 It's infinitely sad that somewhere in the corner there's no, Dima is not looking at me and not asking sarcastic questions about my presentation. Well, I have many personal memories of Dima, but I will probably save them and tell about them on the other occasion, maybe tomorrow. And my talk refers to, well, there will be several concrete results in what I will be talking about. But most importantly, I want to stress that there's a huge field of non-equilibrium field theories which are physically very important and which we understand very poorly. We just scratched the surface of this. And as I believe it's fundamental for our understanding of the problems of the cosmological constant or of cosmic acceleration. It's fundamental for our understanding of turbulence. It's fundamental for our understanding of hierarchy problem in the standard model. So it is a, let me formulate now what the essence and what kind of problems there are. In the problem of the cosmological constant is that in the universe we have two scales. One is its size, I call it infrared scale. The other is the plunk length. What we expect from, if we use some naive equilibrium field theory and calculate the cosmological constant, we typically would get the result that it is, it has a scale L minus two ultraviolet scale. It has plunk size. We will obtain the answer that it is of the plunk size. However, in reality, cosmic acceleration tells us that it is of the order of infrared scale. And the difference between the two is a huge number, 10 to the 60. And that's the problem basically. Now, there was an analogous problem although it was never considered as a problem in the 19th century. We had Avogadro number, a huge number which measures, which roughly speaking relates the scale of molecular physics to the scale of everyday life. And there is some analogy between this ratio of scales and the Avogadro number, but I will probably not expand on this. Another problem with two scales is, so I just, to summarize what I said, it's just that we have the enigma of two scales, two hugely different scales. Now, and so it's interesting to see whether we have some other examples of this. And indeed we do, we have the picture of Kolmogorov turbulence, Kolmogorov cascade, which is not completely correct but it's qualitatively, qualitatively it is correct with some, in any case we have the ratio of infrared and ultraviolet scales which is of the order of some power of Reynolds number. It could be very large also. And the most interesting physics of turbulence develops between the two scales which is called the inertial range. Now, another example of two large scales is the fact that in the standard model we have the hierarchy problems and I will probably say something about that later. Now I shall skip a few simple exercises in non-equilibrium field theory. I only want to, we don't have much time. But I want to convey one feature of non-equilibrium green functions in comparison with the standard green functions to which we get used to. Non-equilibrium green functions which arise when you have the different in and out state than the system develops. It's not remains the same. I define like that. And they do satisfy operator product expansion and all the usual properties. However, if we try to calculate the S matrix, suppose we do, we want to calculate the S matrix. For that, we have to take some time to sometimes of two plus infinity sometimes to minus infinity. And then our field theory textbooks would tell you that the S matrix is the result of this limiting procedure. Modules to some uninteresting factors is just the S matrix. Which satisfies, that's unimportant, which satisfies the usual rules of unitarity, positivity, et cetera. Now, what happens in the non-unitary theory is that when we calculate according to the previous rule the sum of probability we take the S matrix, we will get the answer which is greater than one. So the naive unitarity is actually broken. The reason for that, I shall explain the reason in a moment, but I just go back. Yes. The reason for that is roughly speaking, I'm going to expand on it, is roughly speaking, oh yeah, I have a pointer, that while the green functions, which satisfies equations of motion and all the usual properties, is the ratio of these two matrix elements. The S matrix should be actually, because of this, the amplitude we obtain from the green function is a relative amplitude. It is a probability provided that vacuum doesn't break down. And one of the questions which I will leave unanswered, unfortunately, is maybe we can interpret it as a space-time interpretation of non-unitary fields. Whether we can make use of non-unitary CFTs, which are important and beautiful, but that's unclear. What is important is that another feature of this non-equilibrium approach which I am developing. Now is that the usual cutting rules for which provide us with unitarity are not satisfied in this case. Namely, the non-equilibrium Feynman diagrams are basically the same as the usual Feynman diagrams, except that the time is too valued. We will be talking about the double-valued contours in time a little later, but at the moment it's enough to say that the so-called Schwinger-Keldersch technique, which replaces Feynman diagrams in the non-equilibrium case, the technique which provides us with in or out-out green functions, is that you draw the same diagrams but you associate plus or minus with each vertex. Now, we can try to derive perturbatively the unitarity condition, the optical theorem, which tells us that the imaginary part of the amplitude is actually a product of left and right amplitudes. And indeed it is so here, this gives us the amplitude which is complex conjugate to this one and we have the normal unitarity. But there could be some anomalous pieces in non-equilibrium, which I call spiders, which basically means that vacuum is decaying. Those spiders are very obviously in the stable case than the vacuum is stable, those spiders are equal to zero. But so absence of spiders is a requirement for permanence. And here I can't resist temptation to quote to you some character from the very famous Russian novel who said that, well, people think that eternity is something grandiose. But maybe it's just an old dirty hut full of holes of spiders. But so in fact, eternity probably requires the absence of spiders, contrary to this opinion. But who knows? Well, that's not important. Now, let me give you one more preparatory examples to them, which is very well known. The example of non-equilibrium phenomena, which displays, you see, maybe I should have explained it at the beginning, but my point is that very different non-equilibrium phenomena, they are related by the technique and by the ideology which we are using to describe them. And that's why in order to develop the intuition, it's very important to actually collect and analyze as many examples as you can. So that's the explanation of the fact that I'm actually considering some well-known things, but in a slightly viewing them at a slightly different angle. So let's consider production of particles by the constant electric field, the problem which was analyzed in thousands and thousands of papers, but what interests, and let me first very briefly describe how it is solved. So let's suppose that we have the Klein-Gordon equation as written here, we will take vector potential, the one x component of vector potential to be ET approximately, and when K is large, we take the mode, it's easy to see that the, WKB condition is satisfied, and this is approximately the wave function, and then there is a breakdown of WKB when K is of this order. After that, we have again WKB, but we have not a single exponential, but two exponentials, two waves running in opposite direction. And we calculate the induced current, and it's very simple, this induced current, it's, I forgot to put the reflection coefficient here, which is written here, so there is a coefficient in front, and there is the only difference from the equilibrium, if you calculate equilibrium current, which is in-out matrix element, it will be the same formula, but without this sub-function, and it will be zero. However, the in-current is generated only by the generated particles, and they are generated in this region. As a result, you get the current proportional to the vector potential. Without, as you see, it's very, you don't need to calculate anything to get this, and this presents kind of a new anomaly, and it is actually quite interesting because for the constant electric field, we have time-translation symmetry, obviously, in our problem, and this time-translation symmetry will tell us that the current must be zero. Instead, we get a current which is linearly rising with time, and this is obviously the spontaneous symmetry breaking in the sense that the things become dependent of their past. There's a high non-locality in time. Now, there's an interesting feature which I mentioned very briefly on the blackboard, I guess. Do we have an answer? Oh, thank you. That we can do the calculation in two ways. As I said here, the calculation was done in the gauge AET, A0 is zero. In the second gauge, we can take A0 equal to EX, A1 equal to zero. The results will be different. The difference between two results is related to the fact that if you start in order to get a well-defined problem, you have to start with vector potentials which goes to zero in time and in space, and then you are diabetically turn that approach to the constant electric field. Turns out that in the non-equilibrium situation, the another manifestation of this sensitivity is that the answer depends on the order of limits. In one, if you first take space dimensions to infinity, you get one answer in one gauge. If you take time dimension first, then you get another result. So we have the non-commutative limits. Generally, the non-commutative limits, I think, is the, was always historically the source of highest confusions in theoretical physics. Well, starting from Boltzmann times. Well, it's not all. You see that we got the correction to the polarization or vacuum polarization proportional to time, and therefore we have some infrared divergence. It is possible to see, I will not be doing this, but it is possible to see that if you calculate some non-equilibrium green function using this diagram technique, using this Katkowski, modified Katkowski rules for the Schwinger-Kelch diagrams, you will obtain something interesting. Namely, that well-defined, massive field theory possess the green functions, not only the expectation of current, but all the normal green functions have unusual infrared divergences, proportional to the time scale, proportional to the time scale and to the time from the beginning, so to say. The time from the beginning enters explicitly the formula. And that's one of the summation. I don't know what will happen after summation of these divergences, and that's one of the open questions. And once again, you will never notice those infrared divergences, and people never notice them. If you calculate the in-out matrix element, the Feynman, so to say, amplitudes, but physical amplitudes, which are in-in matrix element, they are full of them. They are full of infrared divergences, strong back reaction to get slightly ahead of myself and so on. So, there's one more comment on this thing. One more comment on this thing is that you can calculate, you can calculate the, in a higher dimensional space, say. Normally we teach students that we have to look only in-out in order to describe physical processes. Why are we looking on things in- Yeah, because you never told the students that they are living in expanding universe. When you tell, when you uncover this surprise to them, they will have to change. What? Subject. Good, maybe. Okay. Yeah, one more point about this, and I will move to the next part, is that if you perform similar calculation in a static field, then you will get the induced density proportional to this formula. Actually, it was thought of my teacher, Arkady McDowell, tried similar formulas many times, trying to derive it from Thomas Fermi approximations and so on. I was always skeptical and I was wrong because, well, he never, as far as I know, considered this exponential factor, but otherwise it seems that we have a very unusual vacuum polarization. Normal vacuum polarization is just, is just the dependence, things depend on the field strength only, while here they depend on vector potential, which is unusual. And I shall just mention, because I don't have time to go into this, although it's quite interesting, that very similar phenomena occur, some phenomena which are related to creation of particles and so on, occur in condensed matter. Well, there is, of course, the naive, the obvious part of it is that there is a zener breakdown which is just particle production, but I mean something more than that. There is a thing called the excitonic insulators in which instabilities due basically to this large back reaction play a fundamental role. And that's, I think, maybe an interesting analogy for the cosmological constant problem to keep in mind. But we don't have time for this. Let me go to the case. Still remember the name, the excitonic insulator. Let me now switch to the case which really interests me, and this is the deseter space. That's, of course, the deseter space is highly interesting because of cosmic acceleration. And it's basically analytic continuation from a sphere. It's, well, it's all written here. You just take a sphere, you make analytic continuation, and you, we are considering massive fields, massive quantum fields on this sphere. And of course, when you have a sphere, you have massive fields, it's, there are no surprises there. It's stable, perturbation theory is working well. Nothing happens. So one might think that we can do the calculation in the deseter space, which, by the way, mathematicians, to mathematics terminology, the deseter space is the image, it's kind of anachronistic, it's, in fact, imaginable by Chesky space, the anti-deseter space is the usual by Chesky space. Anyway, let's look at the metric. Let's write the metric on a sphere in this form now, which after changing coordinates become the metric like that. And you have to consider the only thing which makes physical sense here is the region inside the horizon. Horizon is r equals one by definition. And when r is less than one, it's a complete Euclidean space. And as I said, there is that temptation to calculate things on a sphere and then analytically continue it at the very end, analytically continue it to the deseter space. We can try the same strategy with the black hole. You see the black hole has very similar, in many respects, metric, very similar metric. And in the Euclidean black hole, you have to consider only the space that there is no such thing as region inside the horizon. The complete space is just r greater than 12. And this is geometrically the Euclidean cigar, very well known thing. And again, there are some instabilities which, for the black hole, which I will not discuss, which are not relevant for me now. But basically, if you take a scalar field, everything is stable. And important point is that in this way, which is the way Samsung students would be analyzing this, they would miss a very important physics. There is no sign of evaporation. That's one thing. Then we make this no signs because everything is stable. No corrections through the mass of the black hole. Now, if we're also this metric, which I wrote before, which was nice, complete Euclidean metric, which is chadezically complete space, it's not so anymore when you analytically continue. So you cannot really, you are dealing with an incomplete Lorentzian space. And in this case, unitarity will be lost, generally speaking, because when particles can disappear, so something should be done differently. And although you can't prepare some unphysical conditions in which this Euclidean metric would describe physics, I call it black hole on life support, but I will not. And now I shall explain very briefly what really happened and how the system should be treated. So this is the Euclidean, when we integrate our tau, our angle, it was one of the angles in spherical coordinates. We integrate from minus pi to pi, and those crosses mean that we want to calculate some correlation. This is precisely like calculating some many-body system at thermal equilibrium, in which case you use the Matsubara technique and integrate from zero to beta in the usual case. Here the temperature will be two pi in these units, and you integrate it like that. Now we perform analytic continuation. We want correlation functions at the real times. And for that, you have to deform the contour, and the contour becomes like that. This is almost the Schwinger-Keldersch technique. You have the time contour, and you integrate along the time contour, which is here. So the Schwinger-Keldersch technique would correspond to this contour, not to this one. And this contour, which is physical, this is what is obtained by analytic continuation from the Matsubara, roughly speaking, and it's called the Bain-Kadanoff contour. They actually assumed in these systems, they considered it was equivalent to this one, but it's not equivalent in general. And for our case, the difference is important. You can actually deform this contour, for example, like that. And this will be, in this case, you can represent things as describing this other part of the contour, describes the interior of the black hole. So this one describes only the exterior. In any case, that's a matter of interpretation, but what is not a matter of interpretation is that the correct correlators should be calculated in this way. Mm-hmm. And, well, that basically repeats what I was saying, that the other problem with this Euclidean thing is that it describes only the region, very limited region, and this breaks unitarity. We can improve it a little bit by considering the expanding universe, which corresponds to, we remove the condition that foreign minus and leave the condition foreign plus, it's still incomplete. But what we can do is we can try to attach the Minkowski spaces. We can try to complete it to make it jadezically complete. And we can do it by considering the following sandwich. Let's look at what happens if we're instead of the, instead of purely expanding universe, let's assume that there was a Minkowski space in the infinite past, then there was an expansion, and then we will find something, and we will now answer this question, what something it will be. So we have this sandwich between two Minkowski spaces. We have the expanding DeSitter space. It's easy to see, that's just, of course, the concrete form is unimportant. It's just an example of what you can take. And it's easy to see that with this sandwich is a complete space, although it does not satisfy the Einstein equation, of course, but we do not care now about the Einstein. We just say that let's see how the quantum fields will behave in this or that metric. So once again, you can use the, justifiably use the WKB, and it can be used for large enough case, then you have a single wave once again. And the reflected way appears after it appears when K becomes smaller than, when K becomes of the order of A of T, WKB breaks, after that WKB restores again. It's like a stopping point. And then you get two exponentials, the reflected exponential appears. And now we can ask the following question. What will happen if we actually go to time to infinity? So we started with the safe haven of the Minkowski space. Then there was exponential expanding universe. Then we ended up with the Minkowski space again, but the Minkowski space, which will be generally in the excited states, the field will not be the safe. So the general, you can calculate the green function very easily and it contains two terms. It contains the first term like that, and the second term which is oscillating very rapidly, because T and T prime goes to infinity. So with any coarse-graining, the second term is removed, and we are left with the first term. But what is the meaning of the first term? You can easily calculate the beta, and you find that the beta is given by the Bohr's formula with a temperature, with a Gibbons-Huykin temperature. So what we find is that we started with a pure state, and we effectively evolved into the thermal state, which is not pure. That's in the context of the black hole, that's what's called the information paradox. But there's no paradox, here the paradoxes result very simply. Indeed, this part does not correspond to the pure state. However, the full green function is, it does correspond to the, it's just in the pure state, which is, which was obtained by the unity evolution from the initial state, from the vacuum. However, for any reasonable observer, any observer is a coarse-grainer. So for him it will look like a thermal state, and it will be looking as if information is lost. Of course, yeah, certainly not the fault of the space, spaces above any faults. Okay, now we have another feature here, what we expect now, we expect if we create so many particles and they have gravitational attraction, we expect some instabilities, which is still remained to be seen. But I want to stress that there is a standard view that when you have expanding universe, everything dilutes and nothing remains, no instability remains. In fact, I think what happens is that since you have permanent creation of particles, the density is not disappearing. What is, we have not dilution, but we have approach to the particles become more and more non-relativistic. So it's basically we end up with static particles. As in the limit of large time. But there is this continuous creation which will prevent them. The question is, the question is that those particles become highly degenerate and all of them will have zero momentum, the momentum contribution, momentum tends to zero. And so it's probably, perhaps, it's the way to understand the entropy of the the center space. But just because of the infrared shift, the particles become non-relativistic. So that's what is written here. That there is a standard view. It's an interesting view to just think that we have cases with infinite blue shift like we do in the anti-disseter space. And this blue shift in anti-disseter space is responsible for, it is responsible for the ADS-CFT gauge-string correspondence in general. How it is responsible for gauge-string correspondence. Very simply, because of this scale factor, the mass of the open string is proportional to this. So all massive modes for the open string go away. As a result, in such space with infinite blue shift, the string theory, we have closed strings which have a tower of states, but we also have open strings attached to the boundary. And these open strings, they basically have only massive, only massive modes. All massive modes go away. And so this is for the anti-disseter. For the disseter, we have infinite redshift and that perhaps creates horizon entropy. But that remains to be seen more explicitly. I will now briefly mention one more view which shows that we must expect instabilities and the quantum instabilities in the disseter case. By the way, I should have started, I should have said already that classically the stability of the disseter space, it was, of course, studied very in all possible details by people, and the conclusion is that for small enough perturbations, it is stable. However, in quantum theory, my claim is that it is unstable, so it's just impossible to have a tool when you try to, that's a possible explanation of why we don't see the huge cosmological constant. It simply is unstable. And that's another small calculation which shows basically the same thing. Let's consider the vacuum energy. Let's calculate the vacuum energy and let's do it in the following way. We have some classical metric, we have perturbation and we want to find the effective energy. And what the calculation in the disseter shows is that this is basically the expression for this diagram. Then for, this is done for the Pange Davis vacuum, but for all other states, it's even worse. The Pange Davis, in a sense, is the least unstable vacuum, I would say, put it this way. And it turns out that indeed this is, there's not zero imaginary part. Now, again, let's now go to complete, as I said, it's, I think, quite important to consider the globally complete, the geodesically complete spaces. And those sandwiches, they were geodesically complete, but it would be more interesting for us to start with the Einstein theory to have some solution of Einstein theory and then see what happens to it. And the solution, as of course, very well known, the solution is the global disseter space, which you also obtain from a sphere by some simple analytic continuation and changes of variables. And that's the expression for the metric. And as I said, already it's stable and defined. Let's see what happens to this disseter symmetry in quantum theory. Well, first of all, it's very important whether this space has, of course, a huge symmetry inherited from a sphere. We started on a sphere, then we analytically continued. And of course, all those rotations which were present on a sphere, they don't go anywhere, they just get analytically continued. So it has this huge symmetry. In particular, if you calculate the expectation value, n is the point on the sphere or on this hyperboloid, better to say, this phi square must be constant because of all points of the sphere are equivalent. If you calculate energy momentum tensor, once again, it's some trivial answer if the symmetry is present. So basically, the symmetry makes it impossible to have interesting physics. Its back reaction is always small. Now, however, my point will be, once again, as it was with the electromagnetic production, my point will be that this symmetry is spontaneously broken and we do have a large back reaction in this problem. It is, there is a convenient way to calculate things in the global space. Let me explain what is written here. We have the Penrose diagram, which is just a cylinder in those coordinates. It's a cylinder. And we want, the first thing is that, suppose we want to calculate the one point function. The red boundary corresponds to the inside of this red triangle, the space inside this red triangle is the decider space, is a Poincaré patch turned on its head. The Poincaré patch itself, by the way, would be the same triangle but reversed. So here you see, it describes contracting universe. This is the past. And the universe is big. And in the future it becomes a point. So this is the contraction. This, of course, people never consider this incorrectly because it's hugely unstable. It's not global decider is stable but contracting Poincaré patch is hugely unstable. Even classically. But we see a strange thing which I don't know how to interpret but that's a calculation of fact. That if we want to know the one point function inside the decider universe or two point function with not large separation, we can do the calculation in the contracting universe because not in the global. And that simplifies things. Because when you calculate using Schwinger-Kelder's technique, this technique, unlike Feynman's, is causal. Namely, there are only all the vertices of Schwinger-Kelder diagrams contributing to this one point function will lie all inside the past light cone. But this past light cone lies inside the boundary of the anti-Poincaré patch. Anti-Poincaré, I mean, the Poincaré would be obtained by changing the future into past. And that gives us some calculation advantages because it's very easy. Poincaré metric is, of course, very simple to deal with. And you can perform the calculation. I will only show you... Yeah, I will give you just the spirit of this calculation. You can formulate those... Schwinger-Kelder's rules by using arrows. I called it slings and arrows of time. Each particle has its individual arrow of time. And I hope the quotation is not lost here. You recognize some hidden quotations. But anyway, if you don't, don't worry, it will not impede your progress. Anyway, the rule is that this is the green function and you associate with outgoing arrow with H, with incoming H star. And you calculate diagrams. I will probably skip this part. It's just straightforward calculation of these Feynman diagrams. The important thing is that you get the scaling behavior like that. And as a result, I will give you the result. The result is that the correction to the green function... The correction to the green function contains two terms, which are both logarithmically divergent. They contain explicitly... I should have said this first. When you calculate those diagrams, you see explicitly logarithmic infrared divergence. Again, there is an unsolved problem of summing up in the leading log approximation of all these divergences that have not been done. But the divergences themselves can be interpreted in a nice way. They can be interpreted as the fact that, first of all, you obtain some nonzero because of them. The correction to the green function is such that it can be interpreted as having some nonzero occupation number and some rotation angle. So, matter of fact, the so-called alpha vacuums have been considered before by people and without any... just as possible solutions, possible green functions, not related to this calculation. And they have some weird properties. But here I have... And so because of that, people, as I understand, basically never use them. Here I have no choice when you just do the calculation to obtain those alpha vacuums. Here it's like... This is the logarithm I obtain. Epsilon is the infrared cutoff. Tau is the conformal time. And as I said, we have the fog space over this vacuum. And that's basically the answer. But it shows that back reaction is... if you calculate the matrix elements of the say, energy momentum tensor, the back reaction will be large. Okay. There is some hint that something strange is going on. When you calculate... When you take these first corrections and you do the calculation of singularities on the horizon, you obtain a singularity not at the horizon, but at the antipodal points which are related to the horizon. In any case, there is some topic to be explored. There is certainly something going on. There is this singularity at z equal minus one, which I sketched here, but it's probably we don't have time to discuss the way to obtain these singularities. That's just an interesting point. Yikida, when I started this... What? Oh, good, good, good. Now we will cover turbulence in the standard model. Not yet. So this is the brief picture. There is a picture of what happens in the leading log approximation. As I said, I haven't solved this problem, but I know which diagram, the interpretation of the diagrams which are giving these leading logs. And these diagrams are in remarkable... They have remarkable similarity to the diagrams in the theory of turbulence. Basically, the ansatz, which goes through those diagrams, is this occupation number which depends on the logarithm and some rotations, these alpha rotations, which also depend on the logarithm. And you have to write the kinetic equation, and this kinetic equation contains various vertices. Basically, what is responsible for the instability of the decision space is this A-type vertices, because it is just another picture of a spider, I mentioned, of creation from the vacuum. And this is just an example of the logarithmic term which you obtain from the kinetic equation. Unfortunately, it's not solved, so I don't know the answer for those occupation numbers. Let me just finish with the... Maybe not quite finished. What it all means for the standard picture of inflation. In the standard picture of inflation, we have to consider the... Well, the simplest model of inflation is the model where you have gravity, you have the Einstein term, and you have a scalar field there in floton. And one assumes that the cosmological constant is zero. So one assumes that, say, v of zero is zero. And then one finds some self-consistent solutions for classical solutions for this system. The question which probably should be raised is that why... There is always the lowest, the most stable solution of this equation, namely the Minkowski vacuum with a field equal to zero. In other words, nothing. So why we have... Who and why pumped this field so that it started in the excited state? Why should it be started in the excited state? That's slightly philosophical question. It's not a really... You can really ignore such philosophical questions, but in a different mood you may ask them. So that's some problem, I think. Another problem which I will not mention is that those instabilities we discussed before will be present in this model also. But in fact, what I just wanted to stress is that the calculations we did showed that actually the... If you add the cosmological constant, and it should be added somehow, then there is no choice. The system must evolve. It cannot be staying in the decider. The decider space is not sustainable. It will be decaying. It will be producing various instabilities and so on. So it's in contrast with this picture with the zero cosmological constant. So in this case it's known that Einstein said that his biggest blunder was the introduction of the cosmological constant. And now we see that he was right because it was the cosmological constant which created the universe. So I think I shall stop here. If we go to turbulence, it will take us another seminar. Okay, I shall stop here. Questions? What is the interpretation of those parallel singularities? From which? From the parallel singularities. The production of massive particles. Yes. So could you think about those singularities like linear recognition, degeneracy, and then it's so, degeneracy of what? Well, you see, those interesting letters there just like they indicate that they are secular. They probably should be called secular terms, actually, because they simply means that you, as a system, well, let me, let's look at the simplest case when you produce massive particles. When you produce massive particles, the current, as I said, becomes proportional to time. It can be interpreted in different ways. For instance, you can say that you form, if you produce particles of fermions, they tend to form a fermacy. And in the fermacy, you get some infrared divergences coming from, you have gapless excitations as a result. But in the, basically, I would say that, like in all cases, those secular terms are indication of instability. Like, for example, in normal mechanics, secular terms arise because you have some resonances, and resonances can lead either to chaos or to decay or to various things. So that, and that's purely kinetic phenomenon. As I said, there's no infrared divergences or secular terms in the just standard in-out amplitudes. It's homological. We need to pay attention to this part of that. What? All right. I want to ask about another singularity, or maybe it's the same singularity. You mentioned the singularities which are formed close to the horizon. Could you... I don't know. I mean, could those singular... First of all, the question is, are those singularities behind the horizon? Well, let me... It's a bit complicated. And another part of the question is, could there be a hint there at the final state of the decay? Because you didn't tell us what is the final... Yeah, if you think that I keep it to myself, you are wrong. Well, maybe I missed that. That's another way to put it. And yeah, that was... Yeah, well, that's a tough question. You see, what I... So far, I just strictly speaking what I see is just the singularity. I started with the green function, which depends on this variable z and n prime. This green function has the so-called bunch data's green function. It has singularity on the coinciding points. Now, it turns out, and the reasons I don't really know, don't really understand very well, that there is another singularity at z equals minus 1 is generated. It is actually the singularity if you use the Poincare coordinates. Then I think you can calculate that it is singularity when two points... Then one point crosses the horizon. If you have two points, let's take this here, then you get a singular contribution then one point... You have singularity when they coincide and then one when you cross the horizon. So you need to cross the horizon, too, right? Yeah, you see, I am very back about it, but I really noticed some interesting calculational things, which the details I will display to your pleasure. But I don't have any intuition of what it means, whether it means the fire. It sounds superficially like fire walls, which people are discussing, which I think they are discussing in the wrong way, but I don't have much to offer instead of it.