Universe from Sub-Planckian Bits

0:00 Plankian Bits. Well, the only concession to the main theme of this workshop are the bits in the title of my talk, and I did actually put some in a row of commas in the announcement I sent here to the organizers, I think they have finished the final conference program, and it from bits, because that would have probably been enormously. So instead what this would be about is a piece of quantum gravity and the subplank in bits will just turn out to be, if you like, the smallest building blocks of space-time itself, so it's bits in the sense that you can use it. And, of course, I know that most of you don't know 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 face the problem of quantum gravity in a simple-minded way, which is to say, okay, what would we see if we examine a piece of seemingly empty spacetime at smaller and smaller distances? 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 very small distances.

2:30 Now why is that so? Well, basically, because of Heisenberg's uncertainty relations, which are an integral part of the quantum theory, in which I write in this book of Marshall, but I'm here following earlier speakers, and plus I'm saying it's a principle, which tells you that gravity universally couples to all forms of matter and energy. So, if I probe space at the tiniest distances, here, delta distance, of course, it means that the energy fluctuations I'll be looking at will be constrained exactly by this inequality. my energy fluctuations will become the larger, the smaller the distances are, 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-time in the sense that, well, space will no longer be nice and flat, but be curved. And the higher the energy is, the the geometric fluctuations I'd 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 accelerated physics never sees any deviation from smooth, flat, structural space-time, but now imagine I go down even further, then I start, by virtue of seeing how the energy fluctuations are picking up, I'll start to see most curvature fluctuations, so my space will presumably still like a smooth entity, but it will have non-trivial curvature, which simply means that if I would send light through it, that's an easy way of thinking about it, visualizing it, then it would not move in, well, in a straight line, but instead would move under And of course, if I go down all the way to the time scale, then no one knows what's going to happen. But what we do expect is that there will be wild energy fluctuations, therefore inducing wild fluctuations in the geometry.

5:00 And what this looks like is completely unclear. What quantitatively this ought to look like is completely unclear. So, well, what we quantum gravitate or try and do is formulate quantitative models of what might go on at the highest scales. And one thing that's clear is that, with all likelihood, the way we think about space time and classically everyday decisions will be completely inappropriate at the time of scale. And of course, the question is, what does substitute it? Will we see some space like foam where 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 will exhibit non-trivial structure. Okay so how do I go about this in a nutshell? Here's my five minute crash course of 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 climate proposition is explained. So, for instance, there it will explain to you how to understand or how to compute the quantum dynamics of a non-religious particle in a simple potential and, of course, the way that you have 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, namely if I'm interested in the quantum amplitude of the particle going from the point x1 to the model

7:30 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 are called p, and each of these paths I have to weigh with a complex weight, which is e to the i of s, the action, the classical action of that path. And we are 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 as a particle. So you really consider here like, well, all paths. And also, in addition, in order to really make good mathematical sense of this expression, what you do is you employ a regularization at an 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 to deal with and what you then do, so here is atypical kind of discretized paths if you like, so it's just edges, piecewise straight edges put together instead of these smooth quantities and that is a kind of a short-distance cut of parameter A. It would be able to put the length of a single one or single 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 first grab this object, for all this to class, where, then 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 to do? What are the analogues of the classical particle? They will be here, classical space-time configurations. So, there will be space-time configurations of the analytical variable involved, which is no longer x , 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.

10:00 So what I give here is not an initial and final position x, but what I give you an initial and a final spatial geometry at some fixed time. And what I then do, just repeating this prescription from the particle case up there, is I compute trial and compute its propagator by simply 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 Kerner G. So as a weight, according to the prescription above, you can simply the classical action and this is here where kind of, also of course gravity comes in because we take this with 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 their life easier, and at the same time mathematically not better, you find you approximated 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 always take as triangles. So this is, as I brought it here, if you had two-dimensional spacetime, these would be the objects to sum 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 the Einstein action. So these are these triangles or higher-dimensional triangles which you can pieces cut out of a flat four-dimensional space, we are always interested in this four-dimensional case, will be the bits that you're trying, well, in this regularised framework which you're taking as kind of the smallest 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 the limit as you can kind of put the edge length of your little triangle, or rather the high-dimensional species, as you let those go to zero.

12:30 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 very technical detail, also not to bore you at this late hour. So this is kind of the measured part of what was the sum of my previous transparency, because you would usually write that in a continuum language, as some past integral over your smooth configurations. And g mu is simply the four-dimensional space-time matrix, the matrix tensor. And what you have to do is you have to quotient out by a symmetry which are these four-dimensional diffeomorphisms or reprimaturizations, 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-grave manner. So these are generalized triangulations And what I do is, in order to get it, what I have to do in order to get an interesting, potentially interesting continuum theory out, I have to take the number of these building goals 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, like you need a cover here. So this is kind of the analogy of the formal thing I will do in the continuum and the real kind of sum I'll try and evaluate rigorously. In this way, where quantum gravity gets rephrased in a maybe somewhat unfamiliar language, namely that of a statistical sum over geometries. So you're really replacing, well, this form of continuous path integral by the 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, then it is your task to

15:00 good from these 10,000 legal blocks, all in equivalent space-time geometry, these course-based space-time geometries, and then weigh each of them with the answer and action, each of the answer and action, and then try and perform the sum. And, well, then of course your task may become more difficult with these 10,000, 100,000, but this is where you go, and you try and understand what this object is in a limit. Now for practitioner recorded gravity this well this satisfies the 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 new symmetry such as super symmetry so to use such a framework. Now what does it mean? Background 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 is what background independence means, non-perturbative means that in my sum will be objects that are very, very, very far from any classical smooth flat object like Minkowski space. So there are wildly, wildly curved objects in the sun here. And generically, you see these completely dominated in 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 quantify gravity. Namely, by really making it very explicitly background-dependent and non-determinative. You think we do not have a background structure with respect to which you can measure what you like to measure in gravity. You would like to measure, say, distances. You like to measure curvature. And of course, you do that with respect to a certain metric. And if you do, if you assume a perturbative approach where you just fix, start with the classical geometry and then start looking at what happens when you turn 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

17:30 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 hold to be doing? what one would like to find is to understand what the quantum superposition of these geometries gives you. And they should in the first place, they should give you a full dynamical quantum theory, like they did for the particle phase. And more specifically, they should give you a ground state of quantum geometry. So really, kind of the vacuum, the vacuum of vector, should come out of here. Now, what one would expect from this general setup is that, well, if one looks at very, very short scales and such a sum because of these wild fluctuations, I mean, this would be very ill-behaving, very, very non-classical at very short distances. But what you still hope to find, when you look at this object, we are at much much larger scales than what is here. You hope it will magically start resembling the smooth and structural spacetime we see around us. Well, what we see around us is something very close to the vacuum state of the theory, but, of course, viewed at very, very much larger distances, to say, than the plank legs. Now, the problem here is not generally just hooking up your favorite little model of what little plank-sized bits should do, kind of dynamically. So, well, many people have thought of many nice models they have played with. But the big trouble is then to show, once we have some trouble, that really a classical limit emerges at sufficiently large distances. Now if I say, 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 is absolutely not guaranteed that the generic objects are built in this way from these tiny Lego blocks have an effective large-scale dimension of four. So indeed this is what very interestingly and completely generically happens for building blocks

20:00 that have dimensions 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 100,000 building blocks and you've worked out, 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'll 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, that is the so-called branched polymer. So what does that mean? Well, the little building blocks, they arrange themselves favorably along one-dimensional, thin threads that are still, if you look at a tiny scale, they're still four dimensional, but they become these polymer-like objects from when built at somewhat larger scales. So you see here, the width of these objects would still be characterized by a cut-off length A, and the large-scale, so-called house-dock dimension such an object has, is two. So nothing like what you put in there in the level of the color. Now conversely, and this depends on where, how you tune your back up in constant, this happens. This is a so-called crumpled geometry in which, let's for simplicity, 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 has, they never make the spacetime 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 building blocks sitting in making up a spacetime whose diameter still is basically given by the power of A. And this is an object, if you think about it, which has an effective large-scale dimension of infinity, because you can get everywhere in one step, and that's

22:30 a characteristic of a very, very high dimensional space, if you think about it. And if you'd like a shorter demonstration about effective dimensions, take here a two-dimensional piece this is two-dimensional but okay, let me make it effectively one-dimensional by arranging the little building blocks in a kind of thin tube so it will from the back row appear as a really one-dimensional object. So this is dimensional reduction that is inversely prompted up into an object that I hope from the back row will appear like something dimension. So this is gaining dimensions, if you like. And similar mechanisms underlie these sums over geometries. And they're absolutely characteristic for, well, it happens extremely generically or completely generically in all these non-interruptive 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 sum. And they, of course, will determine also the structure of what I would call the ground state of the theory. And 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 4D geometry. Yeah, question? Even in ordinary Feynman pathological particle moving around, most of the paths are highly non-classical. What one gets is that the idea of how classical future emerges is because the phase oscillates rapidly for non-classical paths, and that is stationary at the classical once, that effectively you get the fluctuations about the classical paths as being the preferred graph. You haven't talked about these fluctuations. Presumably, these would have very, very large fluctuations in the gravitational action, of course, all for dimensions. So, yeah, exactly what happens, I mean, in the past, of course, in the past case, where I'm not talking about the same, you mentioned, of course, the semi-classical picture, but of course, you can actually treat the past sum, this discretized past sum, to get, well, after big quotations, the Wiener measure out.

25:00 And, of course, there, the statement is that the nova-differentiable path dominates the And exactly the same thing is true here. It does not at all mean that you're talking about the wrong physics, but you have to actually look at observables, if you like. So the positive parts are, of course, non-physical. Nevertheless, of course, if you look at expectation-based, suitable observables, you still retrieve, in a suitable limit, your positive parts. So you haven't projected, 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 observers people have looked at in this scheme that have anything whatsoever to do at large scale with quality geometry. And of course you expect some, there is of course something like, well you expect a state design like the normal differential geometries will also completely dominate But this is just too bad. So, now what this does illustrate is, well I already said it in a few words, is that if you let geometry loose, as you do in such a non-interruptive context, well anything can happen and will happen and it will, well, it will screw up your attempts to define a theory that has a correct classical limit, unless you do something quite special. So, in 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 a four-dimensional object, it will be very difficult to get out any other, I mean, to go to further steps and try to recover kind of full-fledged semi-classical geometry from this. So it is actually very difficult, generically, from starting from, you know, as I said, you prefer the model at the Planck scale and using some simple building principle, 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 kind of a completely wrong composition principle, or what is it that goes wrong?

27:30 Well, the message is, don't do it the Cambridge way. All right, maybe not completely politically correct in here, but so don't do it the Euclidean So what does that mean? But the difficulties and these large-scale geometric degeneracies that I talked about on the previous slide, they're all obtained within the so-called Euclidean approach to quantum gravity. 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, I mean, of the correct signature, but instead use Euclidean geometries and sum over all of those. Now, what is classically the difference between Euclidean and the Lorentzian space-time? Well, there's a huge difference. Euclidean space-time 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 space lab. Of course, whether these are appropriate objects to use or some of it's quantum theory, it's up for a year or 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 arrow, yeah? And if I remind you of this kind of just the picture you have from the Minkowski space here, just a three-dimensional drawing. Of course, what is special about any Lorentzian space-time, it will look exactly like this. Namely, if you pick up a point, then there's something like you call the causal past 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 point of XMOD. 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 it's very different

30:00 from the Euclidean case because there's no such thing. There's no such thing as the trajectory of a light ray in the Euclidean space. So the conjecture and, well, a point quite a few years ago of mine was, might the problem of these geometric degeneracies be in any way causally related to the fact that we didn't start from the correct objects in our superposition principle to start with. Namely, looking only at the funny kind of recliving spacetime to the set of the genuine Laurentian objects. Well, 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 four dimensions. I mean, that's of course what you do when you have an idea. You tell everyone and everyone says, ah, this is just not going to work. And you, instead of being completely discouraged, you set out to work and first try with a little two-dimensional model, then you try with a somewhat more difficult two-dimensional model 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 circle position principle? Well, they are still my piecewise flat bits of space-time. So, there are still triangles, tetrahedra, three dimensions, four circles, four synthesis, four dimensions, but they now have space and time-like edges which are just inherent from being pieces cut out of a flattened coffin. So in my language, space-like distances, I don't know, between and time-like red. So these are kind of building blocks which we use to construct now space-times which just have a good, a well-defined local lightbulb structure, 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 Lorentzian and therefore also causal, of course, you know, the lightbulb structure, the causal properties of the spacetime is that such spacetimes will not be

32:30 allowed in our gluing worlds. So this is a two-dimensional example of a one plus one dimensional spacetime where space to start with is a little bit circle which then kind of advances in time and at some stage maybe once the power has pass this red dot here, it splits up into two circles. Now the reason why we do not allow those is because we say, well, if these were kind of classical causal objects, then these points here are very, very bad and something you'd never consider classically because they violate causality in a very bad manner. So if you want to think in terms of the light cones that are well defined 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 point. So these are kind of very bad objects locally, because of the local light cone, if you like, structure, as well as globally. Because why? Well, if it was an observer 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 bigger blocks and glue them together, avoiding this. And then, well, the baby needs to have a name, we call this causal dynamic of triangulation. So this is just the framework within which you now try and construct a theory of quantum gravity from sub-position principle. So the question is, how far can this be pushed analytically? And as I said, well, here, well, I and my collaborator just started doing this in two dimensions. In two dimensions, well, after I've done this, I appreciate, I learned to appreciate why so many people spend the time doing two-dimensional models their entire life flow, because you can actually exactly solve certain sufficiently simple models, and it's very rewarding. You can derive exact expressions for propagators and what have you. So this is what happens in 2D. In 3D, one can make partial analytical progress,

35:00 but here already we had to invoke so this involves this analytic progress involved making these superpositions of geometry onto a certain matrix model about which one knows has an 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 that you meet parameters to go to zero. Now, in order not to exhaust you 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 framework of causal dynamical tribulations in two dimensions one plus one. Space is compactified to 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 an anti-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 as 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-time, 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 matter to this 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, on this two-dimensional geometry, the ground state, you cannot put easy models, one, two, three, four, five, four. So, this is a typical modified ground state you see you obtain when you put on an easy model, a simple spin model that interacts of course with the geometry. Nothing too

37:30 much happens with geometry. I mean never mind the coloring that has just to do with the up and down regions for the spins. Now something drastic however does happen if you put eight copies of an easy model onto the geometry. So here you that clearly the matter 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 run into these funny degeneracies. Now, of course, now the real object of desire is a four-dimensional theory because although you can write down an Einstein time action in every dimension, it's only in dimension four or higher that you have the characteristic local excitations or gravitonic excitations classically. If you do this in lower dimensions, there are no at all, and that makes the series of course somewhat trivial 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 formalism does make some sense, it gives some interesting results for this time model, but now what happens now for the physically interesting case of four dimensions. Is really the requirement of gluing your little Lego blocks in a causal way, for example, that disallows this branching off in the time direction, is that already enough to cure this degeneracy problem? That was of course the original conjecture. And well, the simulations have been running for about a year and very excitingly there are now generally indications that true four-dimensional dynamical behavior comes out of these models. And I explain this on the next couple of transparencies and what the precise evidence is that the quantum ground states on the planes are four dimensional. And moreover, I can even say

40:00 more about it, namely that, well, this ground state, as you'll see in a second, has a certain shape, and this can be explained by a very kind of simple mini-superspace model. It can be matched to such a mini-superspace model, which is highly non-trivial, because what I'm doing here is I'm really doing quantum gravity from first principles, kind of not everything is included. Okay, so here are the results. So, well, they even amazed us, of course, we were quite hesitant to move to the scale of four dimensions, you know, having been so discouraged by everyone around. What we would find. So this is what we find in four dimensions, computing these superpositions of this four-dimensional causally-glued Lego george. 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 the three-dimensional volume of the universe as a function of time. So I'm 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 the really the kind of the dynamically generated dimension. The thing that came up in a completely wrong way before in these polymerized say geometries. Now this behaves now, so what is now the concrete evidence for why this is four-dimensional? Well we simulate these objects first of all at the fixed fall volume. We have to do that because of computer-technical reasons. So what we do then is we generate such ground state at various volumes, and then we check the discrete forward, just the number of little four-dimensional Lego bits I put in there, which are called V4, and then I check how

42:30 this time extension here of where geometry is extended in non-trivial, how this scales as a function of this number v4. And it, 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 full volume to the three-fourths. So this is generally evidence that this kind of block structure is four-dimensional. It's a very large scale. So we just use this in standard order. So and well, what is it? 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 is a function of time. This is time. And I've just made it into a kind of rotation of a curve. I've simply made it into this rotation So the circumference here at any time t is proportional to the three-dimensional volume. So this is here, this is our quantum universe, if you like, that comes up. I just told you one piece of evidence that it has dimension four. Another piece of evidence, well, 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 scale as a function of the radius. So say you pick randomly a point in your geometry then you walk out in steps of one, two, three and of course it all kind of defined topological spheres or multiple spheres 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 the so-called half of dimension. So if you do this, for the three-dimensional

45:00 slices here, they come out, again, within computer accuracy, to be three. if I get another dimension which is a dimension I get from studying diffusion in the space-time I also get what is called a spectrum dimension that within computer activities is actually four so there are very strong evidence that indeed you get something four-dimensional and this is exciting and unprecedented So this is a paper that came out in September in PRL and that, to our great surprise, took 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. But you did assume that the microscopic dimension of your simplicity was four dimensions. simplicity. So you didn't, or actually did get rejected at the most microscopic scale. Yes, but as I told you, 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. My deciding just to be for 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 little bit more detail, but some of the measurements you can make on the geometry give you a very strong indication that at small scales, this has, again, absolutely nothing to do with a nice classical geometry. So, okay. So, this much about the dimension, well, can we do more? Can we say more about this block structure? Yeah? Well, so just to relate to that, I mean, can you relate the microscopic dimension If you had changed the dimension of your microscopic elements, is there any kind of way you can relate to the microscopic elements? Oh, yeah. If I was to come back to your question, of course, you could say, well, it seems somewhat arbitrary. If I, anyway, I'm telling you the dimensions come out dynamically here, you know, why do you start with four-dimensional building goals? Well, it's the first dimension where I might expect to see something interesting at all. Of course, it might well be, so if I start now with five-dimensional building blocks,

47:30 that I again find something interesting, but in the best of all worlds, that's actually my favorite scenario, everything would collapse dynamically back to a four-dimensional object. But that would, for instance, that would be very interesting. It would say, telling us that something's very special maybe about four dimensions. But really, the reason why I'm choosing four-dimensional building blocks here is for convenience, and I said it's a first dimension, but something interesting might happen. So we just, this is why we proceed this way. Okay, 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 generically? Well, and the answer is, yes a very interesting news number two is that one can write down an effective action which gives rise, I mean, effective action of a single variable, where the single variable is basically the volume of the universe and which is completely equivalent to giving what is usually called a 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, where are we just fitted? 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 sections tell you what a mini-superspace model is. super-space model is something quite different from what we've done here. It's mainly super-space model. We start classically by saying, okay, at very, very large scales, our universe is homogenous and isotropic. So it has very, very little structure. And the structure, the only structure that is there, is a scale factor A of T, which I keep track of. So everything I keep track of the entire field contents of Einstein's gravity, so the g minu of x and t, is a single mechanical variable, which basically is again the volume at a given time. So I throw away already classically all these other degrees of freedom and hope it gives me still something interesting after I quantize. And then I quantize this one variable reduction of gravity and

50:00 this is what is called underlies typical quantum cosmology 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 super space 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, not positive. But what I then monitor at the very end of the story, what I found the ground set, is this same variable AFT, if you will. And what one finds is now again an effective action of the scale factor A, which very closely resembles what one has in standard for the with a couple of small but very important differences. So this is kind of the 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, so it basically knows about curvature of the spatial slices. And here have a cosmological constant term, it's basically a void term. So now this is how it comes out of our simulation. So this is now with all degrees of freedom of gravity treated dynamically and 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 science here. So you actually have a minus sign in front of this We also mined a sign in front of this linear potential term. Now this is completely devastating for the quantum cosmologists because, as I said, this is of course then an action that has a kinetic term of the wrong time. And if you the form of the Tritian Parsons, especially in this region of the world, have often put forward, then you put up here a fatal divergence from this so-called conformal mode, so this

52:30 conformal divergence, which you, in some ad hoc manner, have to fix. Now what I'm saying is, if you receive the correct way, what you reckon is to obtain and from all these other degrees of linear gravity is an effective turn around of this sign here into, which then results in a genetic turn of the correct sign. So, in the same pose for this linear turn 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, the effect... What? Doesn't 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. Of course, this describes, then you have to look at what this describes, what this describes, I mean, according to mechanically, right? 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, Planck scale, if you like. So, we still have some, the difference with our mini cosmology, I'll get back to your point, the other difference with our mini super space cosmology is this thin stock here. So there is some, which is not there if you just look at the straight action as I've written it down here. Well, I'm just saying this is what comes out. Of course, from a quantum mechanical point of view, I like this very much because I could never really give this convincing way with this sign here. And in fact everyone loves, would love to have an argument of why this sign is actually not there. So here's the good argument for you, which is 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, but at least you have an overall sign switch. So well this is what it is, that's a published work in a new article of ours, and maybe let me just summarize it.

55:00 So I started off by saying my motivation was to try and understand what space term looks like at very, very small scales, and for that I had to unify I taught them mechanical and general relativistic principles, and I did this in a very simple-minded way. I would say what I'm doing is using 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 I think here what turned out to be very crucial were these causality conditions, or making the whole set up a way to, of course, there are technical difficulties involved in that, which I didn't have the time to talk to, but I'm going to explain to you in private, if you want to learn about it. 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 effective 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 you're simulating Euclideanized geometries, so we still rotate into the Euclidean sector, so you still have to do an inverse on those. So, with a very direct physical interpretation, you should also be somewhat careful. So, what this means? Well, of course, I've been comparing the mini-superspace cosmology in the Euclidean sector, if you like. So, the interesting thing is that these things were really derived from scratch. I mean, really from first principles. And, well, what happened, namely that absolutely nothing interesting was found on this physically relevant case in four 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

57:30 gravity as defined through this core dynamical fragmentation, which is a central theme of a new network by the name of NRAGE, where this stands for UP 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 you should check out this website and thank you very much for your attention What did black hole look like in this picture? I mean, can you say something about the geometry of that? Yeah, it's a good question because, of course, black holes are formals in the minds of many people who respond to gravity. So, the first thing here is kind of quite interesting. You have your 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 the term of geometry. And of course, and well, actually students of mine have approximated a black hole geometry as such a triangulation. And when you look at it, 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 central singularity. So we cannot really, well, we keep some distance from that. anything like the eventual horizon in any immediate way, you know, there's nothing that happens in any dramatic way to a triangulation. I mean, there's no topological invariance associated with it? Well, you can, of course, you can try and figure out, for example, what an apparent horizon looks like in the scramble 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 or black holes? Well, then the point is then, well, you first have to set up a language. You first have to actually pose the correct question in such a completely non-interruptive setting. So, first, for example, you have to say which ensemble of geometries you would like to start with

1:00:00 Again, you have to start to find observables, for example, the nice thing would be to define an observable that has the value one, if that is eventualized in some ways in the geometry, and zero if there isn't. Of course, doing this in a few terms, first you have to really think about what your superposition should be. And, of course, this is, again, since it's background-dependent, again, you must realize that you have to define everything from scope. 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, you have to first define what you would mean by, say, an ensemble of 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