Emergence of a 4D World from Causal Path Integrals

0:00 So this morning is Renata Lowell from Utrecht, who is going to talk to us about causal partners. I should start off by saying what a great honour and pleasure it is to be speaking here. Mine is a very large and diverse crowd of Chris's former PhD students. The subject of my talk is, well, obviously, because it's on quantum gravity, one of the areas of physics, where Chris has had a very great influence, not a lot of his students. So what I will be talking about today is actually a recent piece of quantum gravity research, and since I'm here in front of the crowd that probably has heard a lot before, well, have you heard this one? Well, the piece of research I'll be talking about suggests that causality at the plump 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 plump 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. shift. It is derived, what you do need, as virtually in all these approaches, 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 microcosality causality conditions, which I'll explain in what follows. I will use a gravitational path integral of the right kind, however, which does schematically

2:30 of course insert the expression here of this form, where we sum or integrate in the space of all geometries, equivalence classes of Lorentz-electives gene, and each one, the way we see to the eye the gravitational action. Now, one very crucial aspect is to tell you exactly how I'm going to evaluate this pass integral and particularly how I'm going to evaluate it in a non-determined fashion. The construction here will be very geometrical and to my mind simple and it doesn't require any extra ingredients like these extra dimensions or symmetries beyond those of the classical theory of gravity. Now a few more words about quantum gravity before we get to the nitty-gritty stuff. Because quantum gravity, as we I imagine it should provide not only a hand-waving, but actually a quantitative description of quantum geometry at the planet scale. So about quantum space-time at the planet 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 a much larger scale around us. 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 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 spacetime itself, would curve it more and more as we probe spacetime at shorter and shorter distances. And following Wheeler and others, you may imagine therefore quantum geometry at the plant scale as some kind of wildly

5:00 space-time following. But of course, our task is to make such pictures quantitatively. Now, since four-dimensional gravity, at least in the restricted context I wish we'd describe it, in four dimensions, is not perturbatively defined, we cannot reach a description of this quantum based on at the time scale, from linearly to turning around flat, we call three-phase. 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 of quantum gravity is of course rather special, in that we do not have any experiments to guide us, since in the foreseeable future will not be able to put physics at the Schlapp scale directly. So what we do need then is a non-determinative and background independent quantum theory, which is also 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 space, participate democratically and none is a priori distinguished. And this letter features of course, well both of these actually, are already features of the classical theory of general volatility. And the view I want to put forward is that quantum space-time, the structure I'm looking for, should emerge dynamically from the collective behavior of some fundamental geometric excitations that I still need to identify at the broad 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, in fact have thrown away the yardstick with which to measure geometry or to

7:30 measure what is happening in your theory. So in a sense your non-protective construction first needs to generate its own yardstick, again its own background and something that is not always appreciated by people say they work outside the field, that even extracting a classical limit from subject extraction is usually very hard work and non-trivial. Moreover, in this wildly quantum fluctuating setting, you're building that dispensary, your classical intuition can also easily and the work I will discuss today will illustrate many of these points quite explicitly. So, what I will use today is a non-interpreted version of Feynman's superposition principle. In other words, the path integral and apply this to the case of four-dimensional space-time geology. Now, at this stage you should really say, well, how come I mean, isn't she one of Chris' 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 Serb in those times, it 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 rare discussions we had with him. And I should however say that one thing he did never do 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 a graduate student 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's 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 infinitely. And it did suit me well and I'm very grateful for it. Now it does so also make it difficult for me to assess the influence you've had on my scientific development. Now thinking about this a little 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 Larry illustrates this by saying, well, what it was like to start as a student with Chris. Now, this is me. A long time ago. I was 85. Now, I came in as a beginning PhD student and then we were to have the 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 strings or do you want to work not on strings? Now of course, less by ignorance I said, I do not want to work on strings. Now, so I don't know what would have happened to have chosen string parts, so this is not what I did.

12:30 Now then, having chosen not 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, were gathered in prints 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, for weeks on end. 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 two-level geometries and what I might do with those. Now going back to what Kelly said yesterday night, Chris always being light else, only of course with a little delay of 15 years I realized 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. Now as Gary givens also said last night, well this of course goes back into the direction of the gravitational path integral and particularly about the Ukrainian path integral that then and now is practiced by the Cambridge crowd. Now nothing bad was ever said about the gravitational gravitation pass integral while I was a indicator. Although subliminally, I got some message that there was really something terribly wrong with it. Now, what I'm going

15:00 to do today is actually give you an explicit demonstration that if you do it properly, there's nothing wrong with gravitation pass integral and you may even know interesting things from it. So, now this nitty-gritty stuff, this is my densest transparency, where I just flesh out the way 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 that's a long signature. And here's again my path integral, this form of expression. Of course it's more a statement of intent rather than having any physical context before I haven't told you exactly how I defined this. Now, the integral over geometries, in order to make this well-defined, I really convert into a statistical sum of random geometries, if you like. So I convert this entire very formal integral is a well-defined sum over four-dimensional geometries. So Euclideanized geometries to start with, so this entire thing becomes a discrete sum. Why discrete sum? Because of course I need to choose a regularization in order as part of making this well-defined and being able to discuss its convergent properties. And the one I adopt is one where, well following all work by Reggie, the retrocalculus and what is called the numerical triangulations, I glue my spacetime from these green flat building blocks, so here's one of these flat so-called force indices, and the integral of all geometries is now, well, in general relativity without the orbits converted into a sun over all possible ways in which I can view such building blocks together to obtain four-dimensional manifolds. These manifolds are called off-picks 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 equals the geometry, only remind

17:30 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 silences of these manifolds. 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-range, or x-range squared, which is a squared, just the size of the building one is the number of building blocks you use to construct your children. Now of course what you want to do in the end is to remove these regulators and try and understand well how the theory behaves in a limit where the volume becomes very large, the individual building blocks become very small and you keep some kind of finite physical piece of space happy to read, fix one before this limit. And, well, the whole work is in making this well-defined and looking for interesting limits in here. And I should say, well, from the point of view, of course, it's a non-protective theory, it's somewhat arbitrary but nevertheless a medium starting point to just use as a place. A regi version of curvature and a cosmological which turns out to be leaving this construction. Now this whole business goes by the name of 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-80s. Now, this turns, if you like, quantum gravity into a problem 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, 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

20:00 democratically superpose all these regularized geometries and look for a continuum of such a theory. What we expect, this is a naive expectation but we'd nevertheless be happy 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 core strength scales, some version of a 4D classical space type should emerge. So this is just the classical limit of the construction. Now it turns out the problem in making construction is not with one to get something that wiggles a lot at the plant scale. It's not so difficult to bring up the models. However, the problem is really with two. And more specifically, it is with obtaining a marked quantum geometry that even has the correct macroscopic dimension of four. Now you may ask, okay, how is it possible? I superpose 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, so how come that the final result can 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 of a mathematically different dimension? Well, there are two ways in which I might want to do this. I've crumbed it up and, well, it sounds easy. Or, Or alternatively, I would roll it up as a string theory to the clue and curl it up to obtain object of dimension one, effective dimension one. So this happens in these non-alternative approaches because you allow very strong fluctuations and curvatures at any scale, including very short scales. So you, it actually turns out that typical geometries that contribute to your past

22:30 integral are of this, you know, one of these two screwy times. So going back to this example of Euclidean dynamic triangulations, really one out of two things happens depending on how you to the gravitational, the bare gravitational couple points. So, if you make it small, including making it vanish completely, then you find quantum geometry that is completely crystal, 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 around some central poly. So this is one thing that can happen if you then make the inverse gravitational back up and constantly larger, you run into another phase, which is very akin to kind of the stringy phase where the geometries degenerate into so-called large polymers. So these are one-dimensional objects that branch, well, they can branch anywhere. So this is something that has affected the house 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, people also wrote down exact asymptotic estimates for propagate or partition function, whichever way you might call it, as well as constructing mean-field methods which could just reflect those 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'm happy to explain there are reasons in person, 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, realise that, well, hell is loose out there. So, no aspect

25:00 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 try to do the right thing in this construction, in that we started off by looking at juclidean par-syntegrals, not at the actual physical objects we'd like to look at. So here comes in causality. So now what I will do is to involve micro-causality, and that's something that was not at all present in the clearing form of gravity, I wouldn't have been able to formulate it there because there was just no, the clearing gravity 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 the title was, the way it's going to be implemented in the past integral is by having each history contributing to the past integral be causal. Now, translated to this dynamic of five relations approach, we will require that each regularized past integral history, I sum over in my big sum, should be causal at all scales, down to the minimal scale, regular kind of UV color scale, A, 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 would choose to be in a costume, so they are still flat, but they are, 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 everywhere and it would also change that some of my edge ends square with positive indicating space-like edges and some will be negative minus a square indicating time this is just part of the regularization that should probably be too much. Now the

27:30 essential ingredient now in building up space-time is first of all to use these building blocks consistently, so to use space-like, space-like, time-like to match them up locally, but even more importantly it turns out that causality imposes absence of branchings of space-time in the spatial durations. 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 and building principles that is also present in the whole set approach of the Rappel Sorghuis and collaborators, and I'm sure he will tell you more about, later to talk to me. So I said the essential ingredient here in my construction is that no so-called baby universes are allowed. So I am in a Lorentzian setting, so I have, I know what a time direction is, I mean I have a time arrow, and with respect to the flow of this time, what you cannot allow, if you really want your histories to be causal, you cannot allow for such branches, because you know quite classically that the causal structure, the Lorentzian metric, simply has to be generated at least at least at one point when we have such a cross-abranchis. So these will be the configurations you explicitly eliminate from the pass interval and it turns out by imposing such a condition, well, you eliminate a huge number of geometries which would be there if you just started from, you know, Euclidean geometries in the first place. Because then there would be absolutely no way to say, well, something branches off the time and the Lorentz structure has to go back. There would just be no way of saying this. So you could not possibly, if you insisted on the fundamentally Eclinian formulation, get rid of these baby universes. But you can if you do it in such a framework.

30:00 And this turns out really to make the entire difference and all the difference simplified versions of the quantum gravity model in lower dimensions. And of course this is what we started out doing, just try out 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-interruptive path multiple. So it's a highly non-interruptive condition to impose and it will affect the end result and there's no doubt. It does it also as you can see in four dimensions. Now how exactly it will play out in four dimensions is completely unclear, because you see that the potent geometry fluctuates very much at short scales. So, and what you do, while suppressing many universes, you always say branches cannot occur in the time direction, but of course you still have three spatial directions where geometries can go completely wide. And it's the relative dynamics in kind of the time, in the space direction that will contribute to what your dynamics will finally look like. Now of course, implicit in all this 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, it's very natural, it's an invariably defined version of proper time, I call t. So this is, what Schumann takes is 2 literally because of course the space times that are consumed here, already at the regularized level, are far away from smooth space times. They have curvature similarities and of course even more so if you now take the limit s, the light of space goes to zero, again of course probing a space of completely wild objects. I mean some version, I mean a high dimensional version of the space of all non-differential paths which you know dominate the venal

32:30 measure for the non-righteous particle. So this is what we have here and what I also have to tell you but we'll not have time to go into the detail, is that our construction comes with a map of Lorentzian spacetime to Euclidean spacetime in this triangulated context. And the effect of this map is to convert the complex amplitudes to real falls one way. It's 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'm unable to discuss conversions properties of the infinite sums that appear here in the presence of the factor of I up here. So I really do need that at an intermediate set. Now, the object, one of the central object that the theory constructs is, after Euclideanizing, is a discrete propagandist. So that's an object that depends on an initial spatial geometry, a final spatial geometry, and on the time distance between proper time distance, And this is why I 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 system. So I actually had a transfer matrix, this object, a t hat, whose t's iteration was fixed-wise to the propagator, was called into discrete time distance t. And this is the main object, and of course you can rewrite this, if you show that this transformation matrix has the correct positivity properties, that this continuous e to the minus t h, we have extracted the controlling h of the system. I mean, you have all you need to know, at least with respect to the choice of a particular valid time. Now, after these are more 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 equilibrium approach. Now, of course, as I said, we played around for a long time, this is 4-dimensional models, and they exhibited many beautiful properties about which I could talk for hours, but of course it was with some trepidation that we then looked forward to of the first numerical simulations of this particular particle. 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 they would be talking about a non-terminal structure that emerges at a non-paterpative structure. Now, luckily for me, so far the miracle seems to be happening. We have been conducting Monte Carlo simulations for a few months now These have revealed remarkably the existence 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 up on the last page, whose explicit form, of course, we still do not know. And this is 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

37:30 only one very simple aspect of the geometry and this is its three volume as a function of time. So this is just atypical past integral history and what you're plotting is however only one of the stereometric aspects. So here you see clearly that it is an extended structure, so there is sleep volume here at every slice and And well, now the critical question is, is it an object of the kind we're looking for? And well, the way you do, you look at this, it's going to be first measure by numerical simulations what these properties are. So the evidence, so really what you saw there, as I said, it's a multi-parallel snapshot of the ensemble where I should add that what we have to do actually for technical reasons here in this simulation, we have to keep the four volume fixed. So we are actually, this is the past integral we are looking at as there's a delta function that fixes things to be at a total 4 volume V4. Now the first very non-tribal thing to check is S to see, to check whether that's an object of the macroscopic dimension 4 is to do the following. You pump, if you like, 4 volume into this object and And then you see how the time we're stating here scales relatively with the spatial volume, V3 of T. And well, what is completely non-trivial, because again we're sitting in a long-term formulation, is that it does seem to come out right. So the time as we enlarge the four volume we inject into this object seems to scale by the fourth square root of the four volume and conversely the three volume really does seem to scale with good numerical accuracy as an effective, indicating an effective dimension of four. So, and one way of understanding this is, well here we go, 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 So these correlates volumes and slices that are the distance delta, just discrete time steps apart, 1 to 0.4, where we now we scale this time distance by the fourth square root of the fourth volume. Now, if this object indicates four-dimensional, and you do this exercise at different volumes, we would expect the curves to fall on top of each other. And as you see from here, they do to very, very good accuracy. So that'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 this fellow three-parisonic forms, you can look at dimensions of spatial slices and what you do there, you go walk into spatial slices and, well, basically you compare radii to volumes of balls. And from this, again, you extract the cryptic exponent, house of dimension, which comes out to be three, again, within numerical accuracy. So, wrapping up. Well, as I said, very remarkably, and even for us somewhat astonishingly, what this function produces is 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 by 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 Jordan's, it is actually a very stringent condition to have the classical limit come out right. And, well, I think of this as a very good thing, because it just means Planck scale physics is not just pure speculation, you can just

42:30 not dream up any model you like as a Planck scale, you will have to work very hard in showing that the classically comes out correctly and this could put very stupid conditions on your model. Another condition that you need to satisfy whatever you would play around with models of this type is that, well, when you look at your final theory and you had some like this quantum building or something 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 is not just the classical limit, but the classical limit and all these quantum corrections should better be insensitive or stable and where you are changing around a lot of quantum models at a short length of space. 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 subplanking building blocks. But this is something to be kept in mind in general. Concluding now, in non-interruptive models of formularity, the existence already of this classical limit is not automatic, and I believe it was very specific. So what I showed to you in a very particular instance of a specific causal and non-interruptive class interval, how it dynamically generates a quantum ground state, so a muzzle or macula, if you like, which was methodically for language. What was very crucial here, from the outset, is to put in, geometries of the correct signals, so this is crucial, and the vain microcosmality. So, we've come full circle, and I hope I've convinced also our supervisors that we're passing towards, if only done properly, how we find them can give us interesting information about the public space. Thank you very much. Gary, any questions?

45:00 I think Chris should have the right to do that. Right. These neural results are very beautiful. Could you extend a conjecture to say that these integrals actually really exist? As you say, there really is some measure of some well-defined space of, as you can describe, John Lewis, which really doesn't exist as an integral. I mean, it would seem very tempting to conjecture that. I don't quite say it. No, I understand that. Is it a realistic objective? I think so. I think so. You take. Yeah. Jimmy, go on. I have two questions, actually, which I will sneak in here. If you had started with five-dimensional Murakowski simplices, would you have come out with the effective dimension being five? You could, 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 reggae, that is, by integrating over the edge lines and fixed triangulations, for example, is represented by Hampton, Williams, and others. Okay, we have two other questions. So, regarding, well, the arbitrariness of my initial input, in a way, could I have started with, well, five-dimensional gravity in a building lost historians? Well, one of my pet projects is, my hopes 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 start with dimension seven and then show that everything dynamically collapses to four dimensions. Exactly. That would be wonderful. 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 magic. So, so much for your first question. Second question, is it related to quantum magic calculus? Well, yes of course, 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 differences when we take the continuum of that mass interval.

47:30 Okay, let's not do it here, I'll catch you. Yeah. So, Caroline. In your way of stepping the simplists, the Minkowski events, into the space-time geometry, you use, as you say, a privileged geometrically defined property. My question is, do you expect when you come to the semi-classical picture and the concept of microscopic geometry to find something like that this geometric is different morphism invariant and that it does not depend on the way you slice. Of course you can slice in many different ways in classical gravity. You use something, one particular stacking. Do you expect that out of this diagram you get in the end, in the classical limit, something that mimics the space angle that it is? In short, yes. First of all, of course, before I could start talking about this, I would have to really introduce coordinates, if you like. So, and there, I would think I would first have to know, I would first have to show that, say, this quantum geometry I generated for you here, as macroscopy, something to do, say, with Flakman-Kovsky's face, in some coarse grain approximation. Then I can, or some other classical geology, 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 what other components I may wish to introduce. So, that's one way of answering your question. If you asked it at a more fundamental level, say, okay if you did an 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 would exist but already we know, say, if you do QCD

50:00 and look at various formulations, say in different gauge choices, it is very very difficult to make, to explicitly show that the physics is the same and different formulation. And certainly, here in gravity it will also be very very difficult to show that explicitly. And anyway, an alternative candidate is not yet around, which I hope, you know, compared to. What will happen if you take a continuum limit at the end? Isn't there variance in variance? Is it broken here or not? Because you have a lattice constant in this case. Yeah, of course. I mean, what I can already show you what these simulations do, they will, of course, they are not continually, but what the 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 ought to be a continuum theory. Of course, to write this down explicitly may be quite difficult. I'm sorry, your second question related? 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 stop worrying about, well, recovering symmetries of the classical theory. So, currently I cannot answer. Papa? I do. I'm actually a patient on all of these things, particularly in classical literature. But there does seem to be a tension, right, between your notion of causality and the sex notion or what Cambridge do, you know, 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, you know, which cannot be foliated. But you are assuming, of course, it won't be foliation. So, it would look like throwing away all black hole physics from that perspective. Is there some way of recovering this black hole physics with this kind of thing? The second question is that, what do we know about the dimension, about the ground state? Does it look like recital? Does it look like anti-recital? Does it look like anything at all? I mean, the picture you drew doesn't seem to be, you can walk as an environment at all. Poor dimension, you can walk as an environment at all.

52:30 On the other hand, if it is a ground state, it better be. So what is the same? Yeah, okay. So the first question was causal. Yes, it's a very intriguing question. I mean, the question is, yeah, in all these calculations, in black, black, black holes, over the next, and Euclidean signature seems to be crucial. Now, what I believe is that, well, if you can kind of do this to the end, you will find that in looking at the classical, I mean, there's of course a reason for why these Euclidean methods do encode some interesting dramatic information. It is not always, I don't think it will be straightforward like by looking at it, you can a semi-classic approximation to get this out of such a framework, but I believe one will find, if I could do it to the end, that the certain ways of how one in the Euclidean figure phrases physics, encapture ultimately Louentzian properties of the system. How this could be laid out in detail, I cannot tell you at all, but I believe that's one 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. Yes, yes, so the ground state. We have made some progress in trying to understand this very kind of non-invariant structure one sees there of this bubble of space and time. And we have a, it should be taken with some caution because what one sees here is, first of all, we have the spatial volume constraint in there, and secondly, you still have to retranslate the Euclidean structure you see here to a Eurensian structure. So we believe we have actually formulated a good effective classical logology that depends on the sleep volume that explains the shape you see here. And this may well be some de-sittal type, some de-sittal type

55:00 solution. The detailed cultural properties of the space, and we haven't looked at yet, but this can be done and will be done. 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, if you like, the high bassoon 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 copy course, but here I still have the Bernoulli course, and you ask, how can results depend on this? Now, in the old computing models, because people, you know, saw the various values of k, as I explained, in very degenerative phases, people then looked for a possible second model phase transition at which physics might still come out right, if you like, because it's 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. As it is in lower dimension, it may be sufficient to obtain a continuum theory to just fine-tune one of your errors, 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 that comes out of the world. Second question. The usual intuition of what you're thinking is that the smoothness of the when you approach land scale, and so at land scale you see a geometry, do you see this? Is there a land scale appearing so much larger? Yes, we think so. Well, as far as we can see, these geometries are still very weakly at short scales.

57:30 But at some scale, of course, you would expect that as a dynamically generated scale that knows about One quick last question. Okay, I actually have many questions, but I'll limit myself to one short comment and one short question. The comment is that actually in fixing the volume you're doing the path integral version of what's called unimodular gravity. The question is about the sum over topologies, which you didn't do. There's a potential problem there with some superexponential growth in the number of triangulations. I don't know how well that's controlled by your causality conditions. So can you comment that as a possibility to come over two different components? Yes, it's really from my current point, it is completely ill-defined. It just cannot be done because of all the superexponential growth. I mean, it's as bad as doing the sum over geniuses with the probability. But there's no way, currently, to make this number well sound serious and finite in respect of information. I'm unaware of any successful attempt to do that. Okay, I'm going to make this talk finite at this point, and I'll have a coffee until 11.30.