Limits of QM from GR — A Clash of Principles (contd.) / Universe from Sub-Planckian Bits

0:00 So when I say there's a clash of principles, that's exactly what's going on here. There's a clash, between the principle of quantum linear superposition, on the one hand. And well, not just so much the principle of equivalence, but this sort of Einsteinian point of view would regard the principle of equivalence, which is to regard the 3-4 as the natural inertial frame, and to take some other point of view So we say, take the free fall as natural, and that there's some problem with these principles. So that's a clash of principles I'm primarily talking about. Let me now just say a little bit about other things here. First of all, people might argue, they'd say, surely, quantum gravitational effects are far too tiny to have any relevance at the ordinary scales under consideration, such as cats and so on. I mean, people worry about the Planck length, which is ridiculously different from things in particle physics, let alone ordinary scales, or the Planck time. Likewise, some 20 orders of magnitude below the tiniest particle physics processes. But the thing is that these come about because we multiply together two quantities which on ordinary scales are small, Newton's constant and Planck's constant, each of which is tiny on ordinary scales. and so when you multiply them together you get something ridiculously tiny okay that's right but in this case you're dividing one by the other and then you have to look carefully to see is it small or not it's not obvious so one has to examine the situation in detail to see if this is significant in any particular situation so that's the first point it's something you may have to take seriously The second point is a little bit about, you know, what can I do here?

2:30 Well, I don't have a theory. All I have is sort of what I call a minimalist proposal, which is concerned with quantitative positions of two states, each of which on its own would be stationary. And the idea is to be stationary, you have to solve the Schrodinger equation with an additional gravitational potential term whose source is the expectation value of the mass distribution. And this is what I call the Schrodinger-Newton equation. And then you look for stationary states, which are the negative solutions of this. And it looks as though you have perfectly reasonable things here. And then you take the EG, which is the gravitational self-energy, the difference between them. I should say, you're only using this equation for determining the stationary states. Usually the Newtonian term in practical situations is something you can completely ignore. Or if you like, it just fixes the mass template. It does something fairly minimal. And then you look at the gravitation of self-energy of the difference between these mass densities of the two things in the student position when you get this decay time. I should say it's very close ideas of de Oshio, and stimulated also by work of other people, Pearl, Girardi, Remini, Guervo, Karol Haase, that's what she's having during something. Now, I said this is something that you might not be able to know, so let me say a little bit about the experimental prospects of measuring this. Now, this is something, again, I've talked about many times before, but let me just say it again. There are people who have helped in trying to formulate an appropriate experiment. This is a kind of simplified picture, a general idea. Here we have a source and a beam splitter, and you have to keep both parts of the beam. This one you just, it marks time for a while, you just keep it there. The other one, it does its marking time as well, but by bouncing backwards and forwards on a little mirror.

5:00 of as being about, I should say, this is an experiment which is actually being done, it's being worked towards, I should say, in Santa Barbara by Dick Balmeister and his group, the collaborators are now different from what I wrote here, but never mind, they keep changing, but Dick is the same place. And the idea is to try and do an experiment like this. In their, what they're proposing is that this object here is a little mirror which is about 10 micron cube. So it's about a tenth of the thickness of the human hair. You just about see it. So it's generally macroscopic in that sense. And this photon is a visible light photon. In order to give it enough impact, it has to bounce backwards and forwards. It's jumping up and down on it, and it gives it, if it has something like a million reflections, it can give it enough impact that this object will displace the nuclei by an amount, and I sort of indicated it here, which will be significant, so that the decay time, according to my proposal, would be something of the order of a second, something about like that. So you'll have to keep this thing going for something like a second. So that's a bit beyond what we can do at the moment. You have to get all sorts of things working like a very, very good mirror, probably more than a million reflections, I'm not sure what they're aiming for now, it has to be cooled down, cryogenics are involved, absolute zero, vacuum, all this sort of thing. So it's a difficult experiment, but they are working towards this. They have some problem getting funding, I think it's one of the indices which is holding them up, but nevertheless, they have some temporary funding which is working for the moment. I should finish by saying what's going to happen here. You see, you have this superposition of these two things here, and gradually this thing gets displaced as the photon bounces up and down on it, and so it's now in a superposition of two different locations my proposal says that won't last forever something of the general order of a second it will become one or the other when it becomes one or the other it's entangled with the photon so the photon becomes one or the other too so that means the photon is now either in this or in this

7:30 depending on which way it reduces to and when it comes back again instead of being in a nice position which ideally would mean that it goes back into the laser and the detector here would see nothing. Instead, you get 50% chance of this detector seeing. Of course, this is the simplified version, and in detail what they're trying to do is a lot more detail on this, but this is more the official picture which is in the paper on this. I won't explain what all these things are, I'm an expert on this little thing, but it's all explained in the paper. Now the idea is that you would then have to, if you do see that this thing reduces spontaneously, well of course it might be environmental decadence, decadence of any number of different kinds, So it has to try and pick out the effect that one's looking for here from all kinds of other forms of nucleotide. So you want to reduce them to as small as possible anyway. But having done so, you can do things like vary the... Well, you can certainly vary the mass in each nucleus. You can vary the number of nuclei. This is a more important one in a sense, the spread of the nucleus in the wave function. to know how this thing is, the way functions spread out. The effect depends rather critically on that, so you have to have a good estimate of this. Here you, for instance, if the nuclei are much more localized, you would have a bigger effect than if they're spread out. So this would be another thing you could try and vary. And then the idea would be to see whether the reduction is something which is at least consistent with the proposal I'd be putting forward, or is it inconsistent with that? So at least it's something testable, at least in principle. I should say that where they've got to at the moment is something which, if successful, would fall short of the level that I need by a few orders of magnitude. And the hope is that the experience that they would gain on this would tell them how to pick up a few orders

10:00 However, even at this level, it's something very interesting, because it would give you a superposition which is, ah, no, I forgot them again, I wish to get this here, I think it's nine orders of magnitude more massive than the present record, or is it twelve? That's right, anyway, it's a lot. The present record being the lucky balls that Anton Zeilinger and his group superposed if you do, in fact, get quantum tube positions for these things. But this is far larger. You don't move it as much as the buckyballs move. It's only a diameter of a nucleus, so you couldn't see the displacement. But nevertheless, as far as mass is concerned, it's far huger. Well, it's interesting. There may be other experiments that one could do. And I only mentioned one other possibility that I'm aware of, which would be to use a squid, and the squid one has a superposition of parents going in different directions. The trouble with this is that the mass displacement is virtually zero for two reasons. Well, it's very small because in any case the mass of the electrons in these parents is very small, it's only a surface effect, and there's not much mass. But the second thing is that there's not much mass displacement, because the electrons going one way or the other way, and it's just a momentum difference. It's not a mass displacement. So as it stands, this isn't very good as a test, except that you can confirm quantum mechanics, according to my point of view, with these experiments. They shouldn't give any deviation. But what you might do is introduce some little magnet somewhere, and this magnet would be sensitive to the way that the current is going, and so it would be a magnet which would be put in this information in two locations. So, okay, maybe that's an alternative version. That's not the route that Barmester and his group have been doing, mainly because their expertise lies more in that area than this. But maybe some other people might take this idea up and see whether it's feasible to do I believe some kind of technique, but feasible to do an experiment of just nature, which would test the proposal not going forward. Either way, it would be very interesting because either it shows quantum mechanics still survives in its standard form at a level far beyond any experiment up to this point, or it would indicate something new has to come in.

12:30 or it might give limitations to what kind of proposals one might move forward as changes in front of us. So I think it's very exciting, and I certainly hope that this experiment continues and can be performed in not too distant future. Thank you very much. Thank you. So, of course, of most of the effects we're talking about, we get deviations from standard probability to the first order and in Newton's constant. Yes, sir. But at least in principle, we have a thing called perturbative probability where we can And if any finite order that really comes to the sixth power or whatever, they're completely consistent results. So fulfillment from gravity, as a theory makes sense, is the exception of the extremely high energy immunization effect, with a very small distance effect, where you have to introduce arbitrary numbers that you don't know. So that's just this. Those are just numbers. Once anybody gives us these numbers, to see this exactly what should happen to any finite order in the response. It doesn't mean the theory is right. I mean, this is the point. It's a bit like Andrew's objection, but in a different form. You say, yes, well, look, we can do a calculation in quantum gravity, what we think is quantum gravity, which doesn't give effects that I'm claiming. But the thing is that I would also claim that in some sense those perturbative calculations are not consistent with the principle of equivalence in the sense I'm making that point. But there are many other examples of sequences in, you get a part of it, but there are other kinds of principles of the story, engagement variance, for example. So, just local engagement variance is very similar to, you can also do these things in that case, as an objective experiment. Yes, the claim is different, yes, I mean the claim is that it's People often say, well, you know, why do I say gravity? What about electromagnetism? Shouldn't I again take that force into consideration? Of course, there's two reasons why I don't or shouldn't. One is that I get an answer

15:00 which is probably grossly in contradiction with observation. But that's not a good reason, if you like, because I want a theoretical reason. Theoretical reason is that you don't have a principle of equivalence. So none of these other theories do you have the guiding principle which I'm depending upon. It is similar in many respects, but it doesn't affect, say, the causal structure of space-time. You've got a background space-time for which you do it. But it doesn't give all such spaces only infinitesimally affect. You know, the line's form is wiggling a little bit. Well, no, that's not the point to be regarding it as a more serious thing. But, in effect, if you like, it's a bit like saying, the point I was making before, you know, surely these quantum gravity effects are so small, that why do we worry about that? And I'm saying, okay, they are normally, because we're thinking about it the other way around. You're thinking about how gravity as a force might come in and affect calculations that we do otherwise. But what I'm saying here is, I'm looking at it the other way around, might the principles of general relativity affect the structure of quantum mechanics? And I think there are, as I mentioned, the other reasons, too, the other two reasons, I think, for thinking that the principles of quantum mechanics I'm going to have to change something. And I think the measurement paradox, in my way of thinking, is the major one. But certainly the singularity issue is almost as strong. So I think there are good reasons for believing that quantum mechanics will have to change. And so my point of view is, all right, we look for clues as to where there might be input into how to change those laws. It doesn't say, as you rightly point out, that we can't get away with our present understanding of quantum mechanics and combine that with general relativity producer theory, which at this level is still consistent, which is what you're saying. I think it's a perfectly valid point, but I'm taking a different viewpoint on this and say that that's, in a sense, not what we should be doing. We should be doing something which seeks to look for actual differences in quantum mechanics. So, yeah. So, in your experiment, you imagine this ball, which may be in a superposition, in one position, or displaced, and then a little test particle doesn't really know in which space-time it is. But imagine that the observer is a little amoeba that lives on the ball. So, originally, before the ball would be displaced.

17:30 The observer is a superposition. But then this amoeba will just describe a normal ball, of normal gravitation, only that the test particle is now in a superposition of two locations. And everything would work perfectly fine. Yes, but that's not the situation I'm considering. No, no, but it's exactly the same situation, only described from a different point of view. From that of the a little amoeba, which is saying everything is fine. Which, by the way, cannot be done in the case of fields. cannot be done in the case of the Longo's experiment because there is not just a question of this place. Well, you can think of I thought it was actually I mean, you were talking about a superposition of two mass distributions, but you've got to get them there somehow, haven't you? Yes. So in some sense you put the amoeba could be just sitting on one of the particles of the material Well, we'll see that in the bigger shallow as a molecule. Well, I think one raises the issue of whether an amoeba I would love to be in a superposition. See, it goes back to the many worlds problem. Can you perceive a world which is... See, many worldsists would say, out of me though, if it's allowed to be conscious, or I don't know even whether that's supposed to come into it, we'd only perceive one world anyway, you see, we wouldn't see the superposition. We'd only be perceiving the thing in one place or the other place. I understood that your point was that gravity makes a difference It's not something connected with the observant, which he can't be... Yes, but your amoeba is... Your amoeba isn't going to be the observer. And you will see just the one well-defined gravitational field. There's only one field. No, I don't... You see, that's a different situation from what I'm considering, because I'm considering you're trying to do body mechanics in a situation where... I mean, the situation you're pointing to is driving you into the many worlds viewpoint. Yes, because you're saying there's two perceiving beings, if you like, in superposition. One is perceiving a lump over here, and the other perceiving a lump over there, and somehow it's that superposition of those two perceptions, which is... Both of them, in this case, will perceive exactly the same thing, the lump under their teeth. But what is it? Does it see one lump, or does it see a superposition? What, because their relative distance versus the middle, the big force.

20:00 What would your answer be to this experiment, you see? You say it's in a superposition, it's being held there, you know, for a second. And as you sit there contemplating it for a second, are you, I can't look at it all you, but are you, is your world, has it got the lamp only in one position or in the other position, If I'm sitting on the lamp, on the big ball, the big ball is just right under way, that's in both situations. Only the test particle is in two different places in these well-defined gravitational fields. And then I do ordinary quantum mechanics for the test particle. Yes, but ordinary quantum mechanics, it is because you run into the many worse problems. Ordinary quantum mechanics doesn't treat this. Ordinary quantum mechanics treats it as a superposition, aren't it? There's one thing. At what point are you saying that one or the other happens? I possibly shouldn't do any of this, because it's in the question of how does one treat it. You run up against the problem that's the hydrogen-able reduction. You're doing the equivalence principle the other way. In other words, do a coordinate transformation where you identify the two balls. the balls are not in a superposition. Of course, now the rest of the world is. That's worse. The rest of the world involves a much larger mass distribution. I mean, if you're going to take general covariance seriously, that that should be a valid point of view. I agree. And this is, of course, this is a problem with any quantum gravity theory. To try and move something from here to here, have you done anything? Because you say the principal general professors are the same. You can't adopt that in quantum mechanics get nonsense. Because you say, okay, how do you form superpositions of, how do you form a momentum state? How do you position state? Well, you see all the different locations of things. If all those different locations are the same, then a superposition is just one thing. It's a position state. You can't do a condominium state. But I understand the problem. I mean, as Bill has stated, this is your problem. No, I understand the problem. and of course one has to address that and my point of view is that you do have to take it has to be some average, you look at different ways you do it and you take the one where this mass displacement is the smallest that's what I would do so I'd say, either you can move the whole earth and then the gravitational safe energy effect is huge

22:30 so you have to minimize that yeah, there's more things to say Yeah, no, I just had to tell you a point, Bill. I hadn't realized that's what you were saying. But that's, no, no, no, you have to take that on board. I do discuss it. Let's talk about it a little bit. So maybe we can do the rest of our lunch. Let's thank Roger again. Thank you. Well, the only concession to the main theme of this workshop are the bits in the title of that tour. And I did actually, oops, I went inverted commas in the announcement I sent here to the organizers. I think they have vanished in the final conference program. So, and I also refrained from calling it It From Bits, because I thought it would be enormously. So instead, what this will be about is a piece of quantum gravity and the subplanking bits will just turn out to be, if you like, the smallest building blocks of space-time itself. So it's bits in the center of pieces. And because I know that most of you don't work on quantum gravity, So, this talk will require you to know almost nothing about quantum gravity beyond the fact that you just have no idea how to do it. Now, let me take a kind of simple-minded, or let me phrase the problem of quantum gravity in a simple-minded way, which is to say, okay, what would we see if we examined a piece of seemingly empty space-time at smaller and smaller and smaller distances?

25:00 So what do we expect to see? And, well, we have, of course, some expectations just by putting one and one together, where one and one are the recognized fundamental theories we believe are true. One is quantum theory, the other one being general relativity. So if you put these two together, then you come to the conclusion that something must inevitably happen if you look at space-times as the very smallest distances. Now, why is that so? Well, basically, because of Heisenberg's uncertainty relations, which are an integral part of quantum theory, which I write in this somewhat nonchalant manner here, following earlier speakers, and plus Einstein's equivalence principle, which tells you that gravity universally couples to all forms of matter and energy. So, if I approach space with the tiniest distances, via delta distance, of course it means that the energy fluctuations I'll be looking at will be constrained exactly by this inequality. So, my energy fluctuations will become the larger, the smaller the distances are, the space-time distances at which I resolve, try to resolve my space-time. And of course, then by the equivalence principle, that's how the argument is usually made, we know that energy will couple to space type in the sense that, well, space will no longer be nice and flat, but be curved. And the higher the energy is, the higher the energy fluctuation is, the stronger the geometric fluctuation I expect at the very shortest scale. So, here is my three little cartoons. This is space-time at everyday distances, well, and quite a bit at much smaller distances. I mean, all of the accelerator physics never sees any deviation from smooth, flat, structural space-time, but now imagine I go down even further, then I'll start by virtue of seeing that the energy fluctuations are picking up, I'll start to see also the curvature fluctuations, so my space will presumably still look like a smooth entity, but it will have long-term

27:30 which simply means that if I would send a line through it, that's an easy way of thinking about it, visualizing it, then it would not move in a straight line, but instead would move under the influence of these curvature ripples. And of course, if I go down all the way to the timeless scales, one usually plays along with the physics, the plant scale, then no one knows what's going to happen. But what we do expect is that, well, there'll be wild energy fluctuations, therefore inducing wild fluctuations in the geometry. And what this looks like is completely unclear. And what quantitatively this ought to look like is completely unclear. So, well, what we quantum gravitators try and do is formulate quantitative models of what might go on at the tiniest scales. And one thing that's clear is that with all likelihood, the way we think about space-time in a classically at everyday decisions would be completely inappropriate at the tiniest scales. But of course, the question is, what does substitute it? will we see some space-time foam where not even the topology of space is fixed, where everything is getting ripped up at the shortest distances. We may not even be able to talk about smooth structures. It could be something discontinuous or discreet, as people have suggested. So, here below a little picture, of course, why this is not in contradiction with kind of the large-scale structure. imagine just like a piece of cloth, which when you look at from large distances looks completely smooth, but however when you go very close, it'll exhibit non-trivial structure. Okay, so how do I go about this in a nutshell? Here's my five-minute crash course on how you ought to do it. Well, first you should you should open your quantum mechanics book, well not any old quantum mechanics book, but the one in which finite superposition principle is explained. So for instance, then it will explain to you how to understand or how to compute the quantum dynamics of a non-realtimistic particle in a simple potential. And of course the way

30:00 is often taught and done is say by diagonalizing the quantum Hamiltonian, but a completely equivalent way of doing this is to compute its so-called propagator. So this contains exactly the same dynamical information about the quantum theory and the quantum, well, the prescription of how to compute this propagator has been given by Feynman and it tells you the following. Maybe if I'm interested in the quantum amplitude of the particle going from some point x1 to some other point x2 in some time t, so what I have to do in order to understand to compute this quantum amplitude, I have to take all paths that go from here to here, which I call p, and each of these paths I have to weigh with a complex weight which is p to the i as the action, the classical action of that path. And we are on just two little warnings or two little remarks about this construction, this well-known construction. First of all, these paths are really virtual and no, they have absolutely nothing to do with classical paths of the particle. So you really consider here like, well, all paths. And also, in addition, good mathematical sense of discretion, what you do is you employ a regularization at an intermediate stage. And this is very easy to do. You simply, you don't just consider smooth paths, but paths that are piecewise straight. So these are much simpler algorithms to deal with. And what you then do, so here is a typical kind of discretized path if you like. So it's just edges, piecewise straight edges put together instead of the smooth quantities. And there is a short distance cutoff parameter A in here, which is just the length of a single one of these little bits. So these are the bits here, one-dimensional bits that come into play if you like. And what you do in the end, you obstruct this object for all discretized halfway when you let this parameter go to a go to zero in a controlled manner. Now exercise. Now do the same for quantum gravity. Now what would one try

32:30 to do? What are the analogs of the classical particle? They will be here classical space-time configurations. So that means space-time configurations of involved, which is no longer x of t, but which is a spatial geometry tracing out a certain space-time geometry as time advances. So, this is the analog of a single path of what I had up here for the quantum mechanical example. So, what I give here is not an initial and final position x, but what I give you is an and the final spatial geometry at some fixed time. And what I then do, just repeating this perspective from the particle came up there, is I compute prior to compute its propagator, which is presently adding up for fixed boundary conditions, all the paths that can go in between. And what now a single path is, is a full space-time geometry, which I call this curly G. So, And the weight, according to the prescription above, is again simply the classical action. And this is here where kind of, of course, gravity comes in because we take this as the Einstein action. And what you did for the particle case is true here. You are well advised not to do these over, to take some of the smooth objects, but in order to make your life easier and at the same time, mathematically much better defined, you approximate these objects by geometries that are piecewise straight. In the same way, as these paths up here, they're also piecewise straight. You can imagine, say, if you had two geometries that would be built from piecewise straight objects, which I would always take as triangles. So, this is, as I've drawn it here, if we had two-dimensional space times, these over and then weigh with the gravitational action in this way. And the very same thing can be actually done in any dimension. And there are ways, natural ways to discretize also the Einstein action. So these are these triangles or higher dimensional triangles which you can think of as just pieces cut out of the flat four-dimensional space.

35:00 we are always interested in this four-dimension case, will be the bits that you're trying, well, in this regularized framework, which you're taking as kind of the smallest building blocks of the geometries that go into your superposition. So, this is just very conventional, if you like, project. So, now, what you still have to do then, of course, is to take a limit as you then let this, the little, kind of the edge lengths of your little triangle, or rather the high-dimensional sequences, as you let those go to zero. Now, of course, all this is much easier said than done. So, let me just explain a little bit of how it's done. I will not at all go into a great technical detail, also not to bore you at this late, late hour. So, this is kind of the, the measure part of what was the sum of my previous transparency, as you would usually write that in a continuum language, there's some path integral over your smooth configurations and G we knew is simply the four-dimensional space-time matrix, the metric tensor, and what you have to do is you have to quotient out by a symmetry group, which are the four-dimensional dichromorphisms or geometrizations if you like. So this now gets replaced in my framework by a discrete sum and I already told you what I'm going to sum over. I will sum over kind of discretized objects that where each object represents a space-time in a kind of coarse-grained manner. So these are generalized triangulations and what I do is, in order to get it, what I still have to do in order to get an interesting, potentially interesting continuum theory out, I have to take the number of these building blocks to infinity and simultaneously let each of the little bits kind of shrink also in a controlled manner to zero, where this A is again with my UB, the cutoff here. So this is kind of analogy of the formal signal between the continuum and the real kind of sum are tried and evaluated rigorously. In this way, where quantum gravity gets rephrased and then maybe somewhat unfamiliar language, namely that of a statistical sum over geometries. So, they're really replacing, well, this form of continuous

37:30 part integral by a sum over discretized objects which are all made from these little bits. So what goes... so what goes into your computing the sum is quite simple. So I give you kind of Lego blocks. They are triangular, but nevertheless, let's call them Lego blocks. And let's say I give you... if I choose T equal to 10,000, so it is your task to build from these 10,000 blocks, all in equivalent space-time geometry. This is cross-frame space-time geometries. And then weigh each of them with the Einstein action, each of the Einstein action, and then try and perform the sign. And well, then of course the task may become more difficult with these 10,000 to become 100,000, but this is where you go. And you try and understand what this object is in a limit. Now, for practitioner record to gravity, this, well, this satisfies a number of very nice properties. So, it is background independent, non-perturbative, and it doesn't require any, you don't have to invoke any human symmetry, such as supersymmetry. So, to use such a framework. now what does it mean? Background independent means I don't single out any particular space time metric in my, you know, any space and geometry in my big big sum. So they're all democratically summed over. That's what background independence means. Non-determinative means that in my sum will be objects that are very, very, very far from any classical smooth flat object like Minkowski's face. So they're wildly, wildly curved objects in the sun here and generically we'll see these completely dominate the sun and therefore also the behavior of the superposition. Now these are very nice words but unfortunately they spell out quite a bit of trouble in comparison with what one often does when one tries to contrast gravity, namely by really making it very explicitly background-independence and non-determined, you simply do not have a background structure with respect to which you can measure what you like to

40:00 measure in gravity, you'd like to measure distances, you'd like to measure curvature, and of course you do that with respect to a certain metric. And if you do, if you assume a heterogeneous approach where you just fix, start with the classical geometry and then start looking at what happens when you turn on little wiggles of that, you still have obviously your background geometry to refer to for measurements of length and time, etc. So, you don't have that here. So, doing, taking such an approach gets you into deep trouble, usually. Well, and so it does here. Now, so the idea in such an approach, so what can one hope to be doing? would like to find is to understand what the quantum superposition of these geometries gives you. And they should in the first place say, well I should give you a full dynamical quantum theory, like they did for the particle case, and more specifically they should give you a brown state of quantum geometry. So really kind of the vacuum, the vacuum of vector, should come out of here. Now what I would expect from this general setup is that, well, if one looks at very, very short scales at such a sum because of these wild fluctuations, I mean, this would be very ill-behaving, very, very multi-classical at very short distances. But what you still hope to find when you look at this object now at much, much larger scales than what here dictated by the cut-off, you hope it will, maybe magically, start resembling the smooth and structural space-time that we see around us. And, well, obviously, around us, we assume there's something very close to the vacuum state of the theory, but, of course, viewed at very, very much larger distances, let's say, the plant-based. Now, the problem here is not generally with cooking up your model of what little plank size bits should do kind of dynamically. So, well, many people have sort of many nice models they've played with, but the big trouble is then, to show once you have something else, that really a classical limit emerges at sufficiently large distances.

42:30 Well, what's the trouble, you might ask? Well, one trouble is that in spite of having some kind of microscopic tiny building blocks and putting those together, for instance, four-dimensional building blocks, in my case, it's absolutely not guaranteed that the generic objects are built in this work, in these tiny labor blocks, have an effective large-scale dimension of form. So, indeed, this is what very interestingly and completely generically happens for building blocks that have dimension three or larger. So, higher dimension generalizations of triangles, of flat triangles. Now, if you write down such sums, as I instructed you, you should be doing, and if you then look, I mean, say you have your hundred thousand building blocks well, all the different, of course, in that case, to find a number of ways how you can glue them together to obtain inequivalent space-time geometries, then what you find typically is that generically, they will look like one of those two cases. They will either look like this or like that. Now, the thing on the right-hand side, it's very easy to describe, the so-called branched polymer. So, what does that mean? Well, the little building blocks, they arrange themselves favorably along one-dimensional very thin threads that are still, if you look at the tiny scale, that's the four-dimensional, but they become these polymer-like objects from when viewed at somewhat larger scales. So you see here, the width of these objects could still be characterized when I cut off things eight, and the large-scale so-called house document in such an object is two. So nothing like what you put in there in the first place at the level of the cut-off. Now, conversely, and this depends on where, how you tune your Van Goghling from, this happens. This is a so-called the Cromwell geometry, in which, let's possibly say, talk about two-dimensional triangular building blocks, in which they all condense around one or several, just a handful of seats, of vertices, with the fact that the more triangles, say, or building blocks one

45:00 They never make the space-time bigger, but they will always cluster around one or a few centers, and they will never obtain a macroscopic dimension. So you can have here hundreds of thousands of early works sitting in making up a space-time, whose diameter still is basically given by . So, and this is an object, if you simply value, which has an effective large-scale dimension of infinity. because you are, you can get everywhere in one step, and that's a characteristic of a very, very high dimensional space, if you think about it. And if you'd like a kind of a shorter demonstration about effective dimensions, well, take your two-dimensional piece of paper. Well, this is two-dimensional, but okay, then we made it effectively one-dimensional by arranging these little building blocks in a kind of sin tube so it will from the background appear as a really one-dimensional object. So this is dimensional reduction that will inversely crop it up into an object that I hope from the background will appear like something three-dimensional. So this is a gaming dimension if you like and similar acronyms underlie these sums over geometries. And they're absolutely characteristic for, well, if it happens extremely generically or completely generically, it always wants to have the approaches to quantum gravity. Because they allow geometries that are very, very highly curved at very small scales. And if you look at the set of all geometries, those will dominate your state sign. And they, of course, will determine also the structure of what I would call the ground state of the theory. It inherits this very, very degenerate structure, like the dimension here. So this is, of course, completely unphysical and has nothing to do with the classical large-scale quality geometry. Yeah, question. Even ordinary environment path or particle moving around, most of the paths are hardly non-classical. And there, what one gets is that the idea of how a classical picture emerges is because the phase oscillates rapidly through non-classical paths and so it's stationary at the classical ones that effectively you get the fluctuations about the classical paths

47:30 in the preferred range. You haven't talked about these fluctuations in action. Presumably these would have very, very large fluctuations in the gravitational action for small per So, yeah, exactly what happens, I mean, in the past, of course, in the past case, where I'm not talking about, you mentioned, of course, the semi-classical picture, but of course, you can actually treat the past sum, this discapacitized past sum, to get, well, after big rotation, the Wiener measure out. And of course, there, the statement is that the nova-differentiable path dominates the sum. And exactly the same thing is true here. It does not at all mean that you're talking about the world of physics, but you have to actually look at observables, if you like. So the individual paths are, of course, non-physical. Nevertheless, of course, if you look at expectation-wise, suitable observables, you still retrieve the suitable path. So you haven't rejected, I'm sorry, this looks like a pessimistic slide. Yeah, it's a pessimistic slide by saying, this is just too bad. This is too bad. I mean, there are no interesting geometric observables people have looked at in this scheme that had anything whatsoever to do at large scales with 4D geometry. And of course, you expect some, there is of course something like what we expect to stay inside, like the nova differential geometries geometries would also completely dominate such a scheme, but this is just too bad. So, now what this does illustrate is what I already said in a few words is that if you let geometry loose, as you do in such a non-interruptive context, anything can happen and will happen and will, well, it will screw up your attempts to define a theory that has a correct classical limit, unless you do something quite particular, as I pointed out here, the dimension, I mean, to start with will just come out wrong. And of course, if I don't get anything resembling four-dimensional objects, it will be very difficult to get out any other, I mean, to go to further steps and try to recover kind of through that semi-classical geometry process. So it is actually very difficult, generically, starting from, you know, as I said, you prefer the normal

50:00 and using some simple building proofs like I've done here, to generate anything that has the correct classical limit. Even if you look at very coarse geometric information here, like the dimension. So, the question here is, is there really something fundamentally, is one doing something fundamentally wrong? Is taking such an approach? Is there something either completely wrong with the patterns of composition principle, or what is it that goes wrong? Well, the message is, don't do it the Cambridge way. All right, maybe not quite completely politically correct in here, but so, don't do it the Euclidean So what does that mean? Well, the difficulties and these large-scale geometric generacies that I talked about on the previous slide, they're all obtained, this is the so-called Euclidean approach to Now, what is the Euclidean approach? Well, in the context of this superposition principle, it tells you do not superpose all Lorentzian space-time geometries, and it has a correct signature, but instead use Euclidean geometries, and sum over all of those. Now, what is classically the difference between the Euclidean and the Lorentzian space-time? Well, there's a huge difference. In Euclidean spacetime is a very strange object, geometric object, because you can move backward and forward in time as easily as you can move to the right and the left across this stage. So, it really has nothing much to do with our real spacetime. Of course, whether these are appropriate objects to use or some of the quantum theory is not clear. So, you might be lucky when you do this and get away with it. Now, as I told you, unfortunately, bad things happen. And the question is, well, do better things happen if you do it properly? So, if you work from the start with Lorentzian geometries. So, Lorentzian geometries are geometries that have a time parallel. Yeah, and if I remind you of this kind of just the picture you have from Mikoski space here, it's just a three-dimensional drawing. Of course, what is special about any Lorentzian space-time, it will look lovely exactly like this.

52:30 Maybe if you pick up a point, then there is something like you call the causal part, and the causal future of that point. And that consists of all points in space-time that could have possibly been in causal contact with what happens to you here at this origin problem. Excellent. And of course, what lies outside here could have never been in causal contact. And Lorentzian spaces locally all look like this. They all have such a local light construction. And of course, this is very different from the Euclidean case because there's no such thing. There's no such thing as the trajectory of a light ray in Euclidean space. So, the conjecture at, well, a point quite a few years ago, of mine was, might the problem of these geometric degeneracies in any way causally related to the fact that we didn't start with the correct objects in our superposition principle to start with, namely, looking only at the funny kind of Euclidean spacement instead of the general forensic objects. And this was an idea, as I said, that was born a few years ago. And well, since then, it has passed quite a number of tests in lower dimensions. I mean, that's of course what you do when you have an idea. You tell everyone and everyone says, this is just not going to work. And instead of being completely discouraged, you set out to work and first try with a little two-dimensional model, you try with a somewhat more difficult three-dimensional one before you work your way up to four dimensions which you are really interested in. So now what are my bits, the relevant bits now in this somewhat altered game, this new superposition principle? Well, they are still my piecewise flat bits of space-time. So, there are still triangles, tetra-nitra in three dimensions, or four circles, four synthesis in four dimensions, but they now have space and time-like edges, which is, well, which are just inherited from the pieces cut out of the flat and possible space. So, here in my language, space-like distances are denoted green, and time-like red. So, these are kind of building blocks in which we use to construct, now space times, which

55:00 just have a good, well-defined local light construction, albeit in this kind of discretized language. But this is what you do. And one very simple thing, which you have to observe, I mean, like the gluing rules that come along with the requirement that everything should be nice and the rents and therefore also causal, of course, in the likeness structure, of course, the causal properties of the spacetime is that such spacetimes will not be allowed in our viewing rules. So this is a two-dimensional example of a one plus one-dimensional spacetime where space to start with is a little circle which kind of advances in time that at some stage, maybe once the power has passed this red dot here, splits up into two circles. Now, the reason why we do not allow those is because we say, well, but if these were kind of classical causal objects, then these points here are very, very bad. And something you never consider classically, because they violate causality in a very bad manner. if you want to think in terms of the little light cones that are probably quite everywhere, then somewhere along here, at least in one point, there must be one point where the light cone doesn't know whether it should turn left or right. And you can state this by saying, well, the Lorentzian metric actually has to degenerate in such a way. So these are kind of very bad objects locally, because of the local light cone, if you like, as well as globally. Because why? Well, if it was a that propagates up here, you would, once you've passed this point, you would have lost, in a discontinuous manner, a whole branch of your future, which you still had when you were down here. So these objects are also called causally discontinuous. So, our gluing rule is, take these Minkowski little needle blocks and glue them together, avoiding this. And then, if the baby needs to have a name, we call this causal dynamic of translation. So, this is just the framework within which you now try to construct the theory of quantum gravity from superposition principle. So the question is, how far can this be pushed analytically, and as I said, well, here, well,

57:30 I and my collaborator just started doing this in two dimensions, and two dimensions, well, after I found this, I appreciate, I learned to appreciate why so many people spend the time doing two-dimensional models their entire life long, because you can actually exactly solve certain and sufficiently simple models and it's very important you can derive exact expressions for copper gauges and what have you. So, this is what happens in 2D. In 3D, one can make partial analytical progress, but here already you had to evolve. So, this involved mapping these superpositions of geometry onto a certain matrix model about which one knows has been analytical information, but already here we had to invoke numerical simulations to understand basically, you know, what do generic geometries in the superposition look like in the limit as you make them very, very large and that your UB parameter go to zero. Now, in order not to exhaust it too much with these beautiful propagated formulas, here a picture some of you may have seen. That is a two-dimensional space-time which is or is a typical illustration of the ground state of what happens in this in two dimensions, one plus one. Space is compatriified to a one-dimensional circle. And if you look very closely, you can still see the tiny, tiny triangle. So this object has more than 18,000 triangles. It's generated by Monte Carlo simulation, so which of course has to be finite by definition. And what you see here, time goes this way. What you see here is the fluctuations of the ground, so it is a function of time. So, well, this is two-dimensional quantum gravity for you. It's, of course, a completely, physically, completely uninteresting theory because, well, we don't know what it describes in nature. It's not a two-dimensional space type, luckily. So, well, one can also, I just flashed some pictures so you can relax before I tell you about the four-dimensional story. So, you can very easily couple

1:00:00 metal to the system and then the interesting thing is to understand how matter and geometry interact depending on where you are in your coupling constant space. So, for example, in this two-dimensional geometry, the ground state, you cannot put easy models, one, two, three, four, five, four, five. So, this is a typical modified ground state, you see the obtain when you put on an easy model, a simple spin model that interacts of course with the geometry. Nothing too much happens to the geometry. I mean, never mind the coloring that has just to do with the up and down regions because it spins. Now, something drastic, however, does happen if you put eight copies of an easy model onto the geometry. So here you see has a very strong influence. It deforms the geometry in this very drastic way. It squeezes off space-time here and here. And this is still, well, there's still a well-defined, to the best of all, a well-defined continuum theory of three-dimensional geometry coupled to matter in here. And this is something nice that you do not see at all in the Euclidean regime. But already in three dimensions, we run into these funny degeneracies. Now, of course, now the real object of desire is the four-dimensional theory because although you can write down an Einstein type action in every dimension, it's only in dimension four or higher that you have the characteristic local excitations or gravitonic excitations. If you do this in lower dimensions, there are no wave-like excitations at all and that of course, somewhat turbulent and uninteresting physically. So, the crucial question is now, well, to say, okay, now you've shown after some years of work, you've convinced yourself that this formalization does make some sense, it gives some interesting results for this toy model, but now what happens now for the physically interesting case of four dimensions? It's really the requirement of gluing your little Lego blocks in a causal way, for example, this allows branching off in the time direction, is that already enough to cure this degeneracy problem? That was, of course, the original conjecture.

1:02:30 And well, the simulations have been run for about a year and very excitingly, there are that true four-dimensional dynamical behavior comes out of these models. And I'll explain this on the next couple of experiences. What the precise evidence is that the quantum ground states of the chains are four-dimensional, and moreover, I can even say more about it, namely that this ground state, as you'll see in a second, has a certain shape, and this can be a very kind of simple, mini-simple space model. It can be matched to such a mini-simple space model, which is highly non-trivial, because what I'm doing here is I'm really doing quantum gravity for first principles, non-interruptively, everything's in two. Okay, so here are the results. So, well, they're very nice, they even amazed us, of course we were very, we were quite hesitant to move to the scale of four dimensions, having been so discouraged by everyone around, what we would find. So this is what we find in four dimensions, for computing these superpositions of these four-dimensional causally glued legal geometries. So this is the shape of our ground state and, well, I depict this in a very simple manner, I restrict myself to plotting only one variable as a function of time. So, time runs again along the vertical axis and what I've plotted here is see the three-dimensional volume of the universe as a function of time. So, I'm using three-dimensional, well, this is just language. I haven't yet measured what the dynamical dimension of the space is. So, this is what it looks like. So, you see one finds again kind of a thin stalk here, and then a region where geometry is truly extended. So, this is kind of the interesting part of the geometry, and there you want to understand what its large scale dimension is. So, it's really the kind of dynamically generated dimension. The thing that came out in a completely wrong way before in this polymerized say geometries.

1:05:00 This behaves now, so what is now the concrete evidence of why this is four-dimensional? Well, we simulate these objects, first of all, at the fixed four-volume. We have to do that because of computer-technical reasons. So what we do then is we generate this surprise state at various volumes. And then we check the discrete four-volume, just the number of little four dimensional big bits I put in there which are called V4 and then I check how this time extension here of where geometry is extended and non-trivial, how this scales as a function of this number V4. And, miraculously, it scales its expectation value because we are looking at ensemble averages. It scales exactly with the power, exactly, with what we can measure within computer accuracy, it scales with the power of one quarter. Similarly, the volume of these, what I provisionally call the spatial three-dimensional slices, also scales with the correct form volume to the three-fourths. So, this is generally evidence that this kind of block structure is four-dimensional, it's very large-scale. So, it's not just the same logic. So, and, well, what is the picture that goes along with this that looks quite different from the ones we saw in two dimensions. So this is the universe. This is simply, again, the plot of the three-volume as a function of time. This is time. And I've just made it into a kind of, by rotation of a curve, I've simply made it into this rotation volume. So the circumference here at any time t is proportional to the three-dimensional volume. So this is here, this is our quantum university that comes out. And I just told you one piece of evidence that it has ever before. Another piece of evidence, by now there are quite a few pieces of evidence. Another way to understand, well, to get out dimensions is to understand how the volumes of geodesic balls

1:07:30 scale as a function of the radius. So, say you pick randomly a point in your geometry then you walk out in step one, two, three, and of course it all kind of defined topological spheres or multiple spheres, and then you just check what the volume is enclosed in a sphere of a certain radius. and then you just check how it scales to leading power of R and the exponent of that you call this, that is one way of defining so-called house of dimension. So if you do this for the three-dimensional slices here of this object, they come out again with a computer accuracy to be three. If I look at yet another dimension, which is the dimension I get from studying diffusion in the space time on this outbreak. I also get what's called the spectral dimension that within computer accuracy is actually four. So there's very strong evidence that indeed you get something four dimensional out of this construction and this is extremely exciting and unprecedented. So this is a, well this is a paper that came out in September and that in PRL and that to our great surprise, turned a huge media interest because apparently everyone wanted to know why space time is four-dimensional. And here's now the explanation. Okay. So this was interesting result number one. The dimension comes out right. Yes. But you did assume that the microscopic dimension of your simple C was four-dimensional. You were using four simple C at the bottom. That was a simple, yeah. So, so you did, four actually did get injected at most microscopic scale, four-dimensional this is by far not a sufficient condition at all. And it's still true that, well, this looks kind of now like a smooth geometry, but this is just an artifact of my deciding just to be one variable. If you look really at small scales, these are very wild. And again, some of the, well, I can talk to you in a bit more detail, but some of the measurements you can make on the geometry gives you a very strong indication that at small scales, this So, okay, so this much about the dimension, but can we do more? Can we say more about this block structure?

1:10:00 Yeah. So just to relate to that, I mean, can we relate the microscopic dimension to the microscopic dimension? I mean, if you have changed the dimension of your microscopic elements, is there any kind of way you can relate to the microscopic dimension? Oh, yeah, I want to come back to your question, of course, you could say, well, it seems anyway, I'm telling you the dimensions come out dynamically here, you know, why do you start this four-dimensional universe? Well, it's the first dimension where you might expect to see something interesting at all. Of course, it might well be, so if I start on this five-dimensional universe, that I, again, find something interesting, but in the best of all worlds, in my favorite scenario, everything would collapse dynamically back to a four-dimensional object. So that would, for instance, be very interesting. I would say, tell you that something is very special, maybe about four dimensions. But really, the reason why I'm choosing four-dimensional building blocks here is for convenience. And as I said, it's the first dimension where something interesting might happen. So we just, this is why I'm pushing this way. Okay, so can we say something more about this object? I mean, of course, dimensionality is a very, very coarse way of characterizing geometry. Now, for instance, can we understand why it has this shape here, generative? Well, and the answer is, yes, a very interesting news number two is that one can write down an effective action which gives rise and an effective action of a single variable, where the similar variable is basically the volume of the universe and which is completely equivalent to giving what is usually called the scale factor in cosmology. We can write down an effective action which generates exactly these shapes we see in the computer simulations. So what we actually did, well, we just fit it. Okay, let me just tell you first what it is and why it is so interesting. So, if I now relate this to a mini-superspace model, I should just in certain seconds tell you what a mini-superspace model is. A mini-superspace model is something quite different from what we've done here. In this mini-superspace model, you start classically by saying, okay, at very, very large scales, our universe is homogenous isotropic. So it's very, very difficult structure. And the structures, the only

1:12:30 structure that is there is a scale capital A of t, which I keep track of. So everything I keep track of of the entire field components of Einstein gravity, so the g and u of x and t, is a single, well, a single mechanical variable, which basically is again revolving at a different time. So I throw away, already agreement and hope it gives me still something interesting after I quantize. Then I quantize this very, well, one variable reduction of gravity and well, this is called quantize, this is what is called underlies typical quanticosmology. And what appears there typically is an action, classically, of the scale factor A. Well this action is actually quite nasty because it's unbounded below. So just to say mini-superspace cosmology is quite different from what I do here because what I do here, I do dynamically treat all the degrees of freedom that are there. They're all sitting in my positions, but what I then monitor at the very end of story, one that I found is this same variable A of T. And what one finds is now again an effective action of the scale factor A, which very closely resembles what one has in standard form of cosmology with a couple of small but very important differences. So this is kind effective action of our four-dimensional quantum gravity model. So what you see here, it's a simple mechanical action of a single variable, A. Here you have a kinetic term, here you have a linear term that basically knows about the curvature of your spatial slices, until you have a cosmological constant term, it's basically a volume term. So now this is how it comes out of all simulation. So this is now with all degrees of freedom of gravity treated dynamically non-determinatively. Now what would you have gotten if you had proceeded in a mini-superspace logic here? Well, you get a very similar object apart from the sinusoid. So you actually have a minus sign in front of this kinetic term, but also minus sign in front of this linear potential term. Now this is

1:15:00 devastating for the quantum cosmologists because, as I said, this is, of course, then an action that has a kinetic term of the wrong sign. And if you perform as people, especially in this region of the world, have performed, then you pick up here a fatal divergence from this so-called conformal mode, so this conformal divergence, which you, in some ad hoc manner, will have to fix. Now what I'm saying is, if you proceed the correct way, what you will rapidly obtain from all these other degrees of human gravity is an effective turn around of this sign here, which then results in a kinetic turn of the correct sign. So, in the same pose for this linear term here. And, well, the other interesting difference is that this action in our model is only true at sufficiently large scales, and why it's not true. Well, it's picked. Doesn't that make it wrong? What? Does that make it wrong because we know what the sign of the action is on large scales? We know how gravity operates on large scales. Yeah, it's a good question because this describes, then you have to look at what this describes, what this describes, I mean, auto-mechanically, right? It's, well, when I show you this, I mean, I still have to then understand what the scales are. So, well, this is basically still kind of very small at Planck scale, if you like. So we still have some, the difference with our mini cosmology, I get back to your point. The other difference with our mini super space cosmology, this is thin stock here. So there is some which is not there if you just look at the straight action as I've written down here. Well, I'm just saying this is what comes out. Of course, from the quantum mechanical point of view, I like this very much because I could never really give this convincing way there's a sign here. And in fact everyone loves, would love to have an argument of why this sign is actually not there. So here's a good argument for you, which is

1:17:30 very nice in the sense that it says, now if you really treat all dynamical degrees of freedom simultaneously, well this difficulty goes away. I agree, it also seems to imply that you get just a different result. Well at least you have the overall side switch. That's the whole thing. So, well, this is what it is. That's some published work in the new article of ours and maybe let me just summarize it. So, I started off by saying my motivation was to try and understand what space looks like at very, very small scales and for that I had to unify quantum mechanical and general relativistic principles and I did this in a very simple-minded way. I would say what I'm doing is really a minimalist approach in the sense that I'm not adding anything new to all principles, I'm just putting them together in a specific way and in particular what turned out to be very crucial these causality conditions or making the whole set up the rent and of course there are technical difficulties involved in that which I didn't have time to talk to but I'm happy to explain to you in private if you want to learn about them. So what we found in four dimensions is something for the moment very encouraging. Namely, we could really generally dynamically construct a universe that could be said to be four-dimensional and which was unprecedented to my knowledge, and we also could match the shape of this universe by an objective action of the scale factor. And I should also warn you, if of course you look at this, what you still have to do here, there are two things for why you shouldn't take all the aspects of this too literally, of course you're still, well we're simulating Euclideanized geometries, so we still rotate to the Euclidean sector, so you still have to do an inverse representation on those. So with a very direct physical interpretation, you should also be somewhat careful. So what this means. But of course, I've been comparing the mini super space cosmology in the Euclidean sector. So the interesting thing is that these things were really derived from scratch. I mean, really from first principles. And, well,

1:20:00 what other people have predicted hasn't happened, namely that absolutely nothing interesting was how none of this physically relevant case in full dimensions. And let me conclude that I'm very happy knowing for a few weeks that my collaborators and I are not the only people to find that interesting because the European community has just thrown a lot of money at us to research specifically, quantum gravity is defined through this call, which is a central theme of a new network by the name of NRH, where this stands for European Network on Random Geometry, and it's about to go online, which means I'm just putting these web pages together, but maybe if you or the students are interested in getting a job in this blossoming and interesting area of high energy physics, we should check out this website. And thank you very much for your attention. Questions? What do black holes look like in this picture? I mean, can you say something about the geometries? Yeah. Yeah, it's a good question because of course, black holes are four miles from the minds of many people who look at the gravity. So, the first thing here is kind of quite interesting. You have here a completely geometric setup in the sense that you have no coordinate systems anywhere. So, you really, when playing around with these objects, you have to see in terms of geometry. And of course, and well, actually students of mine have approximated a black hole geometry as such a triangulation. look at it and you say, well, where's the black hole? Because of course, you don't see, well, we don't actually, we don't know what happens at the center of singularity, so we cannot really, well, we keep some distance from there, but of course, you don't see anything like the eventual horizon in any immediate way, you know, so it's nothing that happens in any dramatic way to your triangulations. I mean, there's no topological in there associated with it, or? Well, you can, of course, you can try and figure out, for example, what

1:22:30 in the current horizon looks like in this rainbow for a single geometry. Now, if you now wanted to think big and say, okay, what can this approach maybe eventually say about the problem of, say, entropy of black holes? Well, then the point is then, well, you first have to set up the language, I mean, you first have to actually pose the correct question in such a completely non-interruptive setting. So, the first, for example, you have to say which ensemble geometries you would like to start with that go into your superposition principle. And then you have to start defining observables. For example, that, of course, that, well, the nice thing would be to define an observable that has the value one if there is an eventual rise in some way in geometry, and zero if there isn't. Of course, you have to really think about what your superposition should be. And of course, this is, again, since it's background independent, again, you must realize that you have to define every simple script. So you're not, again, you're not, it's not, you put like a black hole geometry here, and then you let it quantum wiggle a little bit, but on the contrary, and what you would mean by so much all geometries that do contain black holes. Other questions? Let's thank the speaker again. And feel free to hang out a while in the lobby. Thank you. More or less than a million lines of text. Is this...

1:25:00 Yeah, but... I mean, there's very much a perfect comment around the bits here. I mean, these are... I guess you've got a copy there, yes. No, I'm sorry, I can't do it. See me, I'll give it to you for the present in the morning. Thank you. Thank you. Thank you. Thank you.