Emergence of a 4D world from causal path integrals

0:00 This morning is Renata Lowell from the French who is going to talk to us about causal part-interests. I should start off by saying what a great honour and pleasure it is to be speaking here with one of the very large and diverse crowd of PhD students. I think you couldn't mind me. Sorry, I turned around. I forgot that. I hope you nevertheless understood what I was saying. The subject of my talk is, well, obviously, because it's on quantum gravity, one of the areas of physics where Christoph has had a very great influence, not only on his students. So what I will be talking about today is actually a recent piece of contragration, and since I'm here in front of a crowd that probably has heard it all before, well, happy about this one! Well, the piece of research I'll be talking about suggests that causality at the Planck scale May be responsible for the fact that our space-time is four-dimensional. Now I'll explain in the rest of my talk what I mean by causality at the time scale and how I go about expecting such a result for you. Now the result really has been derived from a rather minimal set of ingredients which I would regard as a virtue. It is derived that you do need to make an assumption or hypothesis about space-time geometry at the very smallest scales, and in my case this will take the form of a microcausality condition, which I will explain in what follows. I will use a gravitational path interval of the right kind, however.

2:30 Which does schematically, of course, as I said, an expression here of this form, where we sum or integrate over the space of all geometries, the equivalent classes of Lorentzian active streams, each one the way we see to the eye at times of gravitational action. Now, one very crucial aspect is to tell you exactly How I'm going to evaluate this path integral, and particularly how I'm going to evaluate it in a non-determinative fashion. The construction here will be very geometrical and to my mind simple, and it doesn't require any extra ingredients, extra dimensions or symmetries beyond those of the class of the theory of gravity. Quantum gravity, before we get to the nitty-gritty stuff, because quantum gravity, as you imagine, should provide not only a hand-waving but actually a quantitative description of quantum geometry at the Planck scale, so about quantum space-time at the Planck scale, which includes identifying its fundamental degrees of freedom. The way they interact and thereby give rise, amongst other things, to the space-time we observe at much larger scale than others. Note that these fundamental degrees of freedom, taking into account everything we know about quantum theory and general relativity, will not be the smooth metric fields g and u of x. Rather, of course, what we imagine to happen It is that as we probe space-time at smaller and smaller scales, we find larger and larger energy-momentum fluctuations, which in turn, of course, have a back-reaction on space-time itself, or curve it more and more as we probe space-time at shorter and shorter distances, and following Wiener and others, you may imagine, therefore, haunting geometry. At the Planck scale there is some kind of wildly quantum fluctuating space-time following.

5:00 But of course our task is to make such pictures quantitatively. Now, since four-dimensional gravity, at least in the restricted context I wish to explain, is not perturbatively defined, We cannot reach a description of this quantum space on anthropomorphic scale from linearly to turning around flat, we call it space. So we have to think of some other way of doing this. And of course, although maybe it doesn't require special emphasis, the situation in quantum gravity is of course rather special. We do not have any experiments to guide us, since in the foreseeable future we will not be able to hold physics at such large scale directly. What we need then is non-termitive and background independent quantum theory, which is just a fancy way of saying that you want a formulation of the theory where all geometries Never mind how far away they are from Platonikovsky's place, participate democratically, and none is a priori distinguished. And these latter features, of course, are already features of the classical theory of general relativity. And what I want to inform you is that quantum space science, the structure I'm looking for, should emerge dynamically. From the collective behavior of some fundamental geometric excitations that we still need to identify at the bottom scale. Now these are beautiful words and everyone would agree that this is how we should proceed, but also one shouldn't lose sight of the fact that this attitude to the quantization of gravity does make life difficult. Now, why? Well, because in absence of any background structure, we in fact have thrown away the yardstick with which to measure geometry or to measure what is happening in your theory.

7:30 So, in a sense, your non-perturbative construction first needs to generate its own yardstick, again its own background. Why people say they work outside the field, that even extracting a classical limit from such an expression is usually very hard work and non-trivial. Moreover, in this wildly quantum fluctuating setting, you're building that dispensing. Your classical intuition cannot seduce me, and the work I will discuss today will illustrate Many of these points are quite explicit. So what I will use today is a non-interpreted version of Feynman's sumo position principle, in other words, the path integral, and apply this to the case of four-dimensional space and geometry. One of Chris's former students, the champion of canonical quantizers. Well, of course you'd be right, and this is not quite how I started off. Now, since I'm the only one of his former students who's actually talking at this conference, I thought I should take just a couple of minutes to describe what it was like to be this student. Now of course I can strictly speak speaking only talk for myself and the people of my generation who I knew closely but I believe some of these things will be quite have been quite invariable over time so now although we didn't call him sir it still was true that we all were in awe of his persona and his intellectual capacities. And we loved his great sense of humor. Now, as a supervisor, Chris was a very conscientious person.

10:00 He was always immensely constructive in the sometimes vague discussions that we had with him. And I should say that one thing he did never do The idea was to impose his own ideas and opinions on his students, not even voice them in front of his students. Now this can make life very tough for graduate students because you really, maybe subconsciously, of course you're looking for the easy truths to absorb from the mouth of the supervisor. But if you were Chris' students, these would just not be forthcoming. And that could make life rather difficult, although with hindsight, I think it's a very good way to encourage students to think independently, and it suits me well, and I'm very grateful for it. Now, it does also make it difficult for me to assess the influence you've had on my scientific development. Now, thinking about this a little bit closer... I do get the feeling that it might be considerably larger than I was first inclined to think, although we were given, as I already implicitly said, much freedom as students about how to go about our work. Now, Larry illustrates this by saying, well, what it was like to start as a student with Chris. Me. A long time ago. I was 85. Now I came in as a beginning PhD student and then we had to have a discussion to decide what I was going to work on. Now Chris said, now there is one very fundamental decision you have to make. Do you want to work on Now of course, less by ignorance I said I do not want to work on strings, so I don't know what would have happened if I had shown you string parts, so this is not what I did.

12:30 Having chosen non-strings, he proposed to me three topics I may wish to work on. Now, I indicate these here by three little boxes. Well, these indicate Chris's famous box files. Which in the day before the archives, the really precious sources of information, went together in groups of various topics. So I was given, over the summer holidays, three of these precious box files to work through. And they had to do with various subjects which I might wish to work on. Now the first one which I ended up doing eventually had to do with quantizing constraint systems, looking at the commutator of constraint and quantizing. And well, what I ended up doing before long was poring over a couple of characters' papers on the litigation model and of course the very famous Lesus lecture notes by the master himself. Now, very interestingly, what I recently remembered, I didn't actually remember topic number three, but topic number two, we said, well, you may want to look at this, some very interesting papers that have come out recently that talk about 2D random geometries and what one might do with those. Going back to what Kelly said yesterday night, Chris always being light years ahead of everyone else, only of course with a little delay of 15 years, I realised what a beautiful suggestion that was, and this brings me actually back to the topic of my talk today, which is really very much a development of these ideas. Gary Givens also said last night, well this of course goes down the back in the direction of the gravitational pass integral, and particularly about the Euclidean pass integral that then and now is practiced by the Cambridge crowd. Now, nothing... Now, what I'm going to do today is actually give you an explicit demonstration that if you do it properly, there's nothing wrong with gravitation passing through, and you may even have interesting things.

15:00 So I do this. So in great tradition we start from something not quite right and well this something is Euclidean quantum gravity so it's quantum gravity but has a lot of signature and here's again my past integral, this form of expression. Of course it's more a statement of intent rather than having any physical content. context before I haven't told you exactly how I define this. Now the integral of the geometries in order to make this well defined I only convert into a statistical sum I convert this entire very formal integral into a well defined sum of four-dimensional geometries, so Euclidean, Euclideanized geometries to start with, so this entire thing becomes a discrete sum, a wide discrete sum. Because, of course, I need to choose a regularization model as part of making this well-defined and being able to discuss its convergence properties, then the one I adopt is one where, not following all the work by Reggie, the retrocalculus and what is called dynamical calculations, I blew my spacetimes from this green, flat building blocks. So here's one of these flat so-called forces. The integral of all geometries is now, well, in general relativity without all of this, converted into a sum of all possible ways in which I can view such building blocks together to obtain four-dimensional manifolds, these manifolds that are of fixed topology. It turns out I have to do this in order to make things well-defined. I don't have much time to go into how this imposes geometry, only remind me that curvature in such simplicial objects is encoded in how, well, in deficit angles, so how you pick up deficit angles when you work around lower dimensional simulancies of these manifolds.

17:30 Now, the very nice thing is that It can be shown how you do this by putting in two regulators. One is basically the x-lens, or x-lens squared, which is a-squared, just the size of building blocks, essentially. Another one is the number of building blocks you use to construct your geometry. Now, of course, what you want to do in the end is to remove these regulators and try and understand Well, that's how the symmetry behaves in a limit where the volume becomes very large, individual building blocks become very small, and you keep some kind of finite physical piece of space-time, a volume of capital V, fixed 1 to the 4th of this limit, and throughout the whole... Were 15 making this well-defined and looking for interesting limits in here. And I should say, well, from the point of view of cosmological interpretive theory, it's a somewhat arbitrary but nevertheless convenient starting point to just use as it lays. A Reggie version of the curvature term, and a cosmological concept term, which turns out to be needed in this construction. Now this whole business goes by the name of dynamical triangulations and this is really an outgrowth of this work on two-dimensional random surfaces that people started working on in the mid-eighties. Now this turns, if you like, quantum gravity into a form of a completely different kind, namely into one of combinatorics essentially, where you have To count how many kind of incarnations there are of a manifold with a given value of the action and the fact that you turn that into a very different looking problem enables you to apply very, from the point of view of quantum gravity, non-standard both analytical and numerical tools to this. So, now you ask me, good, so, you do it, I mean, what is the result if you now democratically superpose all these regularised geometries and look at, look for a continuum of such a theory?

20:00 Well, what we expect, okay, this is a naive expectation, but we'd nevertheless be happy to find if... We generate a quantum geometry for such a superposition that is at short scales highly quantum fluctuating as you would expect, but then if you look at it from a large distance at much larger coarse-grained scales, some version of a 4D classical space type should emerge. So this is just the classical limit of the construction. The problem in making such a construction is not with one to get something that wiggles a lot at the plan scale, it's not so difficult to dream up little models, however, the problem is really with two, and more specifically, it is with obtaining a model of quantum geometry that even has the correct macroscopic dimension of four. Now you may ask, okay, how is it possible? I superposed four-dimensional geometries. I just told you I take the four-dimensional building blocks and superpose these in all possible ways, the manifolds are constructed with them, so how come that the final result is something that is not of dimension four? Well, this is the beauty of quantum theory, in a way. So, what then illustrates the effect by, well, a two-dimensional example, think of a two-dimensional sheet of paper. Now, how do I convert this to an object on a mathematically different dimension? Well, there are two ways in which I might want to do this. As string theorists we do, we curl it up to obtain an object of dimension one, effective dimension one. So this happens in these non-perturbative approaches because you allow very strong fluctuations and curvatures at any scale, including very short scales. So it actually turns out that typical geometries that contribute to your path integral are of this, you know, one of these.

22:30 The two screw types. So going back to this example of Euclidean dynamical transformations, really one out of two things happens depending on how you tune the gravitational, the bare gravitational coupling points. So if you make it small, including making it vanish completely, Then you find quantum geometry that is completely quantum, actually has an effective dimension that is infinite, so it's an object that basically has no linear extension whatsoever, but everything is condensed, a very highly dimensional object that runs on a central point. So this is one thing that can happen if you then make the inverse rotation of the back-up constant larger to run into another phase. Which is very akin to kind of a stringy space where the geometries degenerate into so-called branched polymers. So these are one-dimensional objects that branch, well, they can branch anywhere. So this is something that has an effect in the house of geometry, of two. And, well, how do you know this? I mean, how did you manage to extract such information about quantum geometry? Well, you did this by a combination of numerical and analytical tools, so Monte Carlo simulations actually revealed very quickly that such structures were present, and subsequently... He also wrote down exact asymptotic estimates for the propagate or the partition function, which I will later deny to call it, as well as constructing mean field methods which would just reflect the basic dynamics that could give rise to such an effect. Now this is not just a fluke in this particular model, but I believe And I'll be happy to explain my reasons in person to you if you want, that this is a generic behavior for Euclidean quantum gravity models. Approaches of this type. An important lesson you should take away from this is that if you're really determined to do quantum gravity non-interpretatively, well, realize that

25:00 Well, hence those out there. So, no aspect of quantum geometry can be taken for granted in such a formulation. Not even its dimension. Of course, I started by saying I'm not starting with quite the right thing, and of course we didn't quite do the right thing in this construction, in that we started off by looking at Euclidean class intercourse. So, here comes in causality. So, now, what I will do is to involve microcausality, and that's something that was not at all present in the premium quantum gravity. I wouldn't have been able to formulate it there because there was just, you know, we're treating gravity, it doesn't know about light forms, it doesn't know about causality, it treats time as it does space. Now, following a very old idea of more than 20 years ago of how you type you want, the way it's going to be implemented in the path integral is by having each history contributing to the path integral. Be causal. Now, translated to this dynamical calculations approach, we will require that each recognized positive history I sum over in my big sum should be causal at all scales, down to the minimal scale, regular kind of UV kind of scale, a to the small s. Now, what does this now mean? How does it play out in concrete terms? Well, first of all, all my building blocks, I will choose to be in a costume, so they are still flat, but I choose them as pieces of flat Minkowski space and not of the building space, so in principle, of course, you can draw the light blocks everywhere, and what also changes is that some of my etchings Square would be positive, indicating space-like averages, and so would be negative, minus a squared, indicating time averages. But this is just part of the regularization that shouldn't probably be too much. Now, the essential ingredient now...

27:30 In building up space sciences, first of all, to do these building blocks persistently, so to do space-like to space-like, time-like to time-like, to match them up locally, but even more importantly, it turns out that causality imposes absence of rhyme strings. Now I illustrate to you on the next slide what this looks like. Just to say this whole framework I call causal-dynamical triangulations, of course, and you may recognize, of course, the theme of causality being put in as a fundamental building principle that is also present in the causal centric approach of Raphael Sotius and collaborators that I'm sure he will tell you more about later this afternoon. He said no so-called baby universes are allowed. So I am in a, so I have, I know what a time direction is, a time arrow, and this is the flow of the really want your histories to be causal. You cannot allow for such branches. Because you know very classically that the co-construction of the Lorentzian method simply has to be generated at least at one point when we have such a course of branchness. So these will be the configurations we explicitly eliminate from the path interval. And it turns out by imposing such a condition, well, you eliminate a huge number of geometries. If you just started from Euclidean geometries in the first place, because then there would be absolutely no way to say, well, something branches off in time and the Lorentzian structure has to go back. There would just be no way of saying this. So you could not possibly, if you insisted on a fundamentally Euclidean formulation, get rid of these baby universes.

30:00 But you can if you do it in such a framework. The entire difference, I mean all of the difference, if you look at your simplified versions of the quantum gravity model in lower dimensions, and of course this is what we started out doing, just try out, well what effect does the removal of such outgrowth have in lower dimensions? And you can then see very explicitly in two and also in three dimensions that it completely changes. The quantum geometries that come out of the non-intermittent path integral. So it's a highly non-trivial condition to impose, and it will affect the end result, and there's no doubt about it. It does that also, as you can see, in four-dimensions. Now, how exactly it will play out in four-dimensions is up here, right? Completely unclear, because you see that the quantum geometry fluctuates very much at short scales. So, and what you do by suppressing many universes, you only say branches, branches cannot occur in the time direction, but of course you still have three spatial directions where geometries can go completely wild, and it's the relative dynamics in the time, in the space direction that will contribute to what your dynamics will finally look like. Implicit analysis is a specific notion of time with respect to which you complete these eight causal branches. And the notion of time used here in this context is very natural. It's an invariably defined version of proper time. I call it T. So this is, what Truman takes is too literally. Because, of course, the spacetimes that contribute here, already at the regularized level, are far away from sleuth spacetimes. They have curvature similarities and, of course, even more so if you now take the limit as the space goes to zero, well, you're again, of course, probing a space of completely wild objects. I mean, some version, I mean a higher dimensional version of the space of all non-differential paths, which you know dominate the real measure of the non-righteous departure point.

32:30 So this is what we have here, and what I also have to tell you, but will not have time to go into detail, is that our construction comes with a map. There are many examples of momentum spacetime to Euclidean spacetime in this triangulated context and the effect of this map is to convert the complex amplitude to real Boltzmann waves, really in the way one would expect. So such a map is well defined on this category of geometries that goes into my path integral and I do need it because I am unable To discuss convergence properties of the infinite sums that appear here in the presence of the factor of i up here. So I really do leave that at an intermediate step. Now the object, the central object that the theory constructs for me is, after euclideanizing, is a discrete propagator. The final stage of geometry, and on the time distance between, what the time distance is between the key and the distance. This is why, in the way I've constructed it, I can really think of it as a discrete evolution in a discrete time, so akin to what you find in regular lattice history. The duration must be expressed to the propagator, corresponding to discrete time distances. This is the main object, and of course you can then visualize this, if you show that this transform matrix has the correct positivity properties, that this continues into the minus-t age. And of course, once we have extracted the controlling nature of this system, Now, after these somewhat technical remarks, let us return back to our main objective.

35:00 So, is it now the case that if you implement this framework in four dimensions, the causality conditions you've now put in will take care of these geometric degeneracies we found in the Euclidean approach? Of course, as I said, we played around for a long time with these lower dimensional models and they exhibited many properties about which I could talk for hours, but of course it was with some trepidation that we then looked forward to the outcome of the first numerical simulations of this particular path interval. And I've learned to appreciate what a marvelous thing it is to be able just to put this thing on the computer and see what comes out to give you an indication of whether you're talking about the empty set or you're going to be talking about a non-trivial structure that emerges at a non-perceptively for some construction. Now, luckily for me, a miracle seems to be happening. Well, we've been conducting Monte Carlo simulations for a few months now. These have revealed, remarkably, of a quantum ground state, of extended geometry, which moreover, macroscopically, seems to be four dimensional. Now, before going into great details, into some of the technical details, well, this is a baby. So this is a Victorian proof, if you like, that an interesting structure indeed does emerge in four dimensions. So this is a quantum universe that comes out. So the ground state, like of the Hamiltonian I wrote on the last page, whose explicit form, of course, we still do not know. A typical, this is what the Monte Carlo simulation generates, this is kind of a typical version of the ground state, so what do I mean by typical, well, first of all, what we've plotted here is only one very simple aspect of the geometry, and this is its three volume as a function of time, so this is just a typical past integral history, and what you're plotting is however only one of the simulated aspects.

37:30 So here you see clearly that it is an extended structure, so there is string volume here at every slice, and well now the critical question is, is it an object of the kind we're looking for? And well, the way you look at this is again you first measure by numerical simulations what its properties are. So the evidence, so really what you saw there, as I said, is a Monte Carlo snapshot of the ensemble where I should add that what we have to do, I should protect it for reasons here, in this simulation, we have to keep the 4-volume fixed. So we are actually, this is the path integral we are looking at, as there's a delta function that's fixed and seems to be at a total 4-volume V4. Now the first very non-trivial thing to check is, to check whether that's an object of macroscopic dimension 4, is to do the following. You pump, if you like, 4 volume into this object, and then you see how the time we're stating here scales relative with the spatial volume, v3. Well, what is completely non-trivial, because again, we're sitting in a long-term formulation, is that it does seem to come out nice. So, the time expended as we enlarge the full volume we inject into the subject seems to scale by the fourth square root of the full volume, and conversely, the three volume really does seem to scale with good numerical accuracy as an effective... Indicating effective dimensions of four. So, and one way of understanding this is, well here you get the curve, is to look at the correlators for three volumes of adjacent slices. So this is how, I mean you don't just do it by visual inspection, but this is one thing that you look at.

40:00 These are the distance delta, just discrete time steps apart, 1 to 3, 4, where we now rescale this time distance by the fourth square root of the four-volume. Now, if this object really became four-dimensional, and you do this exercise at different volumes, here indicated by the different colors, you would expect the curves to fall on top of each other. Well, as you see from here, they do very, very good accuracy. So there's a very strong indication that indeed the global scaling of this kind of quantum universe we've generated dynamically is of dimension four. And another thing you can do, which suggests itself in this philosophy of quantum economics, you can look at dimensions of spatial slices. And what you do there, you go walk in the spatial slice and, well, basically you compare radii to volumes of force. And from this, again, you extract a critical exponent, a house of dimension, which comes out to be 3. Again, within numerical accuracy. So, wrapping up, as I said, very remarkably, and even for us, somewhat astonishingly, Macroscopic dimension for completely non-determinative. This is a very promising result but of course it can only be a first step in the further exploration of this model. It's only kind of a part of getting the classical limit right. I conjecture that actually But reducing the correct classical limit in any of these models of quantum gravity, not just my own, but whatever your favorite novel of people in the audience, it is actually a very stringent condition to have the classical limit count out right. And, well, I see this as a very good thing, if it is true, because it just means... Planar scale physics is not just pure speculation. You can just not dream up any model you like that's a planar scale.

42:30 You will have to work very hard in showing that the classical theory comes out correctly. And this could put very stringent conditions on your model. Another condition that you need to satisfy whenever you play around with models of this type is that Well, when you look at your final theory, and you had some microscopy brilliant or similar into it, you should make sure that the final result of the theory doesn't depend on the details of how you set this up, and the final theory means not just the classical limit, but the classical limit and all its quantum corrections should better be insensitive or stable under your changing around your little quantum models at a short length of time. Now, in models of the type I've talked about, this to some extent is taken care of automatically because of the way I set it up. I took a large volume limit of essentially sub-planetary building blocks. But this is something to be kept in mind in general. Concluding now, in non-materials models of quantum geometry, this is already of a form. This classical limit is not automatic. And I believe it was very significant for us. So what I showed to you in a very particular instance of a specific causal and quantitative path integral, how it dynamically generates a quantum ground state, so a muscle or vacua, if you like, which was measured from a thin polar mesh. What was very crucial here, from the outset, is to put in geometries of the correct signature. So this is crucial. And obeying microcosmetic. We've come full circle and I hope I've convinced Alton and our supervisors that the class intervals, if only done properly, are verified and can give us interesting information about quantum space. Thank you very much. Okay, any questions?

45:00 I think Chris should have the right of reply. Right. These new air results are very beautiful. I mean, could you extend a conjecture to say that these integrals actually really exist, that is to say, there really is some measure of some well-defined space or, as you described, geometry, which really does exist as an integral, is there? I mean, it seems very tempting to conjecture that. Well, we're not quite there yet. No, I understand that, but is it a realistic conjecture? Yes, I think so. You take it. Yeah. I have two questions, actually, which I will sneak in here. If you had started with five-dimensional Minkowski simplices, would you have come out with the effective dimension being five people and so on all the way up? That's question one. And two is, could you comment on whether there's any relation with the... Work on the other form of regé, that is, by integrating over the edge lines some fixed triangulations, for example, as represented by Handler and Williams and others. Okay, well, two very good questions. So, regarding, well, the arbitrariness of my initial input, in a way, could I have started with, well, five-dimensional gravity, you know, building blocks to stories? My pet project is to, well, my hopes would be, it would be very exciting, let me phrase it this way, if you could show that no matter what you start from, of course, I think you have to go to dimension four and higher to see anything interesting, but it would be extremely exciting if you could just, yeah, start with dimension seven and then show that everything dynamically collapses to four dimensions. Exactly. That would be wonderful. That's what string theory is a string of. I cannot comment possibly at this stage, but it's something that should be kept in mind. It would kind of dynamically show that there's something very special about quantum motion. So, so much for your first question. Second question, is it related to quantum magic calculus? Well, yes, of course, in the case of quantum, it shares many... Of the construction details with quantum magic calculus, however, quantum magic calculus is so far only works for Euclidean signature, and I believe, and I could explain in a little detail from what I've seen people publish, that they run into another incarnation of the geometric dependencies when evaluating, when really taking the continuum limit of their passing tools.

47:30 Let's not do it here, I'll catch you later. So, Carol. In your way of stacking the simplices, the Minkowskian ones, into the space-time geometry, you use a geometrically proper time. And then it does not depend on the way you slice. You can slice in many different particular seconds. If you want to face the gap, you get in here, in the classic, something that mimics. Yes. If I could start talking about this, I would have to read into coordinates. And there, I would think I would first have to know, I would first have to show that this quantum geometry that I generated for you here has macroscopically something to do, say, with Fleck-Minkowski space, in some coarse-grained algorithm. Then I can, or some other classical geometry, then I can only start asking the questions, I mean, I will still have this parameter T everywhere, but then I can just start talking about, well, what other components I may wish to introduce. So, that's one way of answering your question. If you ask it at a kind of more fundamental level, say, okay, if you're entire, if you did the entire quantum construction, With some different notion of time in the first place, what would happen? Would your final result be independent? I don't know. I would hope that such a covariance, you know, would exist, but already we know, say, if you do.

50:00 QCD, you know, and look at various formulations, say, in different gauge choices, it is very, very difficult to make, to explicitly show that physics is the same in different formulations, and certainly, here in gravity, it will also be very, very difficult to show that explicitly, and, well, anyway, an alternative candidate is not yet around, which I, you know, compared to you. Yeah, of course, I mean, what I can only show you what these simulations do, they will, of course, they are not continually, but what study says, of course, how physical quantities behave as one takes this limit of going to larger and larger objects. Now, so, and this is, of course, The final result of what comes out is, or ought to be, a continuum theory. Of course, to write this down explicitly may be quite difficult. I'm sorry, your second question related... Lawrence, in various... Yes, that would be, again, a question to ask of... Well, once I've understood the detailed, more local geometric properties of this final quantum space time, I can then start worrying about, well, recovering... Symmetries of the classical theory. So currently I cannot answer. I'm actually a patient on all of these things, but there does seem to be a tension, right, between your notion of causality and of the sex notion or what the Cambridge group would include and so on. So for example, the Cambridge group would say that the basis of black holes and so on is really to have this So it would look like throwing away all of quantum physics from that perspective. Is there some way of recovering this quantum physics with this kind of thing in the future? The second question is that what do we know apart from dimension about the ground state? Does it look like deciduous? Does it look like anti-deciduous? Does it look like anything at all? I mean, the picture you drew doesn't seem to me you can walk as an invariant at all, for an invariant to become more of an invariant.

52:30 On the other hand, if it is a ground state, it better be. So, what is the statement? Yeah, okay. So, the first question was causal. And black holes. Yes, yeah. It's a very intriguing question. I mean, the question is, yeah, in black, in all these calculations, Now, what I believe is that, well, if you can kind of do this to the end, you will find that you look at the classical, I mean, there's of course a reason for why these Euclidean methods do encode some interesting geometric information. It is not always, I don't think it will be straightforward, like by looking at the kind of semi-classical approximation to To get this out of such a framework. But I believe one will find, if I'm going to do it to the end, that there are certain ways of how one, in a Euclidean framework, phrases physics and capture ultimately Lorentzian properties of the system. How this will play out in detail, I cannot yet tell you at all, but I believe... That's kind of the best spin I can put on Euclidean quantum gravity. It will turn out to encode in a very concise way some properties that ultimately are forensic. Now, your second question was? The ground state. The ground state, yes. So, the ground state. We have made some... It's a progress in trying to understand this very kind of non-invariant structure one sees there of this bubble of space-time and we have a, it should be taken with some caution because what one sees here is what we already have. And secondly, you still have to retranslate the Euclidean structure you see here to a Lorentzian structure. So we believe we have actually formulated a good, effective, classical logology that depends on the three-volume that explains the shape you see here. And this may well be some digital time. We believe this is a hand-de-sitter-type solution.

55:00 The detailed character qualities of the space that we haven't looked at yet, but this can be done over the day. Go on. I have two questions. First of this, in the dynamical demolition, you're conduced to this idea that you have a two-dimensional parameter space, lambda, the unipost, the unipost, and the geological post. And you have to look for a critical surface there, a critical point to look at. These were simply connected, I don't know, in these. Are those results, as Hatch would, equal value of K and lambda? Yeah, that's a good question. What I always have to do in order to take the infinite volume limit is I have to tune the cosmological constant to a specific value and this is exactly like the hypersonic surface in parameter space you are referring to. So I have to do this here always in order to go to the infinite volume limit. Now, I have, of course, at least one more coffee question for you. I still have some very interesting questions. You asked, how do results depend on this? Now, in the old, the clicking modes... Because people saw the various values of k, as I explained, in very degenerate phases, people then looked for a possible second-order phase transition at which physics might still come out right, if you like, of the right dimension. Now, this four-dimensional world doesn't seem to have a second-order phase transition, but we also believe that it just may not be necessary. There is a continuum theory to just fine tune one of your variables, just to go to the continuum limit and the dependence on how you choose the value of Newton's constant, we don't yet know, we believe it will probably end up being some, just some overall scaling which will not change the physics in any fundamental way. Well, as far as we can see, these geometries are still very wiggly at short scales, in the spatial dimension.

57:30 But not at the scale of A. No, but at some scale, of course, you would expect that it's a dynamically generated scale that knows about exactly the lubricants. One quick last question. I actually have many questions, but I'll limit myself to one short comment and one short question. The question is about sum over topologies, which you didn't do. There's a potential problem there with sum over topologies. Can you comment on that? Yes, it is completely undefined. It just cannot be done because of the sum over topologies. I'm going to make this talk finite at this point and I'll bring in the coffee until 11.30.