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, a finitary topology. Incidentally, there is 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 opposed to a partially ordered set. The definition I meant, I had written up before I didn't to avoid interfering with Stephen'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 can 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 there soon. 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 want to give you maybe with 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, and we'll see that that figures in the particular model for causal set dynamics that I'll be talking about later.

2:30 So mathematically, it's a set of elements. Which I can call here, I've called here A, B, and C, and the key property 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 in some that that there's a finite aspect to the partial order since without something like even the real line would be 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 gross dynamics that I'll refer to later. It has to do with a number of ways of adding an element, the new element being born here, if you will, or going back by taking away the element. This, yeah, this is wrong. Okay, so this is wrong. Suppose you had started with this causal set. The causal set is, of course, indicated by a space-time diagram, also known as a Haase diagram. Suppose you had started with this 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. It considers 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. It has to be added with this as 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 that can be identified as the new one, which is, of course, this one.

5:00 But this is not really anything 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, 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-up, well, try and get 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. There'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. And yet, although they're different ways, they produce the same result. One thing that you could notice, though, is that although there were two in-equivalent ways to add a new element, namely here, I shouldn't have drawn the other possibility, which was here, 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 effect, it can be proved as a theorem and plays a role in the analysis of the growth dynamic. In fact, this points out something else, which is in the 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 side remark is that this same causal set illustrates what's called Arrow's paradox in economics, but I can explain to anyone who's interested afterwards how that goes.

7:30 So, I think that finishes with 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, and second of all, also because it gives me the only chance I have in this talk to mention category theory. 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 any role later on. So, a positive standard argument can be thought of as C0 topology. It constitutes a plausible destructure 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 n sets too, and if you want to go back to 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 in every neighborhood of y. This gives us a planetary 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. And a poset, 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 could be used in a way as a quotient of a continuum. Here's the categorical bit. The dual notion to a quotient is dependent, and that goes nicely with the causal interpretation of the order. And that's, that's part of what I'll be talking about. Poset as causal set. Nicknames are puzzles and concepts, 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... These are very old in a sense, 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.

10:00 So the first two quotes I'll read you are from Riemann's Nobular Lecture, and then a letter that Einstein wrote to a friend that John Stachel provided me with. The first one is from Riemann. 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, a quotation from Einstein. It seems to me that the alternative continuum-discontinuum is a genuine alternative, i.e. there is no compromise. In a discontinuum theory, there cannot be space and time, but only numbers. It will be especially difficult to derive something like a spatiotemporal quasi-order from such a schema. And then he goes on and says, I hold it entirely possible that developments will lead to them. So here, in one form or another, are all the basic ideas that we'll see represented because of said theory. First of all, the idea that there might be something, as Bernadette 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 quotes, 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 it be that, say, some continuum with its geometry Topological structures can be thought of in terms of an order. The basic reason is that continuum space-time is already a post-set.

12:30 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 space just like he usually gave axioms to characterize Euclidean space, but those axioms are based on causal relationships and nothing else. More generally, in third space, one is of space-time, well, as we all know, it's a Lorentzian manifold, which means a manifold provided with a quadratic line element like this, and as Rihanna also was emphasizing, it's a Lorentzian signature, ultimately. This signature makes possible light cones, two of them, one for the future, one for the past. 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 origin. With the Euclidean signature, there's of course no like-tones at all. And with more minus signs, more time-honored reactions, we get nothing but closed time-honored curves. And no distinction between past and future. So now still in the continuum, if we know the like-tones, we know... We know almost the whole metric. We know it up to a local conformal factor, so we're missing roughly, well, exactly 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 a discrete context, we can count and therefore get volume that way. So from this point of view, discreteness is the missing ingredient that allows us to put together an order, to derive out an order, a full metric. Such a viewpoint is just exactly the same as what we're saying. We could say that unifying space-time with its metric is not something separate that we add in, but just inherent in the structure of the discrete manifold itself. More extensively, it unifies all the mathematical structures that we think of as somehow being even measures of polydifferentiable 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 border plus number, microscopic thing. But to get this, we need, of course, A finite territory or discreteness condition. And so all the science post-sats here is to be a locally finite, or in general we can just read this as half 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 third elements. Past finite is just a statement that no element which is stronger is a statement that no element has more than a finite denominator of n steps. Of course, there are also many practical reasons that have driven people to want discreteness. What I've argued here is that discreteness actually adds information in a certain sense, because it gives us volume information. But, of course, the other practical reasons have to do with what it subtracts, which is also the divergences and the infinities that are present in continuum theory, in particular in quantum field theory and in... 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 with today, and 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 equations. And there's a further thing in the causal set case which is not... It's quite 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 get that practical meaning? How do we actually read out the 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, Now 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 composing 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 orders, which usually you find in terms of time lines 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 set? 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 Lorentz 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 zone process more formally, a random process of selecting quantities on finite density within the manifold and then taking induced causal order. In contrast, for example, something like this kind of lattice distribution of point, which you might feel is very uniform, is actually not only not Lorentz 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, 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 string thing doesn't show this. So the conclusion is that... To do justice to Lorentz invariants, we seem to need, already at the kinematical level, a randomness. This is the level of interpretation in which we interpret the causal set as a continuum, or actually a manifold.

20:00 Okay, so that's a very general notion of how we think of the correspondence is going. Given that correspondence, how would we get out certain important geometrical information like length, and then topological information like dimensionality? Well, what about proper time? Given two elements of a causal set which correspond in the continuum to two points, how would we measure or estimate the proper time in the continuum between those two points? Well, if you shed your... You might temporarily say the shortest chain, but that's a part of the Riemannian intuition, if you go to the Lorentzian intuition with the twin paradox, then you would say probably the best guess is the length of the longest chain, and this is in fact proven rigorously for string means of Minkowski space in any dimension. For topological information, specifically for dimensional information, there are many ways to perceive 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, which we call the Meirheim-Weier dimension. So suppose you had a causal set that was created by sprinkling some Yankovsky space, and you wanted to know what dimension the Yankovsky space was, and here's a picture of an interval, or an Alexander neighborhood in an Yankovsky space, sprinkled twice. 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 of elements is the volume, but to work out the relation between the number of element pairs and the volume, take the ratio, where the volume is two and two is two, take the ratio and 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 Hauser's dimension, but not exactly, it's something in the same sphere.

22:30 You can analyze it fairly well analytically. You see that you need, on the order of two to the n, 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. Looking by the size, the number of points looked at, 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 10 or 20, you're doing quite well, and by the time you get to 100, you're humming in on the two dimensions. So that's just an illustration. 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 on this talk, anyway, the idea of foam was referred to, which is that at different scales, the topology may be different. The simplest example of that is Collison-Klein's theory, in which the topology is, say, five-dimensional on the small scales, and four are large, or eleven on the small, and four are large, or whatever. Here's an example of this. There's a process of coarse graining, and what I see is how the effective dimension Crosses over from two, two space time dimensions, in this particular case, down to one, as required for ingraining to C. So another nice feature is the possibility of expressing in a precise way this kind of scale dependent topology, although I haven't given you any details of how it works. There's one other concept that I want to introduce for later use at the kinematic level, which is that of post, and then I'll just list some important questions still remaining at the kinematic level. So what is a post? A post has a clear meaning in the cosmological context, in which one thinks of possible cycles of expansion and recontraction of the universe. If there are such cycles, then they can be punctuated by what are called the mathematics literature posts, which is illustrated here by this red point of X.

25:00 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 with the mathematical literature, in connection with the analysis of a particular kind of dynamical law for quantum physics. Which can be called percolation. I'll describe later a little bit what I did. Mathematicians call it a rounded graph order, and they prove that it has an infinite number of posts, so if that were the dynamics, the two dynamics of the causal set, there'd be an infinite number of levels of expansion and recontraction of space-time. Some other topics that I recently referred to and won't say any more about for lack of time are approach training, causal completion, I haven't referred to and I won't say any more about now. Let me open some important questions. We saw that height, which means the length of the longest chain, is a good estimator for proper time. Proposal sets that correspond to flat space facts. Does this estimator work in the case of curved space facts, too? There are reasons to believe it will. Now, Mirheim gave some arguments, but they're also worried, and it needs to be settled. Very similarly, how about the Mirheim-Meyer dimension, or the other dimension estimators, which I didn't have time to talk about? Do they work well? One of them, at least, the Mirheim-Meyer one, provably works in flat. 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 UG scaler? All of these things are well approximated by some manifold. It seems in fact that there is, and I 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...

27:30 I mean, obviously what I've been telling you so far is under the assumption that the causal set is well approximated by some manifold. But if you're just given the causal set, how do you know whether it is or not? Some tests, as I mentioned, are possible, but the exact necessary and sufficient conditions, in other words, a complete set of tests, that's obviously going to be important at some stage. Okay, so that was a very brief lightning review of the situation of kinematics, I suppose. Until a few years ago, there would have been very little to say on this, because there was no good work on dynamics. And second of all, having to do with the dynamics of a scalar field on the background causal set, what would correspond to the field theory in terms of the dynamic. So there are 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 is proved fruitful is that one should 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, which is the only level we've reached so far, it's still a classical growth model. It's stochastic, it's stochastic classical growth model, but not a quantum stochastic growth model. So we see. Although there are many interesting results on this note, the primary intention was to be a stepping stone to quantum sequential growth, which would be quantum gravity. Now 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 quantum mechanics, then this will make sense to you.

30:00 If not, I would have to try and convince you that it makes sense. The natural step is from a classical measure representing a classical stochastic process to a quantum measure or an equal hearing function, which Jim talked about extensively. The interesting thing is that under the assumption of classical stochastic 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. Do whatever you can do, say, to get some generalization of Newton's, of Poisson's equation in the case of general relativity. Both of these can be pursued in the case of the 12th set, but the one that has been pursued most is the first. Try to find a principle to analyze the general covariance and so on. That will produce a more or less unique dynamic. 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 test of any pre-existing element, because that would represent the conflict between the time of the growth process and the intrinsic time of the positive 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 amplitudes should be 1, and then the two key assumptions are general covariance, which is a kind of labeling independence, where 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 the slicing of directions 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 second. Well, I think I'll show you now. There are two reasons for the term postcolon. It's the poset of finite causal sets.

32:30 Here's an infinite poset. Here's a little bit of it. And you see the very simplest finite poset, one element and two element posets. And as each element is formed, you move along some path in this diagram, always upwards. And what defines the causal process, the stochastic process itself, is the transition probabilities for making each step, for making the next step. In fact, general covariance is a kind of passing dependence in this scenario, and it has the consequence that the transition, in particular, that the transition probabilities depend only on where you are and not how you arrive at them. Therefore, it's my quote again. That's the one key assumption, which is a very natural analogue 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 there as such, but there is a causal order. And the idea is that the causal order should limit... It should not be able to go from space-like related and its causally unrelated elements. So, just quickly, the way we implement 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, by Bertrand Russell. Each one is associated with a transition probability, and we want to know the ratio of those two. The assumption or the condition we impose is that the ratio is independent of what's going on in the space-like related region, specifically independent of the presence or absence of a spectator element, which is indicated here in green. Anyway, these conditions almost uniquely imply a family of processes that turn out to be a kind of generalized percolation model. Well, 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 in which, at each stage, it chooses which will be its ancestor.

35:00 The percolation process follows a particularly simple rule for that choice. The element about to be born, so to speak, looks at all the existing elements and goes through one by one, slips a client with probability p and chooses them as ancestors. Then having chosen those, it has to take what comes with it. 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. This absolute value of s, by which I mean the number of elements, the current element, 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-p, where n is the number of periodistic elements. So leaving aside the constant 3 factor, that's just p over 1-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 of that particular set. Well, generalized percolation is just like that, except it doesn't use this particular expression, but it still chooses proto-ancestor steps dependent only on their terminality, but with a width that needn't be of the form some constant to the k, where k is what I call the test. So, rather it can be a free parameter, a non-negative parameter, which I call... It's called here T sub k. And the choice of these parameters, the sequence of parameters, T0, T1, and so on, that defines the stochastic presence. Now, actually many results are known for these. We call these classical sequential growth processes. And a particularly interesting question is, will quotes occur? And also, will an infinite number of quotes occur? What seems to be the case is that there's a critical fall-off rate for these T sub n's.

37:30 If we had percolation, they would just fall off like, there'd be no log n here, they would just fall off like T to the n. And that's known as an infinite number of course. This is a somewhat more rapid form, but not much, because it has a log n in its 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 poles from finite or null poles, and it's, if I remember it right, square root of pi over 3. So, if we choose such a dynamic and we choose t greater than tc, These might have reached the kind of conditions that we need for the Big Bang without invoking any kind of post-Planck era inflation. In particular, one can argue that it will produce a kind of positive set which is spatially homogeneous, spatially isotropic, and also big. This is a big mystery of cosmology. It goes by the name of the flatness problem and so on. But it's the main thing that inflation was involved to explain. So, here it happens automatically, and the deepness is associated with the large number of cycles of expansion and contraction. In other words, on each expansion it grows a little bigger. And so finally, it's as big as you like, it's a complex. The big drawback, it makes it a toy model, aside from the fact that it's not... One of the things that's haunting me is that it doesn't produce something fully related to manifolds. It's something that has manifold features, but it's almost certainly not genuinely 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 recontacts, re-expands, you get a new dynamics of the same sort, after all it's still generally well-branded and so on, but with different parameters, and in fact just by the simple renormalization group transformation. Sort of the new kind of renormalization group associated with it, not just going from the very small to the very large. But from the very early to the more recent, and quite a lot is known about the flow of generalization groups and so on.

40:00 So, this is meant as a warm-up for quantum gravity, but what we lack still is to produce an actual quantum gravity is the quantum analog of the Bell-Fondelli definition. Everything else seems relative, which is clear. The other approach... I'll just allude to it. We're just trying to directly do something that's an analogue of the Einstein approaches, until very recently it wouldn't have been possible if we had no idea how to write something that was an analogue of the Ricci-Skater. 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, it would be a matter in addition to the causal set, A possibly simpler laboratory for exploring some of the issues of dynamics is a fixed positive set, but with dynamical matter. In this case, the dynamics there in the field. So again, review what's known here. And this is very much new things, so I can't completely say this. Let's think about... The simplest case is a massive scalar field in the Minkowski space, and in the 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 Laplacian, minus five over that. How would we transcribe this equation to a scalar field living on a causal set which had been produced by sprinkling n-dimensional Minkowski cells? In particular, how would we deduce the value given an arbitrary scalar field, which just means an association to each element of the Poiseuille set of some real number or complex number? How would we deduce the value of the dollar version at phi at some element of the Poiseuille set? So here I've shown a particular element which we want phi and I've illustrated some of the elements that will enter into the formula for it. It's parents, its immediate ancestors, what you might call its grandparents, the ones that are one level down and all that, and then still one further level down, what you might call its great-grandparents.

42:30 So there's this miraculous formula, which to me is miraculous because for a long time I thought no such formula would even in principle exist, but here it is. At the parent minus twice the grandparents plus the great-grandparents, biomial coefficients here, mixed in with an appropriate factor, the value of the scalar field at the point where we want the dollar variation, and we get something which on average, better than we ever know, will give us, in the limit of high density of string, will give us exactly the dollar variation. What the strict theorem proves is that the test functions of the complex are correct, and notice that this is a fully Lorentz invariant construction because the sprinkling itself is Lorentz invariant, but one problem with this formula, miraculous though it is, which is that it's only true on average, one would hope that as you increase the density of the quantum spherical points, the fluctuations would get smaller and it would converge to the exact answer. In fact, what happens is that the fluctuations get 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 an alphanumeric label could actually have derived behavior and do have fluctuations dying away as these platforms become smaller and smaller, so to speak. So given this, now we're in a position to address these kinds of other things that go by the name of transplanning 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 loads that are important to us now can have been much, much smaller than the time frames early on, and so what are we talking about when we analyze those modes? To finish up this expression from the D'Alembertian, What we need is mainly a four-dimensional version, 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. In fact, Roger was talking about it, in a sense. That needs to be finished. And then, of course, the harder thing, which is less clear-headed, though, is a quantum version. And still less clear is how, if one wanted to have other fields together, is how one would do it.

45:00 Another kind of dynamics, not of a scalar field, but of a particle on a quantum set, has obvious problems. It brings in obvious problems that have to be overcome. In particular, what could be the analog of the geodesic? There could be nothing like an exact geodesic in a discrete structure. What replaces it? One particular class of models, especially introduced by Joe Henson, It suggests that there is a kind of natural replacement and it leads to a Lorentz invariant diffusion in velocities. It could be a signature of the street. 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 a positive step. What kind of positive step is it? This is the kind of graph representing a positive step made by string theory. One thing you can see is that there are direct links going quite a distance away. This is the essential non-locality. Which makes it so difficult to get something like the box operator, because the box operator, of course, is local. So in kinematic dynamics, there's a third stage of any theory, which is phenomenology. Without that, the theory is in the end rather useless, as you might say, predictions, or experimental predictions, or explanations, or maybe post-predictions, explanations of things that we already know. But I want to make a list of some of them, not quite all of them, but most of the things which could be viewed as phenomenology or potential phenomenology as far as I'm concerned. The first one is a post-fiction, so far so good. There can be no causal loops, no time machine. This is purely kinematic reason. The claim is that although I just ruled out by Pheas having closed time-wise curves or causal cycles by this, that weakening it would not allow me to recover space-time to do that. The second one is a genuine prediction. In fact, it's the most known prediction that any theory of quantum gravity has ever made. It's been tested by experiment, and arguably it's not true, so we could say that quantum gravity is getting a thousand so far.

47:30 I'm not sure what the cricket analog is at. It's something better than made it over, but I'm not quite sure where it's actually at. But the prediction is for a fluctuation in the causal logical constant, which is one over root n, n being the number of causal set elements formed in the past up to that stage, it leads to a prediction, since n for the visible universe, since the big bang is on the order of 10 to the 20 to the 40, it's giving a quantity of magnitude for the, for the fundamental life scale, the square root of 10 to the minus 100 to the 20 is the price. You'll recognize as the value of what seems to have been seen as a cosmological concept. I won't say any more about what that prediction comes from. We'll discuss it afterwards. I had a slide that had 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, and that would be... Because of the essential in-built Lorentz invariance of the puzzle-set concept, any deviations in geodesic motion must be realized in some invariant way, and what happens is you have a Lorentz invariant way. What happens is you have a Lorentz invariant diffusion in velocity space. This is a kind of diffusion in space-time, and it's driven by the diffusion in velocity space. It's an interesting problem in stochastic processes. But it also, in this work that Fei and I and Joe did together, there's a potential, as suggested by Fei, there's a potential relevant to the origin of ion-geochosmic rays, there are still some problems with the ion-geochosmic rays. The numbers don't work exactly as one would want. Closely related to that are propagations, not of a particle on a causal set, but a field on a causal set, what I talked about before, and this is what we want to do now, now that we've done the consolidation, we want to see how propagation of light from distant stars might be influenced by the underlying discreteness.

50:00 This is to be compared with some other work in which people have sought evidence of discreteness in the breaking of Lorentz invariance, or let's say, the change-dispersion relation. Here we don't do that, but there may be other effects, some extra extinction or fuzzing, anyway, that needs to be worked out and related also to the previous discussion because, of course, particles can be mechanical, should be thought of as wave patterns, and then it becomes a problem in the field proposition. Finally, there's black hole entropy. So this is at a kinematic level. The underlying idea is that one should think of the black hole of entropy, which is proportional to its horizon area, as in effect counting the number of discrete constituents, what we might call molecules, of the horizon. The question is, can one come up with some common general definition of these molecules that would be proportional to the area? What's new about this, I think, what's different, distinctive about this comparison, other approaches, for example, in blue quantum gravity, is where you also have finite constituents, which are like cultures in the horizon, is that this also works for very far out of equilibrium levels, not just for structural. I think it will extend to four dimensions too, but a small snag as it was in three or four hours later, it's going to get better for itself. Just a side comment here because it's about time. These kind of ideas are closely related to a particular prediction of holography, which is an entropy bound called the Rousseau bound. Let me just state the geometrical fact in a nice little picture because I'm running out of time. Taking this cone as a constant space, and taking this base of the cone, which is a three-dimensional volume, the number of maximal elements in a causal set composing this, to which arguably all the information would flow, all the entropy, is actually proportional not to the volume of that base region, but to its area, which those of you who are familiar with syllogism 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 previous discussions some of them are kind of new on this slide. First, discreteness and Lorentz invariance can live together. It's difficult, but at the cost of a kinematic emptiness it's possible. And this leads, this was crucial in the argument of the cosmological constant fluctuations. This leads to a kind of, what I call, blurring between ultraviolet and infrared effects, in which things that you would have thought of as ultraviolet questions, like maximum velocities, end up dependent on an infrared parallel. Another conclusion, which there was a hint of in the discussion with Donald Erickson, is that locality must be given up, and this will help us to control the superexponential growth of a number of all the sets, which, however, I don't understand. Very closely related to this is another aspect of the cosmological problem. Why is it non-zero? Why is it so near zero, the other half of the problem? The insight here is that I find this to be sort of, and this is very closely related to what Renato was saying before, it should be sort of as an aspect of the problem of why do we get a manifold law? Why do we get continuum of behavior from this fundamentally discrete thing? I think non-locality is going to be essential in that sense. And finally, this I want to stress because, in honor of Chris, it's very eloquent writings on this problem. That we have, in the context of classical sequential rules, we have essentially a complete solution to the so-called problem of time, that is, we can identify certain predicates of questions you can ask about the problem of time, which on the one hand have pure physical meaning, and on the other hand are generally covariant, and from which you can form biological combinations, all generally covariant predicates. Another insight is alternative ways in which cosmological large numbers might be explained without inflation related to a new kind of cosmic reorganization that I described before. And then finally, The idea that growth, that dynamic, should be thought of as the kind of growth that is, at a philosophical level, we might say that general covariance and becoming can coexist, we can have an actual dynamical space-time, even though we have space-time, a four-dimensional object, and full general covariance.

55:00 In the loop 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 and one probably will not. I'll just use three. Why is the mention of spacetime 4? Again, something I forgot I had a comment on before. Why, this is something I've always struggled with. Why is spacetime an uninterrupted continuum? Why are there not holes and edges? And then a final question, which I'll explain if anyone wants. Which is why are life on the ground and not square? Well, I know I've been thrown a lot at you in a very short time, but I hope I've at least given you some flavor of what causes that theory is all about and some idea of where the enterprise stands at present. This is a very naive question from somebody who is looking at the things from the canonical viewpoint. These are always such that they are moving within your causal structure. Now, we know that in general relativity, to get the vacuum Einstein equations, you cannot have only the dynamic equations. You must have the constraints. Otherwise, you are not generating a vacuum Einstein space-time. How can you aspire within the theory which has only gross laws within the structure to arrive at a coarse-grained 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 gross laws would include mostly constraints, generally 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 the 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 relationship. One can take a set of space-like elements that are causally unrelated elements, the 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. Then there is no distinction between initial value equations and equations of motion, there are only equations of motion, and in that context, classically known, 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. I think your cosmological boson, if I understood it well, was larger in the past. 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. Why do you suck at accelerating numbers? What? Why do you suck at accelerating numbers? So I'm not predicting that it was positive now. That's a purely random event. Put equally well, it's 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, and that actually might be a good thing.

1:00:00 I don't want to criticize this, but this is when the wormhole commons mechanism was popular. He took a pure Lorentzian model, and he said lambda was picked around about 1 over a squared, and if a is bigger than c, then lambda is 10 to the minus 120. So there's some similarity in the arguments. There is that the a-squared is a similarity, right? The a-squared is coming here, a by a means it's a scale factor. So this is coming as the square root of a to the fourth, which is roughly the space-time volume. But the crucial difference, I think, related to Juan's remark is that it's not a law of one over a-squared. It's an envelope for the fluctuations. The sine is not fixed. We're familiar with basic elements which have individuality, like Boltzmann particles, and basic elements which don't, like Bose elements. Now, you seem to be talking as if your elements have individuality. Speak about which element is more. And when you draw your diagrams up there, you assume there's a distinction between adding an element on the left and an 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 distinction between those two cases. Now, I think in the dictyomorphism... I mean, to make the analogy with general relativity there, there are people who talk about incoordinated language and talk that the coordinate transformation is an important thing, assuming the elements are somehow inherently individuated by the points, and people who use the manifold language talk about difficult morphisms and point transformations, assuming the elements are not inherently individuated, and I think the analogy here is you should be talking about combinatorics of the elements rather than relabels, because relabels is the analogy of the coordinate language, combinatorics of the elements. This is the analogy of diffeomorphisms. And I think you could throw off your counting. On the other hand, you do assume diffeomorphism invariance in the quarter language, which may save everything. Because it's, you know, it's another way of talking. You re-label them and say forget about them. You act as if they were labeled and 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 are not. Because 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, then there's no distinction 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 of gauge, if you want, which was the label, to get the idea of sequential growth.

1:02:30 And then we had to make a special law that is 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 thought, someplace with dialect will do to it. As soon as it's there, then you negate it. It would be nicer to do it directly without this teacher. But have you ever seen anyone formulate the Einstein equations without the coordinate system or something like this? It's just the same issue. You can, you can. You can formulate it. Okay, we can talk about that. But another more direct criticism, a more typical criticism. I don't understand why your cycles have to involve only one point. Why couldn't everything come together with three points, so every element, three of these just came out of the air, so that every element either perceives one of those three or follows one of those three. Why must it be just one element? And further, in response to your first point, is that there is a counting issue of this RU raise, and it does depend, in the case where you have anamorphisms, 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 and 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 Well, you know, when Einstein, in effect, did the wrong count, the whole argument, he got the wrong answer, he couldn't have done it the way that he did. Well, that's a much more radical, that I think we're taking care of. Yeah, I think you're right. Let's thank Raphael once again.