It's infinitely sad that somewhere in the corner there is 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, or 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 is 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 we have. 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 Planck 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 Planck size. We will obtain the answer that it is of the Planck 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 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, I, 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, then the system develops. It's not, not remains the same, are defined like that. And they do satisfy operator product expansion and all the usual properties.

However, if we try to calculate the F matrix, suppose we do, we want to calculate the F matrix. For that, we have to take some time to, sometimes up to plus infinity, sometimes to minus infinity. And then our field theory textbooks would tell you that the F matrix

is the result of this limiting procedure modulus to some uninteresting factors is just the F matrix, which satisfies, and that's unimportant,

which satisfies usually, 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, that while the Green functions,

which satisfies equations of motion and all the usual properties is the ratio of these two matrix element, the S matrix should be actually, because of this, the amplitude we obtained 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 field theories, 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-Kelash technique, which replaces Feynman diagrams in the non-equilibrium case, the technique which provides us with in or 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 obvious in the stable case, then 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 face

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 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 phenomenon, 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 this 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,

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 x component of that 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 or two waves running in opposite direction. And we calculate the induced current.

And it's very simple, this induced current. I forgot to put the reflection coefficient here, which is written here. So there is a coefficient in front and there's 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 step function and it will be zero. However, the in-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 world. 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. Thank you. That we can do the calculation in two ways.

As I said here, the calculation was done in the gauge, A e t, A zero is zero. In the second gauge, we can take A zero equal to e x,

A one 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 diabatically turn the approach to the constant electric field.

Turns out that in the non-equilibrium situation, another manifestation of this sensitivity is that the answer depends on the order of limits.

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. And now the 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 modified Kaczkowski rules

for the Schwinger-Kelzler 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 a matrix element, they are full of them. They are full of infrared divergences, strong back reaction to run,

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 there in a higher dimensional space, say. Normally we teach students that we have to look on only in-out in order to describe physical processes.

Why are we looking on in-out? Yeah, because you never told the students that they are living in expanding universe. When you uncover this surprise to them,

they will have to change. What? 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 Migdal 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,

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 a 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. Okay, but we don't have time for this. Let me go to the case.

I still remember the name, the excitonic insulator. Let me now switch to the case which really interests me, and this is the de Sitter space.

That's of course of very, the de Sitter space is highly interesting because of cosmic acceleration. And it's defined, it's basically analytic continuation from a sphere.

Yeah, 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, there are no surprises there. It's stable, perturbation theory is working well, and nothing happens. So one might think that we can do the calculation

in the de Sitter space, which by the way, mathematicians, mathematics terminology, de Sitter space is the image, it's kind of anachronistic, it's in fact, imaginary, a Blaschevsky space,

the anti-de Sitter space is the usual Blaschevsky 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 de Sitter space. We can try the same strategy with the black hole. You see the black hole has a very similar in many respects metric, very similar metric.

And in the Euclidean black hole, you have to consider only the space, there is no such thing as a region inside the horizon.

The complete space is just r greater than two m. And this is geometrically, it's a 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 some 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 sign because everything is stable.

No corrections to the mass of the black hole. So now, also this metric which I wrote before,

which was nice, complete Euclidean metric, it's a geodesically 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 prepare some unphysical conditions, in which this Euclidean metric would describe physics, I called 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 or 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 this 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-Kaldesh technique. You have the time contour, you integrate along the time contour, which is here. So the Schwinger-Kaldesh 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 Kadanov contour.

They actually assumed in the 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.

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 for n minus and leave the condition for n 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 geodesically complete

and what we can do it by considering the following sandwich. Let's look at what happens if we're, 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 De Sita space. It's easy to see, that's just, of course, the final 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 wave appears after, well, 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 a safe haven of the Minkowski space. Then there was exponentially 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 same. So 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'll find that the beta is given by the Bose formula, with the temperature, with the Gibbets-Hulking 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, it's what's called the information paradox.

But there is no paradox here. The paradox is resolved very simply. Indeed, this part does not correspond to the pure state. However, the full Green function is, it does correspond to it. It's just in the pure state,

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 we'll be looking as if information is lost. The observer is full of it. Of course, yeah, yeah. Certainly not the fault of the space, space is above any fault.

Okay, now we have another feature here, what we expect now. We expect that we created 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. 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 lunchtime. But there is this continuous creation

which will prevent them. 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 theta space. But just because of the infrared shift, the particles become non-relativistic. So that's what is written here. 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-de Sita space. And this blue shift in anti-de Sita space 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 space, but we also have open strings attached to the boundary. And these open strings, they basically have

only massive modes. All massive modes go away. And so this is for the anti-de Sita. For the de Sita, we have infinite red shift. And that perhaps creates a 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 de Sita case. By the way, I should have started, should have said already that classically, the stability of the de Sita 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, 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 de Sita shows

is that this is basically the expression for this diagram. Then this is done for the Banj Davis vacuum but for all other states it's even worse. The Banj Davis in a sense is the most stable of,

is the least unstable vacuum, I would say, put it this way. And it turns out that indeed this is, there is a non-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 is, of course, very well known.

The solution is the global de Sita 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, undefined, small.

Let's see what happens to this de Sita 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 are 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, that's to say, this phi square must be constant

because of those, all points of the sphere are equivalent. If you calculate, if you calculate, say, 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.

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 deciduous space, is a Poincare patch turned on its head.

The Poincare 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, contracting. It's not, global deciduous is stable, but contracting Poincare patch is hugely unstable class,

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 deciduous 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-Kelters technique, this technique, unlike Feynman's, is causal. Namely, all the vertices of Schwinger-Kelters diagrams

contributing to this one point function will lie all inside the past last light cone. But this inside, this past light cone lies inside the boundary of the anti-Poincare patch.

Anti-Poincare, I mean, the Poincare would be obtained by changing future into past. And that's, that gives us some calculation advantages because it's very easy.

Poincare metric is of course very simple to deal with. And you can perform the calculation. I will only show you, yeah, or I will give you just the spirit of this calculation.

You can formulate those Schwinger-Kelters 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 quotation in this. 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 H with the incoming H star. And you calculate diagrams. I will probably skip this part.

It's just straightforward calculation of these Feynman diagrams. 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 can be, contains two terms, which both are logarithmically divergent. They contain explicitly, so probably I should have said this first,

when you calculate those diagrams, you see explicitly logarithmic infrared divergence. And again, there's unsolved problem of summing up in the leading log approximation

of all these divergences that's 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 non-zero because of them,

you obtain the correction to the green function is such that it can be interpreted as having some non-zero occupation number and some rotation angle. As a matter of fact, the so-called alpha vacua 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 used them.

Here I have no choice when you just do the calculation, you obtain those alpha vacuum. Here, this is the logarithm I obtained.

Epsilon is the infrared cutoff, tau is the conformal time. And as I said, we have the Fock space over this vacuum. 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 this first correction 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 is, 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.

Ikita, when I started, it's, what? You have three more minutes. Oh, good, good, good. Now we will cover turbulence and 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 deciduous space is this A-type vertices, because it's just another picture of a spider,

I mentioned, 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 finish, over there, what it was,

all means for the standard picture of inflation. In the standard picture of inflation, we have to consider 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 flat on. And one assumes that the cosmological constant is 0. So one assumes that, say, V of 0 is 0. And then one finds some self-consistent solutions

for classical solutions for this system. The question which probably should be raised is that there is always the lowest, the most stable solution of this equation, namely the Minkowski vacuum with the field equal to 0.

In other words, nothing. So who and why pumped this field so that it started in the excited state? Why should it be started in the excited state? It's a 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, they 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 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 de Sita. De Sita 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 cosmological constant which created the universe. So I think I shall stop here. Well, if we go to turbulence, it will take us another seminar.

OK, I shall stop here. What is the interpretation of those infra-ringularities?

Of which? Infra-ringularities, as you mentioned. The production of massive particles. Yes. So could you think about those infra-ringularities like Linn-Albrecht initiative, degeneracy, and then the salt, degeneracy of what? Well, you see, those infra-ringularities,

they're just like they indicate that they probably should be called secular terms, actually. Because they simply means that you, as system.

Well, 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 produced particles are fermions, they tend to form a Fermi sea. And in the Fermi sea, you get some infrared divergences

coming from, you have gapless excitations as a result. But basically, I would say that, like in all cases,

those secular terms are an 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. And that's purely kinetic phenomenon. As I said, there's no infrared divergences or secular terms in the just standard in-out amplitudes.

Some are logical. We need to pay attention to this part. What? All right. I want to ask about another singularities, or maybe it's the same singularities. You mentioned the singularities which are formed close to the grison.

Could you, I don't know. I mean, first of all, the question is, are those singularities behind the grison? Well, let me. It's a bit complicated. Or, 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 another way. Well, that's a tough question.

You see, so far, strictly speaking, what I see is just the singularity. I started with the Green function,

which depends on this variable z and n prime. And this Green function has the so-called Banjedev'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 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 are set, 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, then they coincide, and then one when you cross the horizon. So you need to cross the horizon to the right. Yeah, you see, I am very vague about it.

And I really noticed some interesting calculation of 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 firewalls, which people are discussing, which I think they're discussing in the wrong way. But anyway, I don't have much to offer instead of it.

OK, let's thank Sasha again.