit accepts only sample weight and this other metadata, then you should only be able to do that here. Not any other random metadata. And we wanted a good doc stream. Seems very basic, turned out quite interesting. In short, we're going to use different pieces to build this puzzle. We're going to use introspect functions. We're going to use the signature object to read and create a signature object. And then we're going to use a descriptor returning a function to add these methods to the estimators. And then we use init subclass to create the right descriptors attaching them to the estimators when needed. Now we talk about these steps. So inspect and signature. Let's say I have a function f, it takes a, it's type hinted, and then it takes a bunch of position logs, a keyword argument, and variable keyword arguments. Then I have a class a, it has a method with a very similar kind of signature. Now I can check if f is a function. Yes, it's a function. Is a.g a function? No, it's not. It's a method. We'll see immediately what really the difference is. But is a.g like not an instance, but the class a function? Yes, that one is a function. We can do other things. When you get the signature, it returns an object which has information about whatever input your method takes. I can go through them. It has this nice dictionary. I can go through that dictionary. It gives me the name and different things. Like what is the kind of this argument? What's the default value? And are there any type annotations on it? If I run that on f, I get what you expect. A, rs, b, kwrs, and you see like they have different types. Like positional or keyword. Or this one is a keyword only one. Is there any type hint here? Or is there a default? If I do that on a.g, I get everything including self and all the other arguments. But if I do that on an instance of a.g, I get the same thing except self. And that's pretty much the difference between bound, like changed, but like think of it bound methods and not bound methods. There's a lot more to inspect. You can read it like there. But this is what we're gonna need for the rest of the talk. Then next thing, let's say I want to have a function that adds five to my input. Easy. This is how you do it. Now imagine I want to create a bunch of these functions that add different numbers to my input. I can create a function that creates a function inside it and then returns that function. And then I can use that. So here I have a create adder. It creates a function. And then here I return f. I don't return f of something, I return f. And then here the output would be that function which I can actually call. I could do that same thing with a lambda expression. We're not gonna use that because our functions are complicated, but you could do it. Oh, this is a lot of you. But any major questions up until this one, I don't want to lose too many people. If you do have questions, every now and then I'm gonna ask if you have questions, like move to the microphone. We can have it like a little bit more interactive. So now we have a function. We want to change its outfit. Let's say I have a very simple function. It accepts a bunch of position logs and it calculates the sum of the inputs. And here I get help on f. When I run that it says, well, it's f. It takes a bunch of arguments. And this text is what you have here as your doc string. Where are these things stored? Well, first of all, I could call it. It does what it's supposed to do. The donder duck includes the doc string. The donder name includes the function's name. And then when I do inspect.signature, it gives me the signature of the function. Now what I want to do is to change all of these. I'm gonna say, I'm gonna change your name. f.name is adder. I'm going to create a signature object and assign it to donder signature. Basically if a function or a method does not have this donder signature, then signature tries to read the actual signature. But if this exists, then inspect.signature would return this. Which is also used by most of your IDEs and whatever tool you use to develop, which would mean that the hints that you get would be more meaningful now. What I'm doing here is I'm saying I have one parameter called a. It's positional or keyword. The default is zero. And it's an optional float. I have another parameter b. The same thing. And then here in my signature, I say I have these parameters and I'm returning a float. I also changed the doc string of f. Just the whatever string you wanna put here. So I do that. And then I get help on f. Now I get a much nicer thing. You might ask why bother? I could do these things on the method itself, but I'm going to be generating my methods dynamically. So I would need to set these dynamically as well. And now if I try to get the signature of f, I say that I can only pass an a and b. But is that really true? What happens if I do f123? It works. You're not changing the actual function. You're just changing things about the function. That's why my way of saying that is that I'm changing the outfit of a function, not the function itself. You could try, inside your function, you could try to simulate the same thing that Python would do if this was the actually the signature of the function will show somewhat how to do that later. Question still now. Easy. So the Python signature object, the way you're using it, it was introduced in this pep, I think in 3.6. You can read more about it there. At this point, for me, the best documentation in Python are these peps. They're very extensive, really nice. So now the script tours. What's the descriptor? A descriptor by definition is a class that implements don't forget. And it's a descriptor when it's assigned as a class attribute. Not as an instance attribute, but a class attribute. Wait. I feel like something's not working. Gods of demo. I'm going to disconnect and connect again. Let's see what happens. I hope I don't have to reboot. Mm-hmm, mm-hmm, mm-hmm, mm-hmm, mm-hmm. Best thing that can happen during the talk. Yes, because I had to do, I had to, everything collapsed. I'm trying again now. Let's see what happens now. Yep, it should work. We did it. Okay, let's, just to be sure, I'm going to run from top. So, descriptors. It is this thing, it's a class that implements don't forget. And it is, like it behaves as a descriptor when it's assigned as a class attribute. Not an instance attribute, but a class attribute. What happens is that if you do that, and then you try to access it on an instance, then Python will call this dunder get, and it will return whatever this dunder get is returning. I've put a few print statements for us to see like the sequence of things happening. When we run this, first we pass here, then we go to the descriptors in it, here, and then we try to access pet here, and then we access it. And when we access that, we go inside get. So you define that, first it's like this thing runs, and then when you define that, well, you're creating an instance of pet here, and then you go down here. Is this kinda clear? And then what's important here is that your get also has access to the instance that owns you, the instance of the object that owns you, and the type of the object that owns you. So we could also try to use that. Here, I'm changing a bit, I'm this string, I'm say, well, I'm myself and my owner is this. And then here in my instance, I would add a name. And then I set the instances name here, so this would be the constructor, and then it comes to the get of the descriptor, and then I would have access to instance.name. And now it means that I have access to the instance, I can not only just read it, I could also change it, also the type itself. Now we're getting closer to the point where what do we do for the request methods that we had? So this is like much closer to the actual code we are using now in scikit-lan. I'm gonna get the keys, these are the list of arguments that my method, that set fit request method accepts. This is a descriptor, and as we talked about, this descriptor can also return a function, among other things. So we're going to create this function and return that. In this function, I check if the given keyword arguments has anything that I do not accept. And then if that happens, I raise a type error. A type error is also happens to be the same thing that Python raises if you try to call a function that with arguments that the function doesn't accept. So we're kind of here trying to like simulate the same behavior. And then for everything else, we have a side effect on the instance. We just set some attribute. So for example, if you say sample weight equal true, we are gonna set request sample weight equal true. That's what this set attribute does. And then I'm going to return the instance which would result in me being able to chain my methods on the instance. And then here I return F. And then now I can assign this descriptor to an estimator class. So if I have an estimator, I say set fit request is this request method and only accept sample weight. And then when I do that, then I can have an instance of this estimator. Then I try to access set fit request which returns that function that I defined. And then I can call that function and say sample weight true. And when I do that as a side effect, this estimator would have this request sample weight set by this line. And then if I change that to like blah, it would also be blah. If I do that with a parameter that is not expected, I get a type error. If I get the help of that method, not very helpful. So let's try to fix this one now. I'm going to have a helper function to return a signature which would be the desired signature of this thing. It requires the owner, the object owning it and the type of the object owning it and the keys, what does it accept? The first argument is an instance method. So it accepts self. It's a position or keyword argument and the type of it is the owner. Then I extend that for every key in these keys. I say create a parameter. It's a keyword-only parameter. The default is none and it's an optional bool. And then I create the signature object with these parameters and the return type is also owner. That's because I was returning the instance itself. And now I'm going to change my descriptor slightly. Everything that stays the same. I'm going to add the method name here which would be the name of the thing that I'm using in the owner. So if it said fit request, this method name would be set fit request. And here I'm going to set the name of the function, some constant doc string that in reality we're going to do a much nicer one. And then I set the signature with that input function, the helper function that I had. And now when I create it, it's the same except that I also say hey, I'm actually the set fit request. And when I get the help, I get this nice thing. It is set fit request. It takes sample weight as an optional bool. It returns the estimator. We're almost there. There is a really nice how-to on descriptors. You can do a lot more. There's a lot more to descriptors. And you can also have things like what happens if the user tries to set a value or they try to delete something, they're all there. The next thing is init subclass. We're not finished yet. We need to generate these descriptors. Like this is now hard coded. We need to make that dynamic. We need to make that dynamic in a parent class. Again, as a reminder, we had an estimator. It has fit and we want to have this set fit request, but we want this set fit request to be generated automatically. We create a parent class with this init subclass method in it. What this does is that for every class that inherits from you, it calls this thing. And then this thing would have access to the class, not the instance, and then you can do things to it. So here, for example, I'm gonna have, I set the attribute to five and then I call super init class.  That's all. Let's see the sequence of things here. This one is also very interesting. Here, I'm printing the class that I get. Here, I have a print statement in it. Here, I have a print statement. And then I access this attribute. So the way it works is that first, we said yet we define this parent class, but then before defining this, before we get here, we get this print statement. So as you define your class, this init subclass runs. And then once this is defined, then it's done, then you can create your instance and access your attribute, in which case, in this case, it would be five. Now let's try to get closer to what we actually wanna do. Now here in my init subclass, check if my class, this is the child class, has a fit and if this fit is a function. If yes, then I'm going to print the signature of that function. That's all. And then I have an estimator inheriting from parent and then it has a fit method. When I run that, I get the printed version of the signature. So it's kinda working. Now developing on that, we have that. Yes, we have fit, it's a function. Now I'm gonna get the signature of that fit. I'm only interested in parameters that the parameter name is not self x or y. Because in the scikit-learn API, self x and y, I mean self itself, and x and y are not metadata. And we say everything else is going to be some metadata that I need to route. x and y, I have a very explicit routing for them. And then for that, I'm gonna say, well set an attribute on class. The name of that attribute is the set fit request. And the value of that is this request method, which is the descriptor that we just defined. And I say the parameters that you should accept are these parameters and this is your name. And then my estimator would inherit from that parent. Now the estimator dot set fit request just exists. And if I get the help, it's what I wanted. We do in scikit-learn, we do the same thing, except that we also loop around all the methods that we care about, like fit, transform, fit, predict, like all of those. And then this in itself class is a much easier way to do these things that you could also do with the ABC meta classes. And it's this pet for it, you can read, it's really nice. So in summary, we use the inspect and signature. And then we did a descriptor returning a method with the custom signature and doc string. And then we use in itself class to customize all the child classes. Thank you. Can I take questions? I mean, don't be shy, like last time I gave this talk it took an hour and a half because people were confused. Okay, so thank you for your talk. I did have one question, though I'm afraid I already know the answer. I really like static autocompletion, especially for when you start using type hints in Python as you are. Using either Jedi or Pyrite or any other sort of language server protocol. Do I have any options to translate all this to static autocompletion? I don't know about Jedi, but one of the reasons we wanted to have a non-generic signature was exactly that. If I try to get my hint here, I get what I set. So my IDE knows exactly what the signature of the method is. So if your autocomplete is doing the same thing that like here, I Python or VS Code or like any other one does, it should work. Okay, thank you. Is there a way to actually create functions that dynamically not redefine the name in the init subquasm method? Yes, you can make things a lot more dynamic. If you, so when you do like creating functions dynamically, I think then you mean attaching them to a class. Right, okay. You have this thing in Python that you can override two things. One is I think get attribute. This one is called in certain ways and then the other one, def get adder. This one, if you customize this one or like both of them, then you can return anything you want when a user tries to access anything in your class, which means that if you have implemented those and then I do like the estimator that blah, this blah comes to your get adder and then you can be like, okay, now I want to return this thing and that thing could be a function. And then that would be actual runtime. Cause what I'm doing here is like when you parse and then like your class is set. But if you want it to be done runtime, then you can do that through get adder. Hi, thanks. I have a question with the descriptor protocol. Is there a reason why you're not making use of the dunder get name? Or is that because you're doing a set adder on the class that get name is not being called? Yeah, so. Love this question. Somebody asked it. So as I said, descriptors expose also other methods. One of them is set name. The set name would work in this case because what happens is that first you're gonna have this, then you're gonna have new called and then after new is called and then set name is called in new, after new in its subclass is called. And then because in this case, I am assigning the descriptors to my class in its subclass which happens after new, then set name wouldn't be called. Right, okay, thanks. Hi, so you're using a descriptor, but am I correct in assuming that you could do everything that you've done without the descriptor and just creating directly the function there? Or is there anything that does require descriptor and is there any other pros and cons in using descriptor instead of going for it directly? Yeah, so I could have just implemented these. It would have been tricky to change their signature because the signature of these functions depend on the method that you have in estimator. And I don't know what my child estimator would have as signature of fit. So I need some place to introspect that the signature of that fit method and then change the signature of this one. So there is something that I would need to change. Also, inside the function, I would also have to inspect the signature of the other method to make sure that the parameters that I'm receiving are exactly the ones that fit will also accept. So in effect, it would be the same as like doing the descriptor because you need to do some modification on method anyway based on the child class and it's easier to do that in a descriptor. But there are at least three different ways to do the same thing. This is only one of them. To me, it was the easiest one. Is there a way to access the parameters of a function in a way that can't be faked like you could with Dunder's signature? Ooh, not to my knowledge. All right, thanks. And there might be like no, so when I was reading the PEP on signature, my understanding was that there is a way, but we shouldn't do that. If we want people to perceive a function with a different signature than the actual signature, we should be setting the Dunder's signature. But there is an ugly other way. I think my time is over. Thank you very much.