Dynamics of Causal Sets as Basis for Cosmological Model

0:00 I'm Chris May, and I'll give you the second talk to the previous seminar. The talk of the part of the talk will be derivation of causal cell dynamics from discrete generalization and hyperbolic, the classical spiritual growth model. So, as Giannis just said, this is the second of two talks. The first talk, many of you were at, was about a possible explanation for how the causal set dynamics might be able to explain some of the large number puzzles of cosmology. In this case, it was the large number represented today by the ratio of the size of the universe to the size of the cosmic microwave radiation wavelengths. So it's a large number that sometimes goes by the name of the planet. But that was only a toy model, because dynamics was not yet formed. But to the extent that it was interesting, I would say it was interesting because the dynamical in which it was carried out follows from very, very general principles, in fact, a version of Bell's causality, a version of causality, which we call Bell's causality, a version of general covariance, which is appropriate to the kind of discrete structure that causal stat is. So what I want to do today is actually show that to you, show you how, from the general idea of dynamics as a form of growth, which we call sequential growth, and imposing these two general conditions, we're able to derive almost uniquely the causal framework that I call in the title Classical Sequential Growth Model, and that was the basis of the talk last time. This framework also feeds, certain features of this framework feed naturally and would help feed into the prediction that I mentioned last time, and I'll mention again, but not dwell on, about the cosmological constant not being zero. And if I have some time at the end of this talk, then I'll say briefly how that works, and where that prediction comes from, because there was no time to say it last time.

2:30 I'll also mention, if I get that far, that from this prediction, one can make, if you believe everything that into it, you can make an argument that excludes most, if not all, of these so-called brain world scenarios on the grounds that they would predict too large of a fluctuation in the cosmological context. But the main thing I want to do here is just the derivation of the sequential growth dynamical rules. So, however, I'll briefly, just to remind you or tell you in case you haven't seen it before, because you weren't at the other talk, which was a while ago anyway, I just want to remind you of the definition of a causal set and a little bit about generalities of the whole idea. The definition is something simple, a causal set to me is just a set of elements provided with an order relation, which is about the written side in this way. You can read it as So if you say x precedes y, you can say x is an ancestor of y, that's another language that'll be particularly appropriate for this model, or x is to the past of y, all different ways of reading this relation. And to say this, the causal set is to say that it's mathematically a partial order, which involves two axioms, transitivity x precedes y precedes z and y precedes x precedes z it's clear that this is a natural feature of all of those different kind of relations that I use as metaphors for this ancestry or causal influencibility or being with the past and the other one is sometimes called asymmetry, which says that no element precedes itself. That's a convention, but it's a convenient convention in which it's one of the two conventions that are natural. And I'll use this one. And then the third thing is the discreteness axiom, which in this setting, in which we'll be essentially doing cosmology and the universe will have a beginning, it can be expressed in this form, past finite numbers, which says

5:00 that no element should have more than a finite number of ancestors. So given any element A, the set of all elements to estimate should be finite. In other words, only a finite number elements in this set? A. Probably A. Are given any element? Given A. That's fine. We're all A. That set is a finite set. so this is what it is structurally. I haven't said what its dynamics is. I'll come to that in a second. that this causal set will actually arise in a dynamical way, not be all there at once, sort of like what sometimes people call a block universe or a space-time that's laid out under the aspect of eternity, as people sometimes say. But it actually develops. But before doing that, let me just make just a few general remarks on the program, are similar to things that I said yesterday, not yesterday but in the last talk in introduction comics. So even without saying much about the dynamics there are certain general insights I think that have come out one of them is the one that I'll talk about briefly later if we get time, which is the prediction that the cosmological constant would be fluctuating of about 1 over n, n representing the number of elements making a universe up till now. And that gives a prediction of about 10 to the matter of 120 in Planckian units. Another prediction that you may think is either utterly trivial or maybe something that you don't even like, depending on what you think about time machines, is that there are no time machines. This may look like a trivial prediction look like it was built in to the axiom but actually it's not the statement is that there's no natural way of weakening the axioms in such a way as to give rise

7:30 to space-time that do have causal loops another thing that's been done is counting something that's arguably going to represent the black hole entropy when we understand better how to find entropy from the dynamics that is you that I'll call for short, causal links crossing the horizon, and the number of them turns out to be proportional to the horizon. Not surprisingly, somewhat more surprisingly, with the finite proportionality constant, neither zero nor infinity. And still less trivially, I think, the counting has been done also in the case of dynamical horizons, which in that sense is an improvement on something like the something in string field. On the other hand, it should be said that the course of the homework string is not coming up with a dimension of the form of the shape. There's some work on putting a scalar field on the background causal set, and that shows in an interesting way how you can, in fact, have something like a Lorentz invariant lattice. That is, you can combine consistently discreetness, and Lorentz invariants, which is almost impossible, but hopefully is not entirely impossible, or it'll be a problem for the whole causes of that program. One thing that comes out of this is a kind of natural blurring of infrared and ultraviolet effects that I won't say more about now, but maybe has some resonances with other things that people are thinking about these days. a kinematic role for randomness I won't say too much about now but it plays a role in the cosmological constant prediction so I'll say something about it if we get that far and then there's a natural notion at a technical level the notion of coarse graining helps one to say in a precise manner things that people have long wanted to say about gravity to talk about features like for example foam called a scale-dependent dimensionality. What does it exactly mean to say that the topology on one scale is one thing, and a larger scale is another thing? And then this, I wouldn't call this an insight from the causal set program. It's the insight of the causal set program.

10:00 And this is the statement that it's a workable program, at least at the kinematical level. In other words, it is possible, just from this simple idea of the discrete partial order to recover, in an appropriate setting, the geometric ideas of space-time. Of course, that's the basic. And that fact is at the basis of the whole program. If it weren't true, there would be no point in talking about causal sense. So most of the things on this list, with the exception of the cosmological constant partial exception of the cosmological constant prediction are all at a kinematic level as regards the causal set that is they don't tell us about what makes one causal set over some other level or in other words they don't tell us about the equations of motion or the dynamics of the causal set So, in fact, as I said, what I wanted to talk about today is the dynamics and where it comes from, the dynamics that we found for them, and where it comes from. Let me just mention a couple of things first, which is that the dynamics have been called on to do quite a lot in this kind of theory, perhaps more than in some other theories. because although in some sense I think all it's probably fair to say that any discrete theory of space-time is going to face the same question that I've highlighted here and to the extent that most people expect that the right theory will be discrete in some form I think that's a fair statement today everyone is going to face the same issues but in the case of causal set we can quantify how hard the divisions are. The first one is, why does the causal set choose to be a manifold? What does it choose to be like a manifold? In other words, why does it choose to configure itself in such a way that a manifold is equal to approximation to it? One can count the number of causal sets on n elements, and it's something like this. It goes superexponentially with n. One can also estimate the number which could be like a manifold. It's also superexponential, but it's still very much smaller than this. So this is a measure of the gulf separating the generic causal set from the realistic causal set.

12:30 And that gulf has to be planned by the dynamics. If you have some cursory acquaintance with, for example, dynamical triangulations, another approach to a discrete quantum gravity theory, although they usually want to take a continued of movement, then you might worry that a superexponential divergence like this can't be compensated by an action. But it turns out that it can, and this whole family of dynamics shows it clearly. Another related question is, I just mentioned that it's related. Why I think it's related is, why is the cosmological constant zero? That's, of course, another problem that any quantum has to face, the only thing I want to mention is that I think these, in some sense, might collapse into one problem in the case of causal sets. Because in the sense that a large cosmological constant of the natural magnitude, in other words, near something of order of unity in Planck measures, would produce something like a curvature of order of unity. to the discreteness scale, which would mean that you don't have a continuum description at all. So the question why there isn't a manifold description is closely connected to the biological constant values in this context. And some more comments here on how we would get our general relativity. I think I'll skip those in the interest of time. The only comment is that a key, desideratum will be locality but once you have it you're almost home free and it's sort of locating where the difficulty is so what do we want the dynamics to be so from a quantum point of view and ultimately of course we want a theory of quantum gravity I think the dynamics can't I mean there are certain things that it can't be and those are the most common ways that people for a quantum series. For example, it doesn't seem sensible to define it by a Hamiltonian. It doesn't seem meaningful to introduce a Hamiltonian to evolve the causal set, because the Hamiltonian puts you forward an infinitesimal amount in time, and here time doesn't have infinitesimal amounts.

15:00 So out of all the different formulations of causal set, of quantum series in general, the one that seems appropriate to something that has a temporal discreteness, which causal sets certainly do, is a sum over histories. And the particular form, well, a sum over histories can be thought of in different ways, but the particular form that I think would be appropriate in terms of what's often called the coherence function was, which some people here aren't familiar with. Another point I wanted to make, but I'm forgetting. But anyway, so I think what we ultimately want is a decoherence functional, so you call it a set. But the decoherence functional is possibly thought of most naturally as a kind of quantum generalization. Or dynamics expressed in that way is thought of most naturally as a kind of generalization of a classical stochastic process. In other words, something like the Wiener process or something like Brownian motion. The point of view associated with decoherence functions is that I think that quantum mechanics is something like a generalization of classical stochastic mechanics as opposed to a classical deterministic mechanics. So in that spirit, what we set out to do is produce classical stochastic dynamics for causal sets. the aim being that we'll do it we'll do the construction in such a way that the quantum generalization will be obvious or it'll have a big head start towards the quantum generalization so we want a stochastic process but what there are many stochastic processes maybe I should in a second I'll draw you a picture of what the stochastic process the typical beginning of one of the realizations of the stochastic process one process that constantly grows stochastic grows for the causal set and we'll have a huge number of

17:30 free parameters transition probably in order to narrow that down to something even reasonable, we need some further input. And the input that I'm going to suggest are these general principles that I referred to before. A certain form of general covariance, a certain form of locality. Sorry, a certain form of general covariance, a certain form of causality. The idea here is perhaps an analogy with the way that general relativity was or could have been discovered, in which you had the idea of a space-time, a Lorentzian manifold, and you needed some equations of motion for it. And the two principles of general covariance, and in that case, locality, more or less uniquely lead you to general relativity. The hope is to do something similar to the case of the stochastic growth process, which is meant to be the dynamics of the causal set. So let me kind of draw a picture to illustrate what I mean by a stochastic growth process. So as I said, this is the idea, or it's actually built into this framework, that there's an earliest moment, the first element of the causal set. And if you want to go back even before that, you could say that there's nothing. And the causal set arises by a sequence of bursts. So before any birth, so to speak, you have nothing. Then one birth occurs, so there's now one element. There's only one causal set of one element. Actually, last time I showed a little movie illustrating the same point. Some of you may remember that. Now the second element is born. Now let's suppose it now has a choice. I'll label this zero if you remember the first one born. the second one born, it actually has a choice. It could be born with this as its parent or its ancestor, or it could be born here, so to speak, unrelated to the first. But let's suppose it made the choice to have this as its parent. Then the next element is born. It has now a bigger choice of what its ancestor set could be. It could be just this element, both of these elements, or no ancestor. It couldn't be just this, because by transitivity, ancestor, this is also an ancestor.

20:00 So let's say it chooses this for its ancestor. It's these choices, which are being made by some definite transition probabilities that determine the dynamics. Let's suppose it happened to make this choice. Then the third one won't come along. It chooses to have this and this as ancestor. Fourth one is actually. So we're building up the causal set element by element, 7, 1, perhaps, this, and so on. So you continue that process, you'll get a limit, sort of countable number of elements, what you might call a completed causal set. This is the idea of the dynamics as a process of sequential growth. Notice that the elements were born one by one, not in multiple births. These weren't taking fertility drugs or something. So it's a sequential process of growth, and it happened to growth, and it happens in a definite order. For example, this element was born before this element. However, the statement that that was born before the other, or vice versa, is just the kind of thing that shouldn't have physical meaning. Because in the end, it's only supposed to be the intrinsic order relations between these elements that determine what we call physically time. had these labels here, which is a kind of extrinsic time. It was in order to realize the idea of sequential growth. But we don't want these labels to have any more physical meaning than they should. In particular, they shouldn't specify the two space-like elements because the land-related elements had a particular order of verse. That is, there should have been another labeling that was equally good to describe this. And that leads to the condition of general covariance, is what it called discrete general covariance, which is, I think, a natural name because general covariance, I mean, in the continuum, general covariance means, in effect, independence of choice of coordinates. In the discrete case, these labels obviously

22:30 play the role of coordinates. So if you ask about label independence, then it's natural to think of that as a discrete version of general covariance. So that's what I mean by this, this is pretty general . Rafael, except the original thing, 0. And this causal set, all the rest of the vertices, they have at least one of the seed. In this particular, I'm right. But in general, you cannot have a completely disconnected point. well I need something to be born out of the back yeah so it's so you rightly picked up it's not by chance that I do it that way because that's my sort of emotional preference however the maximum generality we have a lot of those other possibilities so another possible cause of sense that could arrive in this dynamic it would be for example that one so this one that has the property you we call it originarian because there's a point that represents an element that represents the origin of it. And one can restrict to this without, I mean, you've got a well-defined scheme by doing a restriction as well. But the scheme I'm actually going to present is not going to demand that. Right, so, but then the irreflexivity there is not true. what supports that vertex the individual yes because we say that well I still say that it doesn't produce itself it's just a convention but if I relax it then I need to say something else in place of this so so general covariance means that the probability of arriving for closet set after the seventh verse is the same, the transition probabilities for the individual verse should be so chosen that the probability to arrive at this is the same if I switch around the order in a way that's allowed by the structure of the closet set. What I shouldn't do is say that this element was born before this element. And if you notice, when I put new elements, they always had ancestors, but they didn't

25:00 have descendants yet, when it was just like ordinary biological processes. In other words, you don't make a birth happen to the past or something else, because it would be happening after that in the sequential sense, but before in the physical sense, and that's an inconsistent. So the statement that the birth process proceeds in that way, with each element being temporarily a maximal element when it's born, that's what I mean by internal temporality. And bell causality I'll get to in a second. First, I want to show you a slightly different way of thinking about general covariance that would be important for us. So this is, if you think of the successive stages of this process, you just have causal sets of ever more elements, 0, 1, 2, and 2. And here is a little bit of a diagram that in principle would consist of all finite causal sets, showing the possible transitions that can be occasioned by a birth, by an individual birth. This is itself a closed set. And so some of the same language that's useful for the causal set, the physical closed sets, also applies to this. So the main thing is too complicated to take in. one thing that you can take it is that it is a very complicated diagram and it makes it hard to study these conditions but if we look for example this process corresponds to a particular path first there was nothing then there was one element then there was the tube chain then there was this knee and so on so a particular particular realization of the process a particular growth corresponds to a rising path through the standard. That's the key point. And the transition correspondence is not quite perfect. There's a few technical issues that I believe in this slide because they're not essential. It's true that every such growth process gives you uniquely to a path, but the numbers is not quite true. But it's true enough for the analysis we need. So in this language, general covariance is now path independence, because different labelings, different orders of birth that arrive at the same final causal set just correspond to different

27:30 paths with this diagram. So if we had to arrive at this, this element could have been born before or after this element. This element here on the right could have been born before or after the top element because it's basically separated. One way you get this path, the other way you get this. So for each line on this diagram, you have a finite number of possibilities. It represents a causal set at some stage of the process. Each arrow going up represents all possible causal sets at the next stage of the process, if we have this at the previous state. And there's a set of transition probabilities, and that is the dynamics. So the dynamics is the assignment of transition probabilities to each of these aeros. Of course, you have to assign them consistent with the meaning of probability, so that the probabilities of all the possible transitions have to add up to one, and that's one constraint. Bell causality, which is path independence through this diagram, represents a second constraint. And what did I say? Genome covariance represents a second constraint. And then those alone, however, are not yet very powerful. However, if we don't want the name causal set to be a meaningless word in some sense, then we actually want these lines to represent the possibility of causal relation. So that means in some sense physical influences should not propagate except along the lines of the diagram. It's like saying that physical influences shouldn't propagate outside the life realm in our continuum space. So how would we How would we implement such an idea? The idea is that the proposal is to do it in the form of what we call bell causality. This is the hardest one to state in a general way, so let me state it instead by way of an example.

30:00 So the idea is that what happens in some region, as I just repeat what I said before, the idea is that what happens in some region of the causes is not influenced by things that are space-like. So let's consider this four element causes, like the three black and the blue element as having arisen at stage four. And now in red and in green are two possible births So there'll be a possible transition from this four-element causal set to that with the red adjoined, or another possible transition to that with the green adjoined. Each will have its own transition probability. Now notice that for both of these bursts, both the red ones, if it's born in the red way or in the green way, in either case is the blue element here relevant causally because it's not to the past of either the red element or the green element. region space-like to what's going on over here. So the assumption that we make is that such an element I'll call a spectator, the spectator with respect to these particular two birds. The assumption that we make is that the spectator being present or being absent doesn't affect the probabilities of the two birds, or rather doesn't affect the relative probabilities of the two birds, because by normalization, So the ratio of the probability of being born this way to being born this way should be equal to the ratio here, where this diagram is just the same as this diagram, only with the spectator element removed. So that's Bell-Cortel. The name Bell should be a hint that it's a rather strong fatality condition. In fact, it rules out the kind of entanglement correlations that violate Bell's inequalities, it's the kind of presumption that's appropriate to a classical theory, but not presumably appropriate to a classical theory, or arguably appropriate. But it would not be good if what you were trying to do was to create a hidden variables model of quantum mechanics as a classical stochastic process. Then you would have killed yourself by doing this, because you would never be able to achieve the necessary entanglement correlations of the correlations that we associate with contaminants.

32:30 However, that's not the aim. The aim is to do a classical model. And the hope is that there will be an appropriate formal analog of the quantum model that will, because it's working with amplitudes and not with probabilities, will it be, in fact, be compatible with contaminants. Anyway, that's the assumption. And those are all the assumptions. The sum rule, the fact that they add up to one, is, of course, trivial. So the first question is, how strong are these assumptions? Do they have any solutions at all? Well, it turns out that they have at least one solution that's very easy to state, and that's what I call transitive percolation. The word percolation is for historical reasons. It's related to some causes. It's just a little mechanic under the name of one-dimensional directed percolation. It's also studied in... The particular model is also studied in combinatorics under the name of random graph orders. But anyway, I'll call it... From now on, I'll just call it percolation. and what is this model well remember that the stochastic process is a sequence of births and with each birth is associated a choice a number of alternatives and what the choice is is you know I'm the new let's say you're all there and I'm born I have a choice of who among you are going to be my ancestors that's the choice so I'm fortunate in the way that real human beings or not but at least i can choose my parents i can't choose all my ancestors because if i choose you for a parent then i have to take your parents as well so that's the transitivity requirement but at least it has a choice of its parents so how does it it's this sequence of choices that is recorded in this structure of the cause of sex What I'm calling percolation is a particular rule for choosing the parent. It's the simplest possible rule that you could imagine. You just go through each potential ancestor, and you flip a coin, and say, I'll take you, and I'll take you, and I'll take you, and I'll take you. Except the coin is weighted.

35:00 So with probability P, you take, this is the percolation parameter. of this model, with probability p, you take each one. And then you may find that some are redundant. They were already implied by other choices you made. And you find that some that you rejected have to be now let in, because they're the ancestors of people that you chose for ancestors. So you choose a sort of proto-ancestor set, or an initial choice, and then by transitive closure, you take all the other ancestors that are implied. So is the scheme clear? So that's one-dimensional. That's the percolation problem. And you can prove, I won't do it because it'll follow from the more general results we'll have, but if you're interested in this, it's a very good exercise to think through carefully why that rule does satisfy the conditions that I told you. Why does it satisfy Bell causality and general covariance? Why is the chance, general covariance, puzzle set, independent of the order of the choices. So we have this one solution. And now the rest, what I have to do is show you that we can get the general solution. And in fact, in doing so, it turns out that we'll use it at a certain stage, we'll use the fact that we know this one solution to solve, to get to identify locations in some equation that represents the general transition. So why is it that we can get, I mean, I think in a way it turned out to be very fortunate, although it's always, with hindsight, you can see a lot of simplifications and understand why it happened the way it happened. But at the time, it seemed very amazing to us that we could actually get the general solution to these conditions. Not only the general solution to a bunch of equalities, which is what all the conditions I've told you about so far are, but even the general solution to the inequalities that tell you that all the probabilities have to turn out to be positive numbers. All of that can be solved explicitly. So why did that happen? First, let's consider a particular transition. I'll just call it capital T. So this is for a transition from a causal set of n elements

37:30 to one of n plus 1 elements that came from a particular group, adding a particular element, an element in a particular way to this C sub n. So remember, we can think of this first as a transition upward move in that diagram that that I have had before of the many, many of all the possible finite causal sets. In this diagram, there is transition. It's backward, but it doesn't matter. So the transition is a particular move, for example, like this. So let me draw a little piece of that diagram just to show an illustration of what's going Because that diagram is itself a post-set, we can use the same language of ancestor and descendant to refer to its elements, although it has a different meaning. the math energy we could use the same and let me talk about when I talk about family like here we're talking about parent and children we mean this one causal set C sub n as the parent and then its children will be all the possible C sub n plus 1s that you can have so let me just have something to look at here would be the parent as the parent causal set and here is There will be a transition probability, so if there's each of these four possible transitions, they must add up to one. This one will play a special role in the derivation, called gregarious. areas, this is just because it turns out to be related to almost all of the others. And this one we call to mu, because it turns out not to be related to any of the others by Bell causality. So the statement is that within a family, which you see an example here, the Bell causality condition, gives you the ratios that would be here, say, T1, T2, T3, and T3. Four transition probabilities.

40:00 The ratios of all of these are determined by Bell causality relations, except for the one involving timid. In fact, the ratio of each of these others to the gregarious one, to the gregarious transition, is determined by Bell causality. That's better, I think, if I just need you to think through why that is, say in words, it's because you have a bell causality relation whenever there's a spectator. But in this case, there's certainly a spectator, because the new element has no relation to anything at all. So it's certainly a spectator with respect to any of these other transitions. And, well, I shouldn't say that's not the right way to say it. Oh, I'm sorry. That's a misleading way to say it. It's not that it's a spectator. It's that it doesn't prevent anything else from being a spectator, because it's not to the future of anything else. And in any of these other transitions, except by definition for the timid one, there will always be at least one element not in the past of the new element. So, for example, here the new element is really this one. And there's an element there not in its past. So there will always be at least one not in its past. And so there will be, and that will be, that can then be a spectator, that could be this one here, say. And that would be then a spectator for both these transitions. So when you have a common spectator for a pair of transitions, you've got a val-pausality relation. And what it does is it relates the ratios of those to the ratios of other transitions fewer elements, so in a recursive way, you build up the information. If you had all the transition probabilities of the previous layer, then this gives you the relations of these. So what we get is the ratios of all the transitions in the family, except for this one, Timid transition. However, we have an extra condition, which is the sum rule, that these have to add up to one, so that provides us an extra equation. And the net effect is that one free parameter remains for the whole family, which we could think of, for example, as being what I'll call q4, that is, it could be the probability of the Gregarian's condition itself. So that's this here, what's written here.

42:30 Markov-Sommerald, in other words, the additivity of the probabilities to 1 supplies the missing one. So that leaves one free parameter per family. Then it turns out that any two families can be related by, I think it's not quite right, a path-independence equation, but it's by a sequence of path-independence equations. In other words, by general covariance, that allows you to relate the Qn of one family to the Qn of any other family at the same stage. and in fact you've got a stronger statement and it's captured in this it can be shown somehow in general I'll try to do that so I just told you that by Q and so we just said that there should be one free parameter per stage of the growth process in particular there's one free parameter per family and then those free parameters are all related to each other leaving just one overall by general covariance their ratios are also determined leaving in the end just one overall free parameter it's convenient to take the free parameter in each family to be this qm which is the probability of the gregarious transition or another way of saying it is a probability of a new component of the universe to start for nothing, the one that you were talking about. It doesn't seem in some way palatable physically. But we're allowing it here. So that one I'll call QN. But there's actually one for each possible precursor set, or each possible parent in this family. So for each possible causal set that you started with, If I've given them all the same name, it's just because I'm about to prove that they all have the same value. So that's what this one is, that the probability of a new component of the universe beginning, so to speak, is actually independent of the structure of the old component, depending only on the number of elements of the old component, which in a way is perhaps natural, since from a causal point of view, it's reasonable because the new element Why should it care what they're doing? But anyway, here's the proof. That's what this lemma states. And here's the proof. So consider two such transitions.

45:00 So this little diagram is going to represent some causal sets. It's an arbitrary number of elements. And this red element is some extra element, some newborn element. It may look here like it has only a single parent, but I don't mean to indicate that it's going to have a single parent. This is just to show it's attached as a new element. Anyway, so then we have this whole puzzle set. And we consider the gregarious transition for it, which is this x, so that's the birth of this new unrelated element. And then here's another one, which is related to this original one by taking one maximal element here, this red, and replacing it with an isolated element. So there's the analogous transition from this for a second isolated element. So the claim, what we're trying to prove is that these two transition probabilities are equal, and then sort of by induction in the end, you'll get it equal to the one that's just a transition from an anti-change to an anti-change. a bunch of unrelated elements to get one more unrelated element. An argument is a combination of general covariance and Bell-Fausality. So let's look first at general covariance. Remember, general covariance is path-independence. So there are four on this diamond, there are four transition probabilities. Going this way, you get A times X, and going this way, you get B times Z. So general covariance tells us, which is this symbol right to show, muting, tells us that it mutes AX equals BZ. On the other hand, look at Bell causality. Look at the transition Y. Notice that in that transition Y, you can think of it as the birth of this element. These two red elements on the right are clearly spectators. If we take them away, we get the transition, if we take them away, we get that transition, which is the same as this transition. So taking away the spectator, y corresponds to b. In the same way, taking away the spectator from z, if you think it's equally easy, gives

47:30 So the statement is that the ratio of Y to Z should be the ratio of B to A, which if we multiply it out is just BZ equals AY. Putting that together with AX equals BZ, you get, and canceling the A, you get that X equals Y, which is what we were trying to show. Or at least you get it if A is not zero. So there's actually a genericity of the assumption here so we're not deriving the absolutely most general solution of these conditions but the generically general solution i mean something if if no parameters happen to vanish or something like that then we'll get the most general solution there are in fact other solutions special cases that originary dynamics is one of them in other words the one in which we never allow any birth of nothing other than the very first one that whole thing also satisfies and there are many examples of that. They satisfy the conditions, but in those, some transition parameters do depend. And so it can also be, so this is the key thing. So for each stage, there's a single parameter, Qn, which represents the probability for a new spawning of a new universe at that stage, and it's independent of the structure of the old universe. And from it, we know, in principle, how to build up, I tried to indicate to you. And we know, therefore, within each family, we know one transition probability. And bell-causality then gives us recursively all the others together . So, in principle, we have the general summation. I haven't proved it here, but the claim is there are no more independent general covariates or bell-causality conditions. There are, however, inequalities that the probability is actually positive. But before getting to those, I've only told you that sort of in principle we can solve them. But now let's see how, in practice, we can actually get the explicit form of the transition probability. This will lead us, finally, to the parameters t sub n that I was invoking in the discussion last time. So it turns out that the equations look simplest in terms of this inverse of this parameter q sub n. And one can, here's a joke, but I'll let you read it to yourself.

50:00 But we will make a mention that q sub 0 equals 1 in this derivation. So the equations are simplest in terms of q, 1 over q sub n. The reason is you can prove by an inductive argument that I won't give, that each transition probability at some stage, so it should be, by this I mean, again, a general transition probability. It can be that it has the form of q sub n. So n is the number of existing elements before the birth. It has the form of q sub n times the sum of some integers times the inverse q's corresponding to smaller x. That you can prove on general grounds. And then if you know, and these are universal coefficients. These are universal coefficients. So if we get them in one case, we have them in all cases. And we have them in one case because we know percolation is a solution. And for percolation, we can compute these coefficients. without getting all the details, you get an explicit expression. And this is actually what the explicit expression is. For an arbitrary transition probability, in terms of the q's, so in this expression there occurs a parameter m and a parameter pi. And that's all that occurs. This is a dummy summation variable. So if you want to know the probability of a given transition, you only really need to know these two parameters, pi and m, on which it can depend. And pi represents numbers. A particular transition is a particular birth, and a particular birth means a particular choice of ancestors. Pi is the number of ancestors chosen. In other words, it calls a language that's cardinality of the past of the new element x. And m is the number of parents, immediate ancestors, the ones that you had to choose. It wouldn't come by transitivity, and that's in post-hut language, the number of elements in the maximal layer of the task of x. So we're almost done. This is a completely general formula, once you know the q's. Q's, however, are not free, because we

52:30 have to make the probabilities always lie in 0, 1. In particular, they have to be always positive. So we could, again, we're lucky. It turns out that a particular, imposing positivity for a particular transition, in fact, it's a timid transition. If you impose that it be positive at every state, you see that in this formula there are minus lines. So it's not guaranteed that you just were in arbitrary shoes of any positive probability. positivity for that particular one, all the others follow. And that particular one, up to an overall, up to some denominator, which is a positive number, is this expression, t. So positivity will follow if all of these combinations of the q's are positive, or turn out to be positive numbers. But it's even better than that. So now that this has to have 20 degrees, in particular, it has the consequence that q0 would be 9 to b1. It has to be greater, of course, than q1, because no probability can be greater than 1. But then q1 has to be greater than q2. So these probabilities form a monotonic decreasing sequence. So the qs are not actually really choosable. In that sense, I don't know. I wouldn't push this too seriously. but in that sense, the choice of the law up to a certain stage, in some sense, helps determine what the dynamic law will be at the later stages. It doesn't leave it completely free. The reason I wouldn't push it too strongly, however, is because we can actually solve these two clauses. So there's an amazing thought. I don't know how well known it is, but if you don't know it, you'll probably find it pretty amazing, which is that we can invert this equation this equation, which gives t, these things which have to be positive, in terms of your And the reason you can invert it is that the binomial coefficients, m choose m, are, it's a big matrix. That matrix is its own inverse, if you add in a few minus signs, then that matrix is of the binomial coefficient matrix.

55:00 So here we see that inverse, so we can, therefore, easily solve this equation, multiplying to about the inverse matrix on both sides. And it tells us that 1 over q is this combination of the t's. And the t's now turn out to be completely arbitrary. So this is the general solution. What I want to do, however, is tell you, this is the general solution for the q's, but it's not yet the general solution for the transition permeability. To get that, we have to take this solution for the q's and plug it in to this expression for the transition permeability, which in the end depends only on pi, the number of ancestors and the number of parents. And thanks to some other identities, you can actually do that. And I'll give you the n one. do you have the transition probability expressed entirely in terms of these free parameters t, which are arbitrary, not negative numbers. t equals t. It only depends on pi and m, and also the number of pre-existing elements. This is a sum sum over k, and another sum over k, which we usually call lambda times And so implicitly I've defined here a function of two variables. Lambda. This ratio is that. I'm running a little for time. So let me not quite justify the next statement. But let's look at what we have. We have a general dynamics given by the sequence of parameters, t0, t1, t2, which are free, but we've assumed that they're all greater than 0 for us, and then it's a parameter.

57:30 So you can actually let now some of them vanish, and you still get to the answer. So you might think that this is a large number of free parameters, and it is still a microscopically small number compared to the number of transitions that we actually had. It's still one free parameter per stage of the process. But it turns out that all these different dynamics are very closely connected to just a simple percolation, because all of these parameters can be thought of as a kind of generalized percolation of parameters. So let me give you the interpretation of the parameters, which is in this formula. It's implicit in this formula. If I had a little more time, I would say it. So the meaning is choose this is, remember that the birth is to choose a subset. So how do we choose a subset of ancestors? We choose S at random with relative probability T sub K. That's what the T's mean, where K equals S, the number of elements in the present ancestor. And then you take the transdisclosure with B4. You And we can see why this is percolation, because in percolation, we go through and choose independences. So what would be the probability for a particular set that has k elements? Well, we had to choose all of those k, that's p to the k. We had to not choose the other ones, which is q to the n-line of k. So t sub k is proportional to that, dividing out a constant independent of k, we get p over q to the k, which I can call t to the k, where q stands for 1 minus q in this case. So percolation is actually just a special case in which there's a pre-parameter t between 0 and 1, the ratio to the probability to its complement, and the t sub n. The t sub n are just the n's power of that parameter t. So let me, knowing that makes it very easy

1:00:00 to think about the meaning of some of these. So let me quickly run through a couple of special cases of this kind of dynamics to understand, you know, get a feeling for the meaning of the t. So first, the simplest case, so the crucial thing to remember is tk, sub k, is the probability of choosing k, the weight for getting a subset of initial ancestors before you take the transitive closure, k elements. So if t sub 0, which is always 1, is the only non-zero 1, then what you're saying is that the only number of ancestors that you can have is zero. The only one that has a non-zero might be zero. So you'll get an anti-chain, a dust, as you might call it. Completely trivial universe. As soon as you get T sub-zero and T sub-one non-zero let's say that none of the others are non-zero you can now choose either zero or one ancestor. That means that each universe either starts a new universe or it has a unique parent. Well, if you think what it means to have a unique parent, let's look at that case where there is a unique parent, never more than one parent, it means that you're building up a tree. So this, but you're also allowed to start a new tree, so you've got a kind of forest unit in that case. If you let only the first n0 be non-zero, have more than some generalization. If you take t sub n equals t to the n, you get, as we just saw, percorations. If you take this particular one, which I mentioned last time in connection with cosmology as an early guess, you get something that has a more uniform dimensionality. I won't tell you more about that now. Let me just close the circle and mention the one that the considerations were based on last time, the one that gives you this Boltzmann-Tolman is this particular chance, which gives us this universe which expands, recolapses, expands each time larger and more rapidly, becoming more like the conditions would have to be in our early universe to give, by expansion, something like what we see today.

1:02:30 So maybe if someone wants to . I won't give this up on a thousand-legical transparency now, unless someone wants to ask you that. So this will be the last transparency. Out of all these things, let me just pick out two. I'm not just at saving time. So what, in a general way, can this teach us? First of all, maybe three. So it can show us that this dynamical law completely undoes this 2 to the n squared over 4. I can tell you what most of these causal sets look like, but they're just three-layer causal sets. They look nothing like any of the causal sets that are generated by virtually any of these dynamics. So that's not to worry. The other thing has to do with another meaning of general covariance. We use general covariance in a very heavy way, but we use it as a constraint on the transition probability. It also represents a constraint on the meaningful question. And this, again, is a problem that any generally covariant theory has to face. And so therefore, I would say that essentially all theories of quantum gravity have to deal with this at some stage. What are the observables, or if you prefer, what are the beables of the theory? What are the meaningful questions you've been asked about? They have to be completely labeled invariant. So here's a proposal for what they are in the case of the causal set. we ask what are the stem we take any finite causal set and ask is it a partial stem in the causal set what that means is was it possible to order the birth in such a way that at that stage that finite causal set is what you have if a certain conjecture is true then all meaningful variant questions can be re-expressed in terms of those and even if it's not true I claim that you'll never think of question of any interest that can't be re-expressed in terms of those sets of measures there with technicality. So this gives us a direct handle on the physical observables of this here. That's one that I think is likely to carry over, just in its exact same form, sort of quantity. And then the last one I want to mention, and this is, of course, in some way, the main motivation for doing this. Since we have it, it's nice to explore it and look ahead

1:05:00 and see what hints it might give us in the future. I think that's worth it while. But the real purpose of doing it, that's what I did last time, but the real purpose is to teach us how to handle these conditions and then give us practice so that we can do it again in the quantum field. And if you say that what you're looking for is a decoherence function, then General Komerick says a more or less obvious meaning label independence. Sequential growth has the same meaning that it did. In the classical case, the hard one is bell causality. There are some natural guesses, but as yet it's not clear whether they're the right one. If they are, and we haven't started working on them yet, we're still exploring the classical ones, but if they are the right guesses, and if they prove as restrictive in the quantum case as they were in the classical case, then it will have pretty well under control some well-defined, well-delimited set of mechanical laws, one of which could be the right law for bond and property. Any questions? You said at one point that growth rules rule out the possibility because of bulk causality you don't have things. I didn't understand exactly what you meant that the entanglement formulation are directly dismissed. Maybe if you could explain a bit. Yeah, so if you go back, so we don't even have to look at the total set. Just look at It's something like an EPR situation. So there's some decay here, and some particle goes over here, and some particle goes over here. So if you ask what it takes to derive the bell of inequalities, which are the things that are violated when the element is present, what you need to assume is that whatever happens here, it may be probabilistic. It may not be totally determined, although in particular, It may not be totally determined, but it has to be independent of whatever happens here once you specify the past. That's the assumption that goes into it. So there may be these hidden variables telling you how will I respond to skin, if someone asks me about S of Z, how will I respond to someone asking about S of X.

1:07:30 They were generated here. Once those hidden variables, once you make that choice, then whatever happens here is totally determined, completely independent of what happens here. So if there's some randomness, it's uncorrelated randomness. That's what goes into proving the Bell-Causs Algebra. And that's a very close analog of what we assume. We assume that once the past is fixed, that is the causal set before the new birth, that the probability for the new birth over here, in this region of the causal state, is independent of whatever may be here, in this case, even whether this region is here at all, which is the presence or absence of the spec station. So what would hidden variables do instead? Well, hidden variables would have to have some non-local relationship, but there would have to be an actual correlation, a built-in correlation, a non-local correlation between what happens here and what happens here. It's not just the result of both having I mean, if you want it, it is a variable series. That's what you have to do. The other thing I had, how do you get that for the number of cosets which are embeddable in manifolds? Is there a quick way to say how I proved that 2 to be n log n? Yeah, it's not to be taken too literally. It's something like an n factorial. So the total number of causal sets was up to sub-leading terms, 2 to the n squared over 4, and you can get that number just by counting the number of 3-layer causal sets. So if you put around n over 2 elements here, around n over 4 here, around n over 2 here, and you ask how many you can make like that, you get something like this. about half of each layer, because of a similar number. But this is, in the sub-leaving way, this is going to prove it. So that's the total number. So that estimate only came from taking some manifold and thinking, how can I spread it? How can, well, I mean, I didn't say anywhere, really, in any exact, or even approximate sense,

1:10:00 what it means for a manifold to approximate . that the manifold is a good approximation to the cause of set if the cause of set could have arisen by making a random selection of points that's the kinematical randomness I referred to before that goes into the cosmological cause of argument in particular so that other estimate comes from estimating, I mean obviously if I move these points around I'll change the causal dimension so that number came from estimating how many different I'm moving them around. How far do I have to move them before the causal relations change? And that gives you an estimate of the number of different causal sets that goes to a single manifold. And then I just use the same number. I don't know how to quantify the sum of the manifolds. I think that this would be the primary. But that's a very big one. What's clear is that it's very, very much smaller than this. I have a question. It's almost the same question that I had in the first talk. I see that the model is in non-local, right? What do I mean by non-local? I mean, for instance, each event that is born carries information by transitivity about all its... It's non-local. It's non-local. It's non-local, but causal. That's right. And if we were, in some sense, this is the theory that, the theory says that the causal set is a real substratum, and the coarser picture is the continuum. So what is our experience of locality? I mean, how would we conceive locality in the manifold? I mean, let us say what is locality in classical physics? Say, perhaps one could argue that locality is the assumption of a differential mannequin, say all laws are differential equations, so effects connect infinitesimally separate, so in the discrete scenario, could we form, I could

1:12:30 see, say, a discrete version of that locality, of classical locality, to be, say, we could because that influences, you know, involve only contiguous words, say, immediately separate. That's a question. Yes, the question is, so it is, although I see that your model carries the parameter, is a fundamental parameter, the pie. So the pie encodes information about all the... And in fact, that was crucial in this cosmological model, because it meant that the presence of those elements in the previous cycles of expansion and collapse acted to re-normalize the effective dynamics in the current era. And it was that re-normalization I think it rise to this very rapid expansion, initial expansion. That's right. So if we were to make a connection between that discrete subtraction and the classical continue we will say that perhaps a going to the continue taking the classical name there is a forgetful events cease to remember there has to be in some sense in some sense it has to be but so I don't remember what I said you said it's the same question you asked last time it is the same I don't remember my answer I don't remember your answer oh good because I'm probably going back to the same answer alright is it a forgetful plan? is it a forgetful plan? forgetful in what sense? what does it forget? forgetful answer the question? it forgets its original say the original event I mean if you take number 6 in the continue in some sense we'll have forgotten where it came from It will remember, actually, 3, 4, and 5. Because these immediate vertices would correspond to infinity of the same. It will get the same effect, taking suborders of the order. I don't know how I would formulate the category. Yeah, I don't know. You want some forgetting of parts of the class. It's a very subtle issue.

1:15:00 things, locality, which is what you're talking about, Lawrence and Wright, which is realized that they find average in the closest set of pictures, and there's greatness, of course, which is fundamental. Those three things almost are mutually conspired with each other. It's almost impossible to put them together. That's why it's hard for you to... So let me remind you have y, that's true. So if you have a literal continuum, so like a differential manifold, then as you say, you can write differential equations. Or if you don't write the mechanics, you can write something like this. And this is a locally formed quantity. When you write a differential equation or a scalar curvature like this, you actually don't need any points. In some sense, every point is dispensable except the point at which you want to evaluate this. true either. You need some neighborhood, but the neighborhood can be arbitrarily small. And it's because you can take that limit of arbitrarily small going to zero that you can have locality. If you were to say well I think that I can't be there's a little non-locality, it may probably come from discreetness, but for whatever reason then you're already stuck because of Lorentz invariants. There's no Lorentz invariant measure of closeness that will do what you want. So we take a point in the crustal space. What points are close to it? Say, time-like related. Well, we know in the answer, those which are close in time-like related are something inside of hyperbole, like this, plus the path. And if we allow space, I just want to. So our neighborhood will be this big cross, or as I already mentioned, surface of rotation. Whereas for locality, what you really want born in a little neighborhood like this. You can't have it in Lawrence's name right now. And that's a reflection of the text. Something that looks, I mean, in some frame of reference, something that looks in this frame to be very far into the past is actually right here. So it's very, very subtle. You can't just say, I should only be influenced by these because then you won't produce Lawrence in very third. then you'll get to know how you lose this theory and you'll say the agility is not quite advanced way out there in a distant path. So the question is, so that's just to make the point out

1:17:30 that you're asking for a lot when you ask for a lot of data. It's very hard to see what this function should be doing. It can't really literally forget any of these. It has to forget selectively, depending on some newspaper we have in this country, especially in the United States. We've got things selectively, so depending on typical state of affairs, it has to be, if the wave is coming along that's relatively short, long wave length in this frame, it has to be influenced by nearby things in that frame. If some very high frequency wave comes, it looks kind of like a blue chip too. But in some other frame is a low frequency wave. Then it has to bring other things from over here, because that's what corresponds to this region in the appropriate frame. If it's a wave which is at high frequency in all frames, then probably we shouldn't be talking about it because it's below the discrete scale. But even something that's above the discrete scale, in some frame, when we're in a continuum, might as well be able to show it at high frequency. So you need something that has a selective frequency. And there's a particular proposal for putting a scalar field on a causal set that seems to work in two and in four dimensions, it might not an action that seems to work, but we still lack the, we just don't know whether it would be in other dimensions, and also we don't, we lack the deeper understanding of why it works, but if we could understand, it's a kind of trick, if we understand why that trick works, then we may be able to say something more enlightening as a more concrete analysis, and how you would manage to forget what you have to do without forgetting too much to your own minds. One clue would be, for instance, not to base your topology on open balls. In some sense, you see that essentially what it says. It's actually the topology, the physical topology is the cause of topology, I think. based on life. Well, maybe, but I mean, we have to recover, we have to recover standard physics, in which the psychology is. Yeah, related to that question, is it, let me ask a question in relation to the issue of remembering. Is it, could one ask for that, you can forget

1:20:00 something locally, but remember it globally in the future? So in other words. Yeah, that's have in this situation, so if you send in a wave packet that's somehow mobilizing this picture here, then you actually want to forget all the local stuff. And what you want to remember is, I don't know if I call it global, but it's somewhere. Right. So is that suggesting some kind of UV-IR mixing there? Well, exactly, yes. So the UV-IR mixing is exactly the . That's right. So, for example, one can say that there's an upper limit on velocity, on a meaningful boost that you can do, coming from the size of the universe. Why is that? I mean, it's still quite a large number. It's probably larger than any boost that we've seen. So we're not, for better or worse, we're not going to see laboratory. I mean, you need too high of an energy at the moment compared to what you have to see this. In principle, the fact that the universe is finite, an IR effect sets a limit on the meaningful boost that we can do locally. And why is it? Well, because this region, if we think in space-time and continuum language, then the smallest region that makes any sense is one prime volume. Because if it's smaller than that, there are no points of correspondence. at the element's correspondence. The smallest sort of experiment you could imagine doing would have to take at least one out of O. Now imagine doing the boosted experiment at the same place, so to speak. So here's our original experiment. It takes place in this region. Now the boosted experiment at the same place, what does that region look like? It looks like this. It starts stretching up. That's the whatever it is, the Lorentz contraction. If you boost it enough, it'll stretch down so far and it'll reach the edge of the universe. I mean, actually, it doesn't need to reach that, really. It'll just reach something where the furniture is great. The Lorentz rate will already be modified. So that's a typical example. By our conditions, imply some sort of local liminar velocity.