Is a Past Finite Order the Inner Basis of Spacetime?

0:00 For me personally, not only have I always tried to follow whatever Chris was doing in physics, but often I found myself on the same scientific road with him. Sometimes, indeed, there were not a lot of others on this road, and I felt like Chris might be the only person interested in the idea I was working on. now one of the times that this happened is when I started to think of space-time as a finite, or better finitary topology incidentally there's now quite a lot of interest in this kind of ideas, for example in relation to image processing by computers but anyway a finite topology is equally well a posan, a partially ordered the definition I meant I've written out before I'm giving to the Voight-Integrator Steven's talk. A finite topology is equally a poset, and what I want to talk about this afternoon is, in fact, a poset, and a poset is the deep structure of space-time, but now with a causal interpretation of the order and not a topological one. Also, in this talk, I'm trying to do what Jonathan asked us to do, which is to kind of give a review of the whole field, so I have to apologize in advance for I think what's going to be the rather rapid fire and incomplete nature with which I'm going to cover some of the things that I will be saying. So, to partly compensate for that, I've given you a website where you could find, at least I've posted some of my papers and there are also links to some other websites and hopefully I'll be able to post a more complete bibliography So, maybe I should start by saying, what is a causal set mathematically, which is just a partial order, and also because I won't have a lot of time to give anything very concrete. In most of the talk, I wanted to give you maybe something that is concrete and simple to take away with. It's kind of cute, just related to the mathematics of partial orders. So, let me start with the definition of a partial order. Partial order is just a set of elements which are provided with an order relationship that can be thought of as the relation of before and after, or even in a kind of genealogical sense, the relation of descent and ancestry.

2:30 And we'll see that that figures in the particular model for causal set dynamics that I'll be talking about later. So mathematically, it's a set of elements, which I can call here, I've called here A, that they have is one which you can easily see applies to descent and ancestry and also to cause a relationship which is transitivity that if A precedes B and B precedes C then necessarily A precedes C we also, in order to rule out cycles it's convenient to have the convention that no element precedes itself and then there's a discreteness condition which for the moment I'll leave aside which is telling us that that there's a finite aspect to the partial order, since without something like even the real line would be a partial order. It is a partial order. So, let me now show you this interesting thing. It's just a thing in itself, the way I'll present it, but it does actually arise in the more careful treatment of the growth dynamics that I'll refer to later. It has to do with a number of ways of adding an element, a new element being born here, if you will, So, or going back by taking away the element, this, yeah, this is wrong. Okay, so this is wrong. So, let's consider, suppose you had started with this causal set. The causal set is, of course, indicated by a space-time diagram, also known as a Hasse diagram. and wanted to add one new element as a maximal element in order to produce this order, or causal set. There's clearly only one way to do it. It has to be added here. And conversely going back, we can see that this was the new element that was added if we had come from this. On the other hand, consider the following pair, or triple, of causal sets. This is a two-chain, as it's called. To go to this, there's only one way to add the new element. ancestor. On the other hand, in going back, there's an obvious ambiguity because I could remove either this one or this one. The opposite case is illustrated when I go on from this causal set to this causal set. To do that, there are two possible ways of adding a new element, either here or here, but going backwards, there's only one way, there's only one element

5:00 that can be identified as the new one, which is of course this one. But this is not really It's surprising, it's just to do with the fact that this causal set has automorphism. It's not possible to tell this element from this element objectively. So of course there's an ambiguity in how I add or subtract an element from it. Here's the cute thing. Consider the following causal set. It's a little more complicated with nine elements. You can see that it has a kind of Z3 symmetry. And then we add this. And now consider going to the follow- Well, let's try and get this to one diagram, this will be the new element. This produces a causal set with ten elements. If I add this here, or I add this here, it's not equivalent. It's a sort of right-left asymmetry, and yet the resulting causal set is the same. Conversely, if I start with this, it's not equivalent by any automorphism to remove this or to remove this. In fact, neither of these causal sets has any non-trivial automorphism. although they're different ways, they produce the same result. One thing that you could notice, though, is that although there were two inequivalent ways to add a new element, namely here, I should have drawn the other possibility, which is here, so there's two different ways of adding it, when I do add it, the set of ancestors that it has after it's added forms the same causal set, namely this one. And that's no accident. In fact, it can be proved as a theorem and plays a role in the analysis of the growth dynamics. In fact, this points out something else, which is in a growth dynamics, as I'll talk about later, a causal set element, unlike Bertrand Russell and Bertrand Russell's ancestors that Carl was talking about last night, has the choice of ancestors. He said it was important to choose your ancestors wisely. In effect, the causal set element, that's the only choice it has when it's born, is what will be its ancestors. And we'll see later that the growth process is precisely the sequence of such choices, or the record of the sequence of choices made. Another very sad remark is that this is, this same causal set illustrates what's called Eros' Paradox in economics.

7:30 but I can explain to anyone who's interested afterwards how that goes. So, I think that's the blackboard. So I want to talk a little bit about, although I said I'm talking about causal sets, I do want to talk a little bit about order as topology. There are two reasons for this. First of all, because Chris has been interested in that interpretation. also to do with Chris, it gives me the only chance I have in this talk to mention Channagor theory. So, I'll just run through it very briefly, basically reminding you of things that you probably may or may not have seen, but it won't play a role later on. So, a clausin-staturn argument can be thought of as C0 topology, and as such, it constitutes a plausible de-structure for space, although I would say not for space-time, the way the correspondence works is that an open set corresponds to a down set. Here's an example. The red set is a down set, something that contains all its ancestors. And if you want to go back from the topology to the causal order, you say that x precedes y, sorry, to the partial order, you say x precedes y, and only if it's contained every name of it is y. kind of topology, has many nice features, but as far as I can ever tell, it did not work well for space-time, and you could not produce a metric, especially a metric with a Lorentzian signature. Now, this point involves a set, it's kind of potent, it's thought of as an approximation to a space-time, goes nicely, or a space, I should better say, goes nicely with covers of a continuum, and it's competing in a way as a quotient of a continuum. Here's the categorical bit. The dual notion to a quotient is a static, and that goes nicely with a causal interpretation of the origin. And that's the next part of what I'll be talking about. So, coset is causal set. The nickname for causal set is coset, and it's intended to be the deep structure of not space, but space-time. Interestingly enough, I'm not sure what to make of this, but all the basic ideas are very old in a sense

10:00 and they can be illustrated with quotations from two of the people who thought most deeply about the nature of space-time, continuum versus discrete, which was Riemann and Einstein. So the first two quotes I'll read you are from Riemann, Riemann's inaugural lecture, and then a quote from a letter that Einstein wrote to a friend The first one is from Rima, for a discrete manifold, the principle of its metric relationships is already contained in the concept of the manifold itself, whereas for a continuous manifold it must come from somewhere else. Therefore, either the reality which underlies physical space must form a discrete manifold or else the basis of its metric relationships should be sought for outside it. Elsewhere, in the same article he wrote, definite portions of a manifold, distinguished by a criterion or a boundary, are called quanta. Their quantitative comparison happens for discrete manifolds through counting and for continuous ones through measurement. Finally, the quotation from Einstein, it seems to me that the alternative continuum-discontinuum is a genuine alternative, i.e. there is no compromise. In this continuum theory, there cannot be space and time, but only numbers. It will be especially difficult to derive something like a spatio-temporal quasi-order from such a schema. And then he goes on and says, I hold it entirely impossible that developments will lead them. So here in one form or another are all the basic ideas that we'll see represented in this puzzle set theory. First of all, the idea that there might be something, as Renata was mentioning also in her beautiful talk this morning, that there might be something different than the continuum underlying the continuum. As Einstein said, space and time cannot actually exist as such in such a theory. The idea that that thing might be discrete, and if it is discrete, then its metric relationships are contained in the structure itself. They don't have to be added in, and most particularly, volume comparisons can be done by counting. And finally, from the Einstein quote, the idea that the quasi-order, or partial order, is a crucial feature in this whole exercise. So why does this work? Why does it have a sense? Why can't it be that, based on the continuum, with its geometry, topological structures can be thought of in terms of an order?

12:30 The basic reason is that continuum space-time is already a post-set. It's not discrete, but it does have a partial order, which is the causal order, the order of the relationship of past and future. In fact, early in the history of relativity, Rob wrote a whole book in which he gave axioms to characterize Wachowski's space just like usually gave axioms to characterize Wachowski's space, but those axioms are based on causal relationships and nothing else. More generally, in third space, what is the space-time? Well, as we all know, it's a Lorentzian manifold, which means a manifold is provided with a quadratic line element like this. And as Renata also was emphasizing, it's a Lorentzian signature alternatively. This signature makes possible light cones, two of them, one for the future line of the past. Therefore, an order relationship, which we normally think of as kind of derived at the end of a long mathematical story, but notice that only for the Laurentian signature does one get an order. With the Euclidean signature, of course, no light tones at all, and with more minus signs, more time on the reactions, you get nothing but closed time only first, and no distinction between the past and the future. So now, still in the continuum, if we know the light tones, we know almost a whole metric. We know it up to a local conformal factor. So we're missing roughly, well, we're adapting in four dimensions one-tenth of the information, GAD having ten independent components, one of which is the volume element or conformal factor. Here's where the discreteness comes in, a la rima. Although the information is missing in the continuum, in the discrete context, we can count and therefore get volume in that way. So from this point of view, discreteness is the missing ingredient that allows us to put together order, to derive out of order, the full metric. Such a viewpoint is exactly the same other thing. We could say that unifying space-time with this metric is not something separate that we add in, but just inherent in the structure of discrete metaphors itself. More sensibly, it unifies all the mathematical structures that we think of somehow being We need to measure this quality, differentiable structure, and so on, all in one simple mathematical concept.

15:00 As a slogan, we might express this by saying, in geometry, the macroscopic thing is order plus number, microscopic thing. But to get this, we need, of course, a finite character to treat this condition. And so I'll define post-sat here to be a locally finite, or in general we can just read this as past finite for most purposes, post-sat. So it's a post-sat which are these axioms that I've written here supplemented by the last axiom, which is local finiteness, which says that between any two elements, A and B, there's only a finite number of served elements. Past finite is just a statement that no element, which is stronger, is a statement that no element has more than a finite number of ancestors. Of course, there are also many practical reasons that have driven people to want discreetness. What I've argued here is that discreetness actually adds information in a certain sense because it gives us value of information. But, of course, the practical, other practical reasons have to do with what it subtracts, which is also the divergences in the infinities that are present in continuum series, in particular in quantum field series and in gravity and in quantum gravity in connection with black holes in thermodynamics. Now, in order to make a theory out of this, one has to go roughly through two stages, at least schematically, two stages that seem to be present in every kind of theory that we're familiar to date. Those are what you might call the kinematical stage and the dynamical stage. First, kinematically we have to understand what kind of object is a causal set. You might think in connection with general relativity of having to understand differential geometry before you can write down the Einstein equation. And there's a further thing in the causal set case, which is not perhaps so obvious, which is that we have to understand how to make the translation between order language and geometrical language when it's appropriate. The slogan that geometry is order plus number is one thing, but how do we give that practical meaning? How do we actually read out metrical and topological information? So I want to just run through several topics in this direction.

17:30 The very first one is the nature of the correspondence between the continuum and the discrete. When would we say that a given continuum radical is a good approximation to a causal set? Remember the basic idea that if I put them in a formula here, that the number of elements equals space-time volume, a certain region of space-time has a volume that basically counts the number of elements imposing it, and then the fact that the microscopic order, the order in relation to finding the causal set, corresponds to the macroscopic relation of causal order, as it usually defines in terms of time-wise and causal curves. So when should we say that these conditions are satisfied? Given a manifold with a Lorentzian metric and a causal set, when would we say that they're approximately the same thing? or better, that the manifold forms a good approximation to the causal sets. Well, without giving a whole argument, I would say, in practice we found only one way to do this that's compatible with the uniform distribution and with local invariance. And that is that you would say that they're approximately the same when the causal set might have been made by a process of sprinkling, or a Poisson process more formally, a random process of selecting points with some finite density within the manifold and then taking the induced causal order. This is, in some sense, locally, Lorentzian variant, and it can be made precise in the context of Jankowski's face, where the Poisson process is, strictly speaking, exactly Lorentzian variant. In contrast, for example, something like this kind of lettuce distribution of points, which you might feel is very uniform, is actually not only non-Lorenz invariant, but not at all uniform for related reasons. If you imagine what this looks like when it's boosted, then it looks very different from one. So it not only distinguishes it, but it's very different and it's very non-uniform, with big boys and areas of very high density. So, unlike where the random shrinking doesn't show this. So the conclusion is that, to do justice to Lawrence's invariance, we seem to need already at the kinematical level, a randomness. This is the level of, like, an interpretation in which we interpret the causal set as a continuum of Moranian nanopause.

20:00 Okay, so that's a very general notion of how we think of the correspondence as going. Given that correspondence, how would we get out certain important geometrical information like length and that's topological information like dimensionality. Well, what about proper time? Given two elements of a causal set, which corresponds in the continuum to two points, how would we measure or estimate the proper time in the continuum between those two points? Well, you might temporarily say the shortest chain, but that's the part of the Riemannian intuition. If you go to the Laurentian intuition with the twin paradox, probably the best guess is the length of the longest chain, and this is in fact proven rigorously for sprinkling the Minkowski space in any dimension. For topological information, specifically for dimensional information, there are many ways to proceed. rather than give all three of them just to conserve a little time let me just focus on the second one which is in some ways the most interesting we call the Mirheim-Meyer dimension so suppose you had a closet set that was created by sprinkling some Ninkowski space and you wanted to know what dimension Ninkowski space was and here's a picture of an interval or an Alexander neighborhood in Ninkowski space and it sprinkled points There are two invariants of that. You could count just the number of elements, n, and you could also count the number of related pairs, whose maximum number would be n choose 2, of course. Well, if you work out the expectation values with respect to the sprinkling of the number of relations and the number of related pairs, and take the number of elements, sorry, which is just the volume, of course, because n is just the volume. The expected number to work on the relation between the number of romantic pairs and the volume. Take the ratio, rather it's time to choose two. Take the ratio, you get a function. This is just some gamma function. It's a binomial coefficient. And the important thing is it's monotonic. So you can invert this relationship and solve for the effect of dimension. It's kind of Hausdorff dimension, but not exactly. It's something in the same spirit.

22:30 You can analyze it fairly well and really, you can see that you need, on the order of two to the M, single points to recognize dimension M, or two to the D to recognize dimension D. So let's just see an example of this, which is from David Meyer's PhD thesis, plotting, looking by the size of the number of points looked at and the volume of the interval looked at the effective dimension computed this way, and you see that for a handful of points you're already seeing three dimensions versus four or two. This is just two, but it's a spatial dimension, so it's three space-time dimensions. By the time you get up to 20, you're doing quite well, and by the time you get to hundreds, you're humming in on the two dimensions. So that's just an illustration that that technique actually works. Here's another thing that illustrates something I won't really have time to talk about, which is coarse graining, but I think, I can't remember which talk was it, but the idea of foam was referred to, which is that at different scales of technology may be different. The simplest example of that is Pellussi-Pryden's theory, in which the pathology is, say, five-dimensional on a small field, and four are large, or 11 on a small, and four are large, or whatever. Here's an example of this. There's a process of coarse-graining, and one sees how the effective dimension crosses over from two to space-time dimensions. In this particular case, bounds in one, as the coarse-graining just sees. So another nice feature is the possibility of expressing in a precise way this kind of scale-dependent typology, although I haven't given you any details of how training works. There's one other concept that I want to introduce to later use at a kinematic level, which is at a post, and then I'll list some important questions still remaining at a kinematic level. So what is a post? A post has a clear meaning in a cosmological content, in which one thinks of possible cycles of expansion and retraction of the universe. If there are such cycles, then they can be punctuated by what are called the mathematics literature post,

25:00 which is illustrated here by this red point of x. Formally, it's an element of the causal set X such that all other elements are either before or after it. They're either its ancestor or its descendant. So this concept came up in connection, in the mathematical literature, in connection with the analysis of a particular kind of dynamical law of the causal set which can be called percolation. the mathematicians call it a rounded graph order, and they prove that it has an infinite number of posts. So if that were the two dynamics of the causal set, that had to be an infinite number of plans of expansion and retraction of space-time. Some other topics that I briefly referred to and won't say any more about for a lifetime of time are portraying causal completion, I haven't conferred you, and I won't say any more about that. Let me go just to open some important questions. We saw that height, which means the length of the longest chain, is a good estimator for proper time for causal sets that correspond to flat space times. So I'll let this estimator work in the case of curved space times, too. Now, Mirheim gave some arguments, but they're also worried, and it needs to be settled. Very similarly, none of the Mirheim-Meyer dimension, or the other dimension estimated, which I didn't have time to talk about. Do they work well? One of them, at least, the Mirheim-Meyer one, probably works in flat context. Does it work well in the context of curved face time as well? The others I should mention by simulation for you, although there's no analytic control of them. I'm going to skip this one and go, I told you before about how to estimate length information, dimensional information, how about curvature information? Is there a way to get something like the huge scalar of a causal sequence as well approximated by some manifold? It seems, in fact, that there is, and I'll refer to that a little bit later on. And then let me skip to the next and come to the most important one, which is how do you know... I mean, the other thing I've been telling you so far is under the assumption that the causal set is well approximated by some manifold.

27:30 But if you're just given the causal set, how do you know whether it is or not? Some tests are possible, but we lack necessary and sufficient conditions in order of a complete set of tests. That's obviously going to be important at some stage. Okay, so that was a very brief, slightly review of the situation. Kinematics are closed. Let me now talk about dynamics. Until a few years ago, there would have been very little to say on this because there was no good work on dynamics. The situation is different now and for two reasons, first of all, having to do with the dynamics of the causal set itself, what would correspond to the continuum to quantum gravity, and second of all, having to do with the dynamics of a scalar field on the background causal set, what would correspond to the continuum to quantum field theory in curved space-time. So there's two different aspects of dynamics. You can think of the causal set as an actor or just a background or some other actor. Let me go first to the first one. The dynamics of the causal set itself. Here, the idea which it proves fruitful is that when you think of dynamics as a kind of growth process, which we call sequential growth because the elements are thought of as being more one by one. This, such a model, if we start from no elements, provides us with a kind of discrete cosmology, although at this level, at least in the old level, we preach so far it's still a classical growth model, a stochastic, a stochastic classical growth model, but not a quantum stochastic growth model. So we see. Although there are many interesting results on this note, So its primary intention was to be a setting stone to quantum sequential growth, which would be quantum gravity. But why do I say that? This makes sense. If you believe that quantum mechanics is better thought of as a generalization of classical stochastic processes rather than classical hematognizing mechanics, then this will make sense to you.

30:00 If not, I would have to try and convince you that it makes sense. and that the natural step is from a classical measure representing a classical sarcastic process to a quantum measure or an incoherence function, which we haven't talked about extensively. So, the interesting thing is that under the assumption of classical sarcastic sequential growth, we have a simple set of principles that lead us to a more or less unique theory. So let me just step back. If we're trying to construct quantum gravity, or even if we're trying to construct classical relativity, there would have been two possible ways to proceed. One, to think of a set of principles that would lead more or less uniquely to the answer. The other, to try and do whatever you can do, say, to get some generalization of Newton's equation general relativity. Both of these can be pursued in the case of causal steps, but the one that has been pursued most is the first. Try to find principles analogous to general covariance and so on. That will produce more or less unique dynamics. In fact, one of them is general covariance, but the complete set of assumptions is the following. First of all, sequential growth, elements born one by one in a random way. Second of all, no element born to the because that would represent the conflict between the time of the growth process and the intrinsic time of the quality relationship itself, which is meant to be the only physical time. It's classical, so we use classical probability calculus. And going along with that, we have the Markov sum rule that the sum of all the transition amplitude should be 1. And then the two key assumptions are general covariance, which is a kind of labeling independence. When it says that at stage N you produce the particular causal set, the probability of getting that is completely independent of the order in which the elements were born, insofar as that element is ambiguous, which means insofar as there were space-like elements. This could be thought of as a kind of free analog of slicing endurance as well. And it has the important consequence that the growth process forms a Markov process in a certain space, which I'll show you in a fragment. Well, I think I'll show you now. It's nicknamed for a few reasons, post power.

32:30 It's the post set of finite total sets. Here's, of course, it's an infinite post-set. Here's a little bit of it. And you see the very simplest finite post-set, one element, and the two element post-set. And as each element is formed, you move along some path in this diagram, always upwards. And what defines the causal process, the sporadic process itself, is the transition probabilities for making each step, even to make you the next step. The fact of general covariance is a kind of passing dependence in this scenario, and it has the consequence that the transition, in particular, that the transition probability depends only on where you are and not how you arrive at it. Therefore, it's Markovian. That's the one key assumption, which is a very natural analog of general covariance. The second one is a little harder to state. It's a causality condition that says that information shouldn't travel faster than light. Well, there's no light ones here as such, but there is a causal order. And the idea is that the causal order should limit the transmission of information to space. It should not be able to go from space-regulated to its causally unrelated elements. So, just quickly, the way we implemented that here is by assuming the black element, say, already exists, there's two possible, here's the new element, it has the red element, it has, say, two choices on how to be born, and you want to know, each one is associated with the transition probability, and you want to know the ratio of those two. The assumption or the condition we pose is that the ratio is independent of what's going on in a space-like related region, specifically independent of the presence or absence of a spectator element, which is indicated here in the green. Notice that we've nowhere mentioned the assumption of locality, although we have said causality and the words are sometimes confused. I think that would be important that it would have to do without locality. Anyway, these conditions, almost uniquely in my examining processes, turn out to be a kind of generalized percolation model. I referred to percolation before, but I didn't tell you what it was, so now let me tell you. Remember that the growth process can be thought of as a sequence of choices by the new foreign element,

35:00 in which at each stage it chooses which will be its ancestors. The percolation process follows a particularly simple rule for that choice. So the new, the element about to be born, so to speak, looks at all the existing elements and goes through one by one, slips a coin with probability P and chooses them as ancestors. Then having chosen those, it has to take what comes with it. So if you're my parent and he's your parent, then he has to be my ancestor as well. So the ancestor of an answer has to also be an ancestor, that's the transitivity condition that you see over there. So it makes the transitive closure after making its initial choice. The probability of getting any particular set of initial ancestors before the transitive closure is very easy to see. The probability, if it has S, if it is set as S, then it has this actual value of S, by which I mean the number of elements, the current L, if it has that many elements, then you have to choose those with probability P, and the other thing you have to not choose with probability 1 minus P, or add the number of P-existing elements. So leaving aside constant pre-factor, that's just P over 1 minus P to the S. In particular, it depends only on the number of elements in the set of ancestors we chose, but not on any other feature about that particular set. Well, generalized percolation is just like that, except it doesn't use this particular expression, right? But it still chooses proto-ancestor steps dependent only on their terminology, but with the width that needn't be of the form some constant to the K, So, rather it can be a free parameter, a non-negative parameter, which I called here T sub K. And the choice of these parameters, the sequence of parameters, T0, T1, and so on, that defines the so-called free presence. Now, actually, many results are known for these, as we call these classical sequential growth processes. And a particularly interesting question is, will quotes occur? And also, we wrote an infinite number of quotes. What seems to be the case is that there's a critical fall-off rate that these teeth

37:30 come in. If we had percolation, they would just fall off like, there'd be no log n here, they would just fall off like c to the n. And that's known to have an infinite number of quotes. This is a sum of more rapid qualms, but not much, because it has a log n in the denominator. What seems to be the case is that the critical value of this parameter t in the numerator, which separates the case of infinite number of posts from finite or no posts. And it's, if I remember it right, square root of pi over 2. So if we choose such a dynamic and we choose t greater than tc, we get kind of an interesting toy model of how the universe might have reached the kind of conditions that we need for the Big Bang without invoking any kind of post-Plunk theory and inflation. In particular, one can argue that it will produce a kind of closed set, which is spatially homogeneous, spatially isotropic, and also big. The bigness of the universe at the Plunk time is somehow the big mystery of technology. problem and so on. But it's the main thing that inflation was involved to explain. So here it happens automatically and the bigness is associated with the large number of cycles of expansion and contraction. In other words, on each expansion it goes a little bigger. And so finally it's as big as you like. It's a constant. The big drawback that makes it a toy model, aside from the fact that it's not quantum yet, is that it doesn't produce something fully like a manifold, it's something that has semantical features, but it's almost certain not genuinely a manifold. Related to this is the new notion of renormalization, which is that after, and the reason that it changes in cycle to cycle is that after the universe bounces and reconnects, you get a new dynamics of the same sort, after all, it's still generally logarithms and so on, but with different parameters, and in fact this by the simple renormalization group transformation, sort of the new kind of renormalization group associated with it. That was going from the very small to the very large, but from the very early and with more recent. And the point of that is known about the flow of the renormalization

40:00 group and so on. So this is meant as a warm up for quantum gravity, but what we lack still to produce an actual quantum gravity is the quantum analog of the bell fondality condition. Everything else seems relatively clear. The other approach, I'll just allude to it, was just trying to directly do something that's an analog of the Einstein approach. Until very recently, it wouldn't have been possible if we had no idea how to write something that was an analog of the reaches scale. Now we do, so it may be that that approach can be explored as well. So that was a review of dynamics of the causal set itself. The other thing which might need to be added if we need matter in addition to the causal set, which in any case, a possibly simpler laboratory for exploring some of the issues of dynamics, is a fixed causal set, but with dynamical matter. In this case, a dynamic scalar field. So let me, again, review what's known here. And this is very much a new thing, so I can't do complete statements. Let's think about the simplest case of the NASA scalar field. In Minkowski space, that depends from Minkowski space, the equation of motion is just zero equals box five, where box is the Lorentzian analog of the Laplacian, which is the spatial across the line, it's five out of the back. How would we transcribe this equation to a scalar field living on a causal set which had been produced by sprinkling 10-dimensional in the policy space? In particular, how would we deduce the value given an arbitrary scalar field which just means an association to each element of the causal set of some real number or complex number? deduce the value of the dalimation at phi at some element of the causal extent. So here I've shown the particular element at which we want phi, and I've illustrated some of the elements that will enter into the formula for it. Its parents, its immediate ancestors, what you might call its grandparents, the ones that are one level down below that, and then still one further level down, what you might call its great-grandparents.

42:30 Well, there's this miraculous formula, and to me it's miraculous because for a long time I thought no such formula would even, in principle, exist. But here it is. We take the value of the scalar field at the parents minus twice the grandparents plus the great-grandparents, of the binomial coefficients here, makes in with an appropriate factor the value of the scalar field at the point where we want the dollar bearish. And we get something which on average, that is when we ever know, strictly gives us, will give us, in the limit of high density of string, will give us exactly the dollar bearish. What is the strict theorem proof is the test function of the context of 4. And notice that this is a fully Lorentz invariant construction, because the string of itself is Lorentz invariant. There's one problem with this line, the miraculous bone is, which is that it's only true on average. One would hope that as you increase the density of the quadruple of quadruple, the quadruple would get smaller and it would converge to the exact answer. In fact, what happens is that the fluctuations become bigger and bigger. So that, in itself, wouldn't be very good. But fortunately, there are other things that are harder to write down, following the same idea that there are far more layers that actually have derived behaviors and do have the fluctuations dying away, and these platforms become smaller and smaller, so to speak. So given this, now we're in a position to address these kind of puzzle things that go by the name of transplanteing of the puzzle, both in the case of Hawking radiation and in the case of cosmology, just to say briefly what the puzzle is, is that wavelengths of modes that are important to us now can have been much, much smaller than splanteings early on, and so what are we talking about when we analyze those modes? To finish up this expression of the delimberian, what we need is mainly a four-dimensional version of which we already have. This was a two-dimension. There is a four-dimensional analog. In fact, there is an analog in any dimension, using sort of homogeneity operators, which Roger was talking about in this sentence. And then, of course, the hardest thing, which is less clear out of the world, the quantum version, and still less clear is how, if one wanted to have other fields and scalars,

45:00 how one would do it. Another kind of dynamics, not of a scalar field, but of a particle on a positive set, has obvious problems. It brings in obvious problems that have to be overcome. In particular, what could be the analog of the geodesic? I don't think exactly that thing in the discrete structure, what replaces it? So one particular class of models, especially introduced by Joe Henson, suggests that there is a kind of natural replacement, and it leads to a Lorentz invariant diffusion in velocity space, which could be a signature of discrete. I'll come back to that in a second. I just want to show you a picture in the two-dimensional case of what we're talking about when we talk about the scale of the field of a positive set, what kind of positive set is it? This is the kind of graph representing a positive set that's made by strings and excessively more elements One thing you can see is that there are direct links probably quite a distance away. This is this essential non-mocality, which makes it so difficult to get something like this box operator. Because the box operator, of course, is low. Okay, so we've got kinematic dynamics, there's a third stage of any theory, which is phenomenology. Without that, the theory is read in the end rather useless. As you might say, predictions or experimental predictions. or explanations of, maybe post-dictions, explanations of things that we already know. But I want to make a list of some of them, not quite at all, but most of the themes which could be viewed as phenomenology or potential phenomenology. The first one is a post-diction, so far some of them, but there can be no causal loops, no time issues. This is a purely kinematic reason, and the claim is that although I just ruled out my fiat having closed time-like curves or causal cycles by this, that weakening it would not allow me to recover face time as I do have closed time-like curves. The second one is a genuine prediction. In fact, it's the, my most known prediction in any theory of quantum gravity has ever

47:30 experiment, and it arguably has come true, so we could say the quantum gravity is getting a thousand so far. I'm not sure what the cricket analog is at. It's something better than made it over, but I'm not quite sure whether it's happening. But the prediction is for a fluctuation in the cosmological constant, which is one over root n, ending the number or a positive set element in the form of the past up to that stage, it leads to a prediction, since the end for the visible universe, since the Green Bang is on the order of 10 to 240, it's simply a clonking magnitude for the, for the presentable life scale. The square root of 10 to minus 120 is the first, you'll recognize that the value of what seems to have been seen I won't say anymore about what that, what that prediction comes from. I'll discuss it afterwards, I'll have a slide and have more time. It summarizes the basic ideas. Another thing is a particle on a causal, propagating on a causal set, which I referred to before. That means, because of the essential inbuilt Lorentz invariance of the causal set concept, Any deviations from geodesic motion must be realized in some invariant way. What happens is you have a Lorentz invariant way. What happens is you have a Lorentz invariant diffusion in velocity of space. This is a kind of diffusion in space-time. It's driven by diffusion in velocity of space. It's an interesting problem with phocaster processes. The change of time functions, I think, that's all. But it also, in this work that Tay and I and Joe did together, as a potential, as suggested by Tay, as a potential relevance to the origin of high-energy cosmic rays, there are still some problems with that idea in general. The numbers don't work out exactly as one before us. Closely related to that are propagations, not in a particle on a causal set, but in a field on a causal set, what I talked about before, and this is what we want to do now, now that we've got a consolidation, we want to see how propagation of life and existence powers might be influenced by the end of the realm of discreetness.

50:00 This is to be compared with some other work in which people have saw evidence of discreetness in the breaking of Lorentz invariates, or let's say change dispersion relation. Here we don't do that, but there may be other effects on this extra extinction or fuzzing. Anyway, that needs to be worked out. Related also to the previous discussion, because of course, Particles contra-mechanically should be thought as wave captives. And then it becomes a probability of field propagation. Finally, there's black hole entropy. So this is at a kinematic level only, but the plan is that one can identify... The underlying idea is that one should think of the black hole entropy, black hole entropy, which is partial to its horizon area, as in effect counting the number of discrete constituents, what you might call molecules, of the horizon, because it can one come up with some common general definition of these molecules that would be proportional to the area. The answer is yes, at least in supermancy. What's new about this, I think, compared to... What's different, distinctive about this, compared to other approaches, for example, in blue quantum gravity, is where you also find that constituents, which are like cultures in the horizon, is that this also works very far out of equilibrium, not just more structural. I think it looks like it's more of the entrance too, but a small snag has lived in which I'm going to see if I had later. Just a side comment here, because it's related, these kind of ideas are closely related to a particular prediction of holography, which is an entropy bound, called the . Let me just state the geometrical factor in this picture, because I'm running out of time. The statement is that taking this cone in the constant space, which is taking this base of the cone, which is a three-dimensional volume, the number of maximal elements in a quasi-set, proposing this, through which arguably all the information would flow, all the entropy, is actually proportional not to the volume of that base region, but to a theorem, which those of you who are familiar with the logics will recognize.

52:30 Let me finish with some insights that I think we've gained from this exercise. Some of them I've tried to prepare, but in the previous discussion, some of them are kind of new on this slide. First, discreedness and Lorentz and Greg can live together. It's typical, but the cost of kinematic emptiness is possible. And this leads, this was crucial in the argument of the psychological constant fluctuations. It also leads to a kind of, what I call, blurring between ultraviolet and infrared effects, in which things that were the thought of as ultraviolet questions, like maximum velocities, end up dependent on an infrared color. Another conclusion, which there was a hint of in the discussion of the Down version, is that locality must be given up, and this will help us to control the superexponential who brought us a number of laws of sense, which I would rather understand, because we discussed it earlier. Very closely related to this is another aspect of the cosmological problem. Why is it non-zero, and why is itself near zero, the other half of the problem? The insight here is that I find this should be sort of, and this is very closely related to what I was saying before, this should be thought of as an aspect of the problem of why do we get a manifold at all? Why do we get continuing-like behavior from this fundamentally discrete thing? I think non-locality is going to be essentially that same thing. And by this I want to stress, because in honor of Chris's very eloquent writings on this problem, that we have, in the context of classical sequential growth, we have essentially a complete solution to the so-called problem of time. that is, we can identify certain predicates or questions you can ask about the public setting, which on the one hand have pure physical meaning and on the other hand are generally polarizing and from which you can form biological combinations, all generally polarizing predicates. Another insight is alternative ways in which cosmological virus numbers might be explained without conflation, related to a new kind of positive renormalization that I described before. And then finally, the idea that growth, that quantum set dynamics, the analog of the Einstein equation, if you will,

55:00 should be thought of as a kind of growth. Because at a philosophical level, we might say that general covariance and becoming can coexist. You can have an actual dynamical space-time, even though you have space-time, a four-dimensional object, and full general mobility. And so, having a new structure, of course you can also, aside from the insights, ask new types of questions to which they lead. Some of these will be familiar in one comparable dot of this tree. Why is the dimension of space-time before? Again, something that we've not had to comment on before. This is something I've never, I've always troubled me. Why is space-time an uninterrupted continuum? Why does it not have holes and edges? And then, the final question, which I'll explain if anyone wants, which is, why are life on the ground and not square? well I I know I've been thrown a lot at you in a very short time but I hope I've given you some flavor of what causes that theory is all about and some idea of where the enterprise stands at present Questions? This is a very naive question from someone who is looking at the things from the canonical viewpoint. Your gross loss are always such that they are moving within your causal structure. Now, we know that in general relativity, to get the vacuo-Einstein equations, you cannot have only the dynamical equations. You must have the constraints, otherwise you are not generating a vacuo-Einstein spacecraft. How can you aspire within the theory which has only grown close within the structure to arise at a post-grain microscopic level to something like the constraints? Well, I think the answer is there's no distinction in this framework between equations of motion and constraints.

57:30 The growth laws would include both the constraints and the equations of motion. In particular, when I say there's nothing like a distinction between constraints and equations of motion, There's not even anything in a fundamental like a space-like geometry. There are only the causal relationships. A space-like geometry, insofar as it exists, is only induced by the temporal relationships. One can take a set of space-like elements, a causally unrelated element, a so-called maximal anti-chain, and in itself it carries no structure because it has no relations. If you thicken it a little, you can induce a kind of geometry on it, a three geometry, but that plays no role in the foundations of the theory. So perhaps the best analog would be something like regi-calculus, where there are not, if you do it in a four-dimensional way, not a three-plus-one-dimensional way, which is what, of course, Renato was doing at a quantum level, then there is no distinction between initial value equations and equations of motion. are only equations of motion. And in that context, classically enough, the initial value equations just amount to solving a sufficiently large number of the equations of motion, setting things up consistently, and then it continues. I need to talk with you to understand how this is happening. Okay. Yes. I think your cosmological points in the time is to be well. Those labs are in the crust. Yes, absolutely. No, it fluctuates in sign. So unlike most other models, say, Tracker models, it's kind of a Tracker model in the sense that the magnitude of the cosmological constant is always around the ambient matter density, but the sign is fluctuating. So that's crucial. Without that, it would be completely incompatible with observations. So why did you start accelerating that? What? So I'm not predicting that it was positive now, that's a purely random event, could equally well have been negative, it just happens to have been positive. In fact, I would rather, I would tend to predict that probably at the time of nuclear synthesis, we could claim that it's probably negative, but that actually might be a good thing. Strominger made the same point in the 1980s. He kept getting lambda is 1 over a squared for the probability of lambda. I didn't know Andy made this point.

1:00:00 He proposed a fluctuation. This is when the Wormhoff-Cullman's mechanism was popular. He took a purely Rensian model and he said lambda was peaked around about 1 over a squared, and if a is bigger, lambda was 10 to minus 120. so there's some similarity in the argument the a squared is a similarity the a squared is coming here a by a means it's a scale factor this is coming as the square root of a which is roughly the space but the crucial difference I think related to one remark is that it's not a law of one over a squared it's an envelope for the fluctuations the sign is not We're familiar with basic elements which have individuality, like Boltzmann particles, and basic elements which don't, like both sides. Now, you seem to be talking as if your elements have individuality, speak about each element of this form, and when you draw your diagrams up there, you assume there's a distinction between adding an element on the left and the element on the right, which if they were individuated, would be true. But if they're not individuated, then it's not true. There's no difference in those two cases. Now, I think in the diphtheromorphism, I mean, to make the analogy with general relativity, there are people talking about recorded language and talking about coordinate transformation would be important to assume the elements are somehow inherently individuated by the points. And people who use the mouthful language talk about diphtheromorphism point transformations assuming the elements are not inherently individuated. And I think the analogy here is you should be talking about the minotaurics rather than relabelings. Because relabelings is the analog of the accorded language, the binatoris of the elements is the analog of diffeomorphisms. And I think, we could throw off your counting. On the other hand, you do assume diffeomorphism in the accorded language, which may save everything. Because it's another way of talking, you relabel them and say forget about them, you act as if they were labeled, you forget about them. But I think it's better to start out from being and say they're not labeled, and therefore what you call two distinct situations all the properties of the elements of the set are relational properties, aside from being the elements of the set. And if you take that point of view, then there's going to speak between those two cases. I couldn't agree with you more, in principle. But in order to construct this dynamics, we found it necessary to introduce this element

1:02:30 of gauge, if you want, which was the labeling, to get the idea of sequential growth. And then we had to make a special law that was to be taken out, which was the general covariance. So in the end, I don't think, I think we are doing justice to the idea. That's what I call, it's like the dialectal teacher, as soon as they're there, they need a gate. It would be nicer to direct it, but have you ever seen anyone formulate the Einstein equation with a coordinate system or something like this? It's just the same issue. You can, you can. You can formulate. Well, we can. Okay, we can talk about that. We can talk about that. But another more direct criticism, more typical criticism. I don't understand why your cycles have to involve only one point. together with three points, so every element, three of me just came out of the air, so that every element either precedes one of those three or follows one of those three, why must it be just one element? Oh no, those are just convenient, the posts, you can have even posts of no elements, posts of one element, and generalizations of those. What I wanted to say in further response to your first point is that there is a counting issue of the story you raised and it does depend, in the case where you have onamorphisms or this more general kind of ambiguity that I showed you in the beginning, it does affect the counting. And in principle, there might be slightly different variances of dynamics depending on whether one treated the elements as truly distinguishable or indistinguishable. In the end, I think it wouldn't make that much difference because almost all causal sets don't have automorphisms and won't have this ambiguity. Well, you know, when I said in effect did the wrong counting, the whole argument, he got the wrong answer, he couldn't have general opinion. I mean, that's a much more radical thing we're taking care of. This isn't too much. Yeah, I think it would actually. Let's call it a subsidy or how do we do it? Let's thank Raphael once again. Thank you.