Time & Deep Structure of Dynamics — Part 2

0:00 And this again is not really controversial. I'm trying very hard to tell you when I'm saying something which is controversial as what is not controversial but is not normally expressed in the language I've been using. I've given you a picture here. There's a very nice model of what shape space for triangles look like. I told you it's a two-dimensional surface, so there's a very nice way of representing it, the surface of a sphere. We had the mayor of Kirchberg saying that, what did he say? On a sphere, nobody is far, what was it? Anyway, any point on the sphere is important. Yeah, democracy of every point on the sphere. Kirchberg is just as important as the end. Now there's a very nice representation. Each point here represents a triangle. So the north pole, the triangles in the northern hemisphere are mirror images of the triangles in the southern hemisphere. So if you go from any point here down to the mirror image south of the equator, you will find the mirror image of the triangle you're talking about. The north pole is an equilateral triangle, which is the most uniform state, and the south pole is its mirror image down there. And the equator are the points where all three particles are on a line. It's the collinear configurations. and then there's these, it's studied there are points on here 1, 2 means this is where particle 1 and 2 have collided they're sitting on top of each other and the other particle is somewhere else but that's where 1 and 2 and that's 2 and 3 and 1 and 3 and these are points, I won't go into any detail about them but these are famous points that Euler discovered and they play an important role in the three-body problem. And then this line of longitude going down from the north pole through this point here where one and two coincide, these are isosceles triangles. So there's three lines of longitude of isosceles triangles. So that's a very beautiful mathematical representation

2:30 of shape space for triangles for the three-body problem. And some beautiful work has been done using this by an American mathematician called Richard Montgomery, who has actually helped me in my work. Now, I'm sure you've all heard of wave functions that Schrödinger introduced. Very appropriate. Look how the Austrians are dominating this discussion, Bach and Schrödinger. Schrodinger of course made well I think actually Schrodinger made the biggest revolution in physics by a long long way and I think the Copernican revolution just pales into insignificance compared with Schrodinger's revolution which was the introduction of a wave function now normally you talk about an electron having a wave function which gives the probability of where you will if you try and find where it is but Schrödinger knew from the word go and interestingly he hated it that his wave function was much more significant his wave function is not for individual particles if you have a three body system it's a wave function which gives you a value of the wave function for each possible triangle each possible triangle has a value of the wave function so classical physics is talking about curves shape space. Quantum mechanics of the universe is talking about a wave function defined on the whole of shape space. Every point in shape space has a certain value of the wave function if you naively try and apply Schrodinger's ideas to the universe as a whole. And Schrodinger found two equations. He first of all found an equation which is called the time independent Schrodinger equation because time does not occur in it. He thought it was describing a stationary state. He thought that in the stationary states of the hydrogen atom that he was describing there was a tremendous amount of activity but overall the state was what physicists call stationary. not change its shape. The distribution of the wave function was unchanged. And the first thing that he found was the equation which describes that. And that equation, in

5:00 principle, describes every single structure you find in the universe. Absolutely staggering. Molecules, well, that's a bit of a... I don't think that's far short of the truth. Every single structure. The structure of every molecule, DNA. I mean, this history goes into how the DNA molecule comes. But basically, that equation is about how structure is determined. Staggering, absolutely staggering. And then, a few months after he got that equation, and with some prodding from people like Lorenz, he found what's called the time-dependent Schroding which describes how the wave function evolves in time. And when they were creating quantum mechanics, and still to this day, quantum mechanics uses a pretty old-fashioned Newtonian notion of time. People doing quantum mechanics have not really learned how to get rid of the notion of time. Now, in the quantum mechanics of the universe in a Leibnizian situation, it is absolutely nonsense to talk about the evolution of a wave function in time because there just isn't any time in which it can evolve. That's not part of the picture. Now, I'll give you two quotations from Feynman, both of which are wrong. Somebody pointed out that everybody likes to quote Feynman because they think that gives them authority. If Feynman said it, then it must be right. But I'm going to give you two quotations from Feynman the first one is very funny Feynman said on time time he said is what happens when nothing else does and that's just certainly not going to be accepted by Leibniz and Muck you can be sure of that and in this case I would rather have Leibniz on my side than Feynman the other one was he said that if you describe the universe if you give the universe a wave function it will just be one universe configurations of the universe which is evolving in time and that again is fundamentally wrong in my view and this by the way is not really controversial, there's a thing called the canonical approach to quantum gravity where this has been recognised that, and this was

7:30 there's a thing called the Wheeler-DeWitt equation which was discovered in 19, oh I can't remember now so I won't bother to find it, but certainly quite a lot 35 years ago or more by Bryce DeWitt probably by John Wheeler called the Wheeler-DeWitt equation, in which time does not appear at all. So the quantum mechanics of the three-particle universe will just be a distribution of the wave function on there which is utterly timeless, utterly static. Nothing will change. It will be like a distribution of mists on which nothing changes. And because there is a rule which takes you from a given form of classical mechanics to the corresponding form of quantum mechanics there is modulo, a lot of technical issues a unique prescription from going from this dynamics of pure shape suggested by the ideas of Leibniz and Mach to the law that gives you the distribution of that wave function on shape space I could write it down if you could stand the thought of looking at those equations but I'll spare it so that's actually the situation there is no time at all in the questions perhaps we can ask how we might conceivably make some sense of that but I think it's time to stop he says I've got two to three minutes but the train is coming early alright thank you very much thank you very much the paper is open for this I think probably the speakers, the questions should come up to the microphone. Take the mics around. Yes, if you want a scale invariant theory, aren't you going to end up doing to the expanding universe much what Mark did to Copernicus, saying, well, strictly speaking, it doesn't really matter whether we say the Earth goes around the Sun or vice versa, but it's easier to think of the Earth going around the Sun.

10:00 a scalar theory, you could say, well, it doesn't matter whether you think the universe is expanding or we're all shrinking. But it'll just be more convenient to say the universe is expanding. I mean, so the point of my question is, if you could just say, well, it doesn't matter whether the universe is expanding or we're all shrinking, would in fact the scalar theory that you are obviously rather fond of have the sort of consequences you think it would because although the Big Bang wouldn't be a Big Bang anymore, it would still be something that could be as easily described that way as described in terms of us shrinking in a universe of fixed size. No, I can give a definite answer to that. It will have a very profound effect, I can assure you of that. And I can give you an argument for that. And this goes back, interestingly, to Newton's failure to do what he promised in the scolium in the Principia, show how you get from the purely relative quantities to the absolute quantities. Now, if I just give you two successive triangles, in Newtonian theory, that tells you absolutely nothing about the angular momentum in the system. Those two triangles by themselves cannot tell you anything about the angular momentum in the system. You need to see a third triangle and a fourth triangle before the effect of the angular momentum shows up in the relative quantities. So, in fact, there's a very beautiful paper of 1772 for which Lagrange won a very big prize for his three-body problem where he actually did do what Newton called for. But by then Lagrange wasn't interested in the problem of whether absolute space exists or not. He was just interested in solving the problem of the three-body thing. But he actually equations in terms of relative quantities and you can see explicitly there that what we call angular momentum and rotation shows up because it interacts with just the separations. You can see the separations change in such a way that they're unambiguously revealing their rotation in absolute space but Mark would say it's their rotation to the distant stars. Now so you can see in some your hand on invisible things from what you can see and this was essentially Newton's argument

12:30 he used the bucket experiment but this was essentially Newton's argument for saying that invisible thing is out there because I can see its effects now when you look at Einstein's theory the fact that it's not scale invariant you have exactly the same effect the the shape degrees of freedom interact with that one solitary volume degree of freedom of the universe and the shape degrees of freedom by themselves do something different from what they would do if that volume degree of freedom wasn't interacting with them and from the way the shape degrees of freedom are misbehaving you can deduce what how the volume is changing just as you can deduce I mean as Poincaré said if the astronomers had a dust cloud all around the earth so they couldn't see the distant stars they could still deduce the angular momentum of the solar system from observations purely within the solar system without seeing anything of absolute space or the distant stars. And the key prediction that these Leibnizian-Machian theories make is that the universe will not have any angular momentum and in addition, there won't be any obstreperous size variable interacting with the shape degrees of freedom. So there will be very definite predictions, no question of it. But the theory doesn't exist yet. Well, we have a theory, which is scale invariant, but we don't think it's the correct one, because it has an element of action at a distance in it. I've qualms about instantaneous states in a relational view. I mean, instance would be absolute entities, like points of time. Maybe you don't have instance, but it seems at least that your view of duration would be absolute. is um well first of all definitely the instant of time leibnitz talks about a plurality within a unity something that's held together the instant of time is my triangle or my shape of my triangle that is the leibnitz would call that the true atom of existence because that's indestructible the shape of the equilateral triangle has not changed since euclid was around you know That's something that is absolute within the scope of mathematics. Now, dynamics, as I have developed it in this talk, is based on taking that as the absolute foundation.

15:00 Those are the atoms of existence, the shapes of the universe. And it's very interesting that this is still actually the way the dynamics of general relativity works. Although action at a distance and instantaneity is not meant to have any meaning in general relativity, that's actually quite wrong. Four of the fundamental equations of Einstein's field equations are action-at-a-distance equations. They're called the initial value constraints. You can't send any signals with them, but they nevertheless determine things. They are crucially important parts of the equation. So, does that answer your question? So, you don't define these shapes as instantaneous, but just introduce them or enumerate them. These are the... Well, that's my starting material. That is my starting material. I think the human mind has a wonderful ability to conceive shapes. We wouldn't be able to do geometry if we couldn't. I would like to bring forward two short questions. First of all, your statement, you believe time doesn't exist at all. But you can observe, as an empirical fact, that there is cause and effect. In other words, you observe something before and something later in physics, and I will leave out philosophy of time. So only the concept of time in physics. How could you base earlier and later in physics on something if there's no time? That's the first question. And the second one pertains to your, I think, perhaps plausible, skeptical view on the Hubble constant. Actually, the distances in the universe are measured by this measurement. Now, if you have a singularity and a big bang, and you take now the measured distance of the farthest objects, about 13.7 billion light years away, but in two directions,

17:30 Are these distances correct if it's to be assumed that on one point, no time, whatever, all matter or energy were together? Perhaps let me just start with the second question, if I may, because that's quicker and easier to answer, I think. you shouldn't think about if you take scale out if you take scale out of a triangle it is meaningless to talk about all three particles being on top of each other three particles sitting on top of each other is not a shape it does not belong to shape space now in the Newtonian three body problem terrible things happen when all three particles collide and hit and come together that's called the three body collision and it's the most interesting topic in the three body problem but it's an immensely difficult one. Now, if you go to shape space, so the singular point, every configuration space, particularly these relative ones, has a distinguished point. And in Newtonian theory, the distinguished point in the configuration space, which really governs, dominates everything, is where everything is sitting on top of each other. In the three-body problem, it's a three-body collision. Now, in the shape space, it's completely different. triangle becomes the most distinguished is the distinguished point it's the most uniform state and it has no meaning to say that it is sitting on top of each other and there's a very interesting fact the thing that really counts in newtonian dynamics is not the newtonian potential but the newtonian potential made scale invariant dimensionless by multiplying it by the square root of the moment of inertia that gives you a quantity which is a pure number and only pure numbers dimensionless numbers can have meaning in physics and that quantity has an absolute maximum at the equilateral triangle so the most uniform state is the one there is a that is very interesting property about configuration space is that there is a distinguished point and in in in in the Einstein standard theory it's where everything is sitting on top of each other that's where the big bang starts it is a completely different picture although it looks very like it i mean the einstein big bang cosmology starts off extremely uniform with everything close to each other in the scale

20:00 invariant theory everything will start off very uniform and it will have no meaning at all to say how close or how far they are apart because that isn't a concept which has any meaning when you're talking about pure shape so that's the answer to to i hope to that question um i i could perhaps say i didn't say in my talk is that both notions of distance and time can be recovered as emergent quantities which emerge from an underlying dynamics of which they are not a primary part now as to your question about how we can talk about the past and why we think we had a past and why earlier and later not past earlier and later okay um you certainly couldn't have any concept of earlier and later with just three particles, but give me billions of particles, then what look like records can be formed. I've got a picture in my wallet of my wife taken about ten years ago, which I carry around with me, just to remind me what she looks like. And when I look at that, I see a structure, and that matches with a in my brain, a shape in my brain, which is my recollection of what my wife looked like ten years ago. And if here there are structures which have these, what I would call, mutually consistent records, then I will say there was an earlier time, if they all conspire and all are so wonderfully concordant, that the easiest interpretation is that there was a past, there was an earlier and there's been an evolution to there and I'm now here then that's okay and that's, so you need a lot of structure but all of the evidence for the past is coded in structure, it's coded in records. Now my conjecture about what happens in static quantum cosmology is that because of this remarkable shape of the configuration space that I've been telling you about with its distinguished points The wave function, the Schrodinger wave function, the static one, is not at all distributed at all uniformly over the shape space, but that it picks out configurations of the entire universe, which I would say just reek of a past.

22:30 They carry, I call them time capsules, they seem to be loaded with mutually consistent records which tells you there was a past of this configuration, that this configuration had a past. Now, epistemologically, that's all we know about the past is from records. We do not, we cannot put our hands on our hearts and say that Henry VIII had six wives. All I can say is every history book I read tells me that Henry VIII had six wives and I see portraits of them and so forth. But whether he really did and so forth, I don't know. That's all here in the now. Everything we know about, that we say about the past is here in the now, in a very richly structured now. Well, I wasn't so sure about the connection between Mach and Leibniz you were. I think it's worth it. The connection between Mach and Leibniz you are drawing. I mean, you're also someone who went on record saying general relativity is perfectly Machian. I see it as very Leibnizian when he said, well, there's a distinguished role of the equilateral triangle in your structure theory. I mean, that's very much Leibnizian. Maximization of determination, so one way of saying the best possible world. But this way of arguing with possible worlds and with maximization of determinations, it's not so much machien, because, I mean, you had the quote on your transparency that Mach rejected any argument by way of possible worlds and told us the world is just the ones and it might well be one of these triangles you showed us and there was no way of an argument that the best possible world or any world is the one corresponding to the equilateral triangle. and the same is how they interpreted the principle of least action that Mach rejected talking about the possible worlds and of course Mach would also have insisted that our concept of time has to do with our biological physiological experiences of time which was other way of talking about it that wasn't really meaningful for a Leibnizian Yes, I think a lot of what you said is totally fair make one point the equilateral triangle is not the best of all possible

25:00 worlds, the best of all possible worlds in my view that distribution of the Schrödinger wave function which gives high probabilities to structured things, if you read the monadology, Leibniz when he gets really mathematical, does not But we live in the most varied of all possible worlds. Article 58. I've actually, with Lee Smolin, written some papers on realising this idea of Leibniz, that we live in the most varied of all possible worlds, which is mathematically formulatable. and I think that there may be quite a connection because it is a fact and we don't fully understand it that the world is wonderfully structured I mean Professor Thierry will be talking about this I presume on Saturday but generally Mach was not at all sympathetic to a lot of Leibnizian ideas he hated a priori ideas he was very sceptical about the principle of least action he didn't like theory and things like that The reason why, but he, I think there's two things. First of all, there's a deep intuitive feeling that only the things you see are real. And of course, Mach took this to idealism like Berkeley. I mean, he says quite openly, the really fundamental things are sights and sounds and sensations and tastes and things like that, which is way beyond what any modern scientist would go for. That's, I mean, that's one aspect of Mach. I've lost the, and, but he had that gut, feeling, which, why have so, I mean it's very interesting you just get this polarity between people who are totally happy with a vast expanse of uniform space absolute space, and those who react against it, and I would say through the history of thought it's a respectable minority, something like 30% of thinkers have been Leibnizian in that respect, or Machian in that respect, saying only relative things count, and people like, and the others have been more going in the other direction but the great thing that Mach did was that he came up with a plausible counter to Newton's evidence for absolute space this was what Einstein called Mach's principle although Einstein I think made a bit of a muddle of these things not in creating general relativity which is a wonderful theory a lot of what he was doing there he knew exactly what he was doing and got it spot on right but what Mach said

27:30 was that inertial motion is not inertial motion relative to absolute space it is somehow caused by a very much sort of Newtonian force which actually guides things and it's the distant masses in the universe which are guiding things in their local motion those are the things that are guiding local motion and that's why they seem to be straight lines relative to the distant stars now that was a huge insight and nobody had it before Mark Leibniz, when you look at them closely, they didn't have an inkling of that. Actually, the only person who had a little inkling of that was Newton. There are one or two passages in Newton where he has a little hint in that direction. your curve is continuous Big upon? It's working. your curve of the points is continuous and if it's If the dynamical change is continuous too, then I think it is necessary that some points are to exist as fixed ones. If I am right, what status of such fixed points is? No, I don't think there's any reason to... You mean that this comes to an end? No, in the middle. No, well, first of all, it's just there. This is not a path that we're walking along and coming to a stop. It's a path that exists out here. When I read the leaflet about Kirchberg, it says there's 821 kilometres of footpaths that you can walk in the area. They are all out there without anybody walking along them or stopping to have their lunch. Is that what you're trying to suggest? Correct. I think there's some fixed point, I think. You have to assume some fixed point. Well, I mean, there's a turning point. There's a point where, in Newtonian theory, you can have a trajectory which comes to a point and then comes back again. These are the turning points where, in quantum mechanics, you have tunneling. Is that what you're meaning? Well, I don't think that does happen in this theory

30:00 because basically I'm almost certain that the shape space dynamics will be chaotic and every point in the configuration space will be visited. It's very interesting. Angular momentum, I don't know how much you know about the three-body problem, but there are things called precluded regions, hills regions in the three-body problem where the angular momentum stops the system reaching various points in the configuration space. These are the tori of Kolmogorov, Moser, and whatever the chap was. And then there's the turning points where the energy is zero. Where the kinetic energy becomes equal to the... Where the kinetic energy is zero. Now, those are what you call fixed points. Those cannot exist in a theory like this because in a sense, there's a very deep sense in which the energy is exactly zero, the angular momentum is zero, and therefore I don't think these points will exist. So I think that such the fixed points are connected with so-called physical constants. Yes, yes, and all of the constants, I didn't say this in my talk, but all of the physical constants are exactly zero in these theories because they would be at a front against the principle of sufficient reason because there's no reason why the constants should have this value rather than another value. Thank you. Thank you. Thank you. Is somebody there? Thank you. Not the next one. Yeah, over there. The events list is the next one. Ah, no, there was someone at the back there. Oh, yeah. I had a question also about the geodesics, maybe similar to that one. If there is no absolute time-space, what's the, I mean, what's the meaning of the line being curved and not, say, straight, this kind of geodesic you drew? what i want to say is that i see it makes a difference if you look at space globally then whether space is a sphere or a donut you that has some explanatory power but if you look at it locally what stops us from mapping a local space which is curved by some transformation not a linear one onto a plane and if we can do that I mean since there is no absolute time and space this is just a model so it seems to me

32:30 why not do it and then what's the meaning of that line being curved what's the explanatory power of saying that this line is curved I said everything here rests on geometry the things that ultimately into this theory are as indestructible as the natural numbers. Descartes showed us how we can build up all of the facts of Euclidean geometry from pure numbers. So those are indestructible. Now what I'm defining is a metric on shape space. I'm defining a metric on shape space and that is as indestructible as the real numbers are. When I've got my point here. When I say that the distance from the north pole to this point here along the geodesic I'll give you a real number for that. It will be, if you like I don't know, I haven't done the calculations, but it might turn out to be 17.3 and nothing can change that. That is an absolute quantity and it's as absolute as 13 as the prime number. So this is defining a dimensionless metric on shape space which way by transformations as long as you don't change the underlying thing you're talking about I'm talking about shapes in Euclidean space or Riemannian three geometries but only the conformal part of it, the part that describes the shape I'd like to ask for a little bit of philosophical clarification about two statements which seem to lie behind your talk or seem to be important to One of the statements is absolute time doesn't exist and the other statement is time doesn't exist. And I'm a bit puzzled about what connection you see between the two statements. It seemed a little bit towards the end of your talk as if what you wanted to say was, well, timelessness, which might make us sympathize with the first Trillinger equation, is what a thorough Leibnizian should really want and what Leibniz should really have wanted. But I'm not so sure about that. Leibniz might have said, well, of course there is such a thing as time. Time is an order of successive states, and this order exists.

35:00 So it's not clear that relationalism explains away time, but it's a theory of time rather than a theory of explaining away time. the connection between those two things you picked up the distinction between my saying there's no absolute time and there's no time very well and the reason I make why I said both of those things were that I didn't make clearly the distinction that the change comes with Schrodinger's revolution when Schrodinger introduced the wave function he completely changed the way we think about the world there's just, if quantum mechanics is really fundamental I do think Feynman was right when he said the universe has a wave function. At least that's a conjecture. I mean, of course, this is a big conjecture, whether you can extrapolate it to it. But it's the only immediately obvious one. And it's very interesting that a huge range of people have conjectured that the universe has a wave function. It's a standard part of a great deal of modern research. Highly respectable scientists are following along this line because they see no alternative to it. Now, the Leibniz didn't know anything about the shredding of wave function. That was truly forced upon physics by empirical facts, the amazing facts of atomic physics that slowly became clear over 25 years. The Leibnizian idea that there's no absolute time is summarized in the notion that there is a curve like that. now moreover which I didn't go into once you've got that geodesic curve there is a way in which you can recover something which looks exactly which is indistinguishable from Newtonian absolute time but it is derived from those more primitive concepts that I've put into it when I say that there's no time at all I'm referring to that static wave function on the universe where there are no curves but just a mist of that wave a static mist defined space of the universe I have two more people to speak I think we should stop I have two more first you and then I wasn't quite clear about what you meant by saying that your theory had a greater predictive power than Newtonian theory especially since you

37:30 said things like it recovers certain parts of Newtonian theory which is a much weaker statement then you say angular momentum so you can kind of account for turning things so it's weaker in that respect too and you also said there's no not just an explanation for redshift so what did you actually mean by it has a greater predictive power oh right um yes i i couldn't cover every point you you certainly were what i should have said is that the leibnizian machian theory is much more predictive than Newtonian theory if you're talking about the complete universe and this was the point that the the question there was clarified there according to the Newtonian theory of a globular cluster it can have angular momentum and it can expand or contract overall the Machian-Leibnizian theory says that is not possible there will be no angular momentum for the complete universe however that doesn't stop in any way subsystems of the universe having non-zero angular momentum so all of the observed facts of newtonian theory can be recovered exactly for subsystems of the universe the predictability applies to the total universe now when you when you come to einstein's theory the predictability is far far stronger because in some senses these entire universe are translated locally and there's analogous conditions to that global condition that holds for the whole universe that hold at every space-time point. And those are actually precisely Einstein's field equations. Those are the Einstein's field equations, exactly. So in that sense, it is more predictive. Did I completely answer your question with thank you I wonder about your conceptions conceptions of closed and infinity isn't it possible that the universe will be in the same time closed and infinite it means if you want some wording for that I would say that it will close if there is nothing outside the universe and it will be infinite if everything the universe. And the second suggestion is about the time and the early and later. I think we confuse

40:00 duration with time. There is a duration and we measure the duration in time, but the duration is not time. Yes, can I answer the second question first and I may have to ask you to remind me of I would say there are at the level of the talk I've been doing there are two aspects of time in classical physics where we talk about curves there is just that continuous succession and of course you can go in either direction and that is what mathematicians call topological it does not have a metric on it Newton in addition put a metric onto this succession and you can say that what Leibniz was objecting to Now what the theories that I've been describing do for you is that they derive that metric as an emergent property. It's not put into the foundations but it emerges and it is a complete theory of clocks. And by the way, this was worked out already in the 1890s, but not in my terms, by astronomers talking to Poincare. And they introduced something called ephemeris time, which was actually the official notion of time for about 15 years before atomic clocks came in. Now, ah yes, I remember your first thing is, is it more a matter of semantics about talking about the universe? there's a mathematically precise definition whether a space is compact or non-compact and I'm talking about compact spaces and one can think of it being like the surface of the earth that's a compact space, the surface of the earth that's rather different from saying I can go off in all directions to infinity I don't really like that terribly much because, as I say, that leaves something outside. But nevertheless, because of that point I made about Einstein's theory, Einstein's theory is really absolutely miraculous because it does transport those relationships, which in the Newtonian concept only apply to the whole universe, to every single point in space-time. So that in some senses, to verify Mach's principle, you just have to look locally in the solar system and check that Einstein's theory is verified, which is getting now quite impressive,

42:30 particularly if you have the binary pulsars. So all of that is actually, in some senses, verification of the Machin-Leibnizian nature of general relativity. At every space-time point, we don't need to go to the distant galaxies to confirm it. I think we have to stop here. I'm sorry. There were two other hands, I think, raised. But I have to shift this to the restaurant, I think, to the discussion in the restaurant. Thank you very much for the discussion again, and thank you for the paper. Thank you. Thank you.