Two possible routes to a quantum dynamics of causal sets

0:00 Hey, do you mind if I do this? No, not at all. Yeah. What's up? No, Yannis is busy with his children today, so we've got him trying to visit people. So it's very awkward, she's from Kilmerion, Westfield. And she's going to be talking a lot of Coulson sets. Maybe you can see that it's two possible groups, the quantum and the Coulson sets. Thank you, it's a pleasure to be here, as always, amongst good friends, and I'm going to talk about cause and set theory, and in particular, two possible routes to discovering a quantum dynamics for cause and set. Not the only routes. There are, as you'll see, only suggestions for directions of research. I'm not going to show you what the dynamics are, because we don't have one yet. So what I'll do in the talk is first introduce you to causal sets, although I know that the concept is familiar to some of you already. And then I'll outline the two different directions which might lead to a quantum theory for causal. It's called quantum measure theory, Raphael Sorkin, who I should mention is the main proponent, main champion of this particular approach to quantum gravity. The second route is along the lines of what I've called a discrete collapse model, and it's something that I'm working on with my student Joe Henson. I'll just summarise at the end. We can try to characterize quantum gravity, different approaches to quantum gravity, as

2:30 either what I call top-down or bottom-up. Theories that we know and love, namely general relativity with its continuum spacetimes and quantum field theory and quantum mechanics in general, and you just try to push them together without changing, trying to preserve as much structure in both general relativity and quantum mechanics as possible in trying to put them together. Theories of physics in a sort of layer structure, then that's trying to get at quantum gravity by starting at a sort of higher level and aiming downwards towards quantum gravity. But one can also, and in fact most people think that it's going to be necessary to approach quantum gravity from the bottom up, namely by... Proposing new fundamental principles different to the ones that underlie quantum theory and general relativity and building up a fundamental theory of quantum gravity in that way. You can try to plonk whole approaches to quantum gravity into one of these two categories and also individual calculations. Paradigm, I would say, of a top-down approach to quantum gravity is the whole canonical quantum gravity program, including Ashtakarni variables and loop variables, etc. So there you take the structure of general relativity as it is, try and make it into a Hamiltonian theory. You try to apply the rules of Dirac quantization that we know and follow your nose, trying to maintain all that structure. String theory, I would say, is something of a top-down theory at the moment, although it's striving to be bottom-up, and maybe if people can discover what M-theory is, it might truly deserve the title of a bottom-up theory of quantum gravity, but I don't think it's there yet.

5:00 Individual calculations, again, you can place them in the two categories, and mostly calculations in quantum gravity tend to be top-down. So I've just given a few examples here. The pair production of black holes, so far, that's all described in a top-down way where we talk about continuum space-time. Quantum cosmology, so far again, is a top-down theory. We try to take quantum rules as we know them. Space-time as we know it and put them together without keeping as much of those theories as we know. And again black hole thermodynamics so far is all the calculations of black hole thermodynamics take place in a top-down framework. There isn't much in fact that happens in a bottom-up way in quantum gravity but I think that the causal set proposal is such a theory. It is an approach to quantum gravity in which we take, we try to postulate new principles, basic principles, in which there is no such thing as continuous space-time, and hope that we'll be able to derive, so let me say in a bit of detail what the causal set hypothesis is. So the basic conjecture is that the underlying quantum substructure of space-time is discrete. And it takes the form of a causal set, which is with a relation such that x, y, and z are elements that precedes y, and y precedes z, so the relation is transitive, and x can never precede itself, which means the relation is irreflexible.

7:30 Together, these two conditions describe what's the third condition on a causal, which is that A , which is a set of all points Which lie between X and Y. If you take that set, then the cardinality of that set, that is the number of points in that set, must always be finite for any X and Y. And this condition is called local finiteness. Partial order is then locally finite. And together, these conditions mean that P is the cause of X. It's a locally finite partial order. Why do the second one, I mean the first one is the fundamental idea, the third one is really fundamental, but the second one seems not always to be why you think that, it might just show that it means you don't have to do it in the quantum theory, that's what it excludes. It does, yeah. It's a good question. I mean, if you relax that condition, then you can show that if you do have closed loops, In the set, so you relax the condition, you don't have a partial order anymore, you have a transitive diagraph. So if you relax the condition and you have a loop, a causal loop in a transitive diagraph, you can show that all of those points are equivalent as far as the causal relation is concerned. So if you've got, say, A, B and C are in a causal, then any relation that A has, then C and B, B and C must also have a causal loop. Causal relation, yeah. So if you have a loop like this, then there's no way to distinguish A, B, and C purely in terms of the relation. They're indistinguishable.

10:00 Oh, I understand that. Yeah. Presence of liberty should be additional structure. Why do you want to assume that you find it? Right. Okay, that's right. So it's possible that you might use such a thing, but this is the simplest thing you can have. If you don't have extra structure, then... Having this irreflexive condition means that you eliminate these so-called non-Hegelian points. Yes, we've coined a new terminology. We've been Hegelian all along and we've been speaking prose. Another thing about that locally finite is that if there's a big bang point and a big crunch point, you know, a first and last point is the whole thing, then in fact it's just finite, isn't it? Because you can, you know, take x and y to be that and that and it just says the whole thing is finite. I'm not saying that's either good or bad. There is a so-called Cassidyram where you represent only the links by lines and the points by vertices and a link is a relation which can't be deduced by via transitivity so it's an irreducible relation it's one that's not implied by transitivity. So here's a little example. And the convention is that a rising link from A to B, for example, means that A precedes B. So just as the relativity time goes upwards, our relation here is that a rising link from A to B means... And there's all sorts of ways of saying it in words. A precedes B, A is an ancestor of B, B is in the future of A, etc. So how can... and that's it, that's what a causal set is. So the question is how can this possibly hope to succeed? How can such a very bare structure, simply an order relation, give rise to all the structure that we have at the continuum level? So continuum of space-time, Mg, where M is a manifold and G is a metric, has a topology, has a differential structure, has a metric.

12:30 How can such a space-time, Mg, emerge from just one of these bare causal sets with nothing other than its order relation? To this question, how does a space-time emerge from a causal set, P, is that it does so when P is a random Planck density sampling of the causal structure of that space-time energy. And I'll say more about each of these words, random Planck density sampling and causal structure. And that's true both of your interpretation and of Sorkin's interpretation of both? This is Sorkin. This is all Sorkin. I should have, I should have said that more clearly. Yeah, this is all, all this background is due to run. So, what do I mean by the causal structure of the spacetime? Well, the causal structure of a spacetime is, it's also a relation, it's also a partial order in fact, which I have denoted by less than with an ST prefix, defined by the causal curves in the spacetime. The first term is in this set J minus of Y, meaning the causal past of an X and Y point in the spacetime. If there exists a future-directed curve from X to Y whose tangent vector is nowhere spacelike, that means that Y lies in the causal future of X, or conversely that X lies in the causal past. Then we say that X precedes Y in the spacetime. Are you requiring an effect by using that wording that the tangent vector exists at every point because one could imagine sort of playing tricks by coming up with one of these very jagged snowflake or fractal type things such that it ends up being space-like even though technically it never has a tangent vector that's space-like because it never has a tangent vector at all and therefore in particular... Well, you could make it piecewise, piecewise smooth. Okay, there's a finite number of exceptions. So...

15:00 In this relation, left-bound spacetime is in fact a partial order. It satisfies the condition of transitivity and irreflexivity if the spacetime contains low closed-causal curves. But it's not locally finite, of course, because there are infinitely many points in most intervals. Now, there's a powerful result, not very well known remarkably, not well known because it's a spectacular result. Due to Hawking, King and McCarthy, tightened up by Malamute, finished off by Raquel Garcia and Marc Marceau. Which says that the causal structure is almost everything, is almost all the information about the space-time that there is. And more precisely, it says that if two space-times and two Lorentzian spaces have the same causal structure, Then they are the same up to a conformal scale. And again, let's make that even more precise, if there exists a bijection from m to m prime, so bijection because it's a one-to-one and on-two map, such that is in the past of f of y, if and only if x is in there, then you can prove that f is actually a diffeomorphism. And the callback of g prime is related to g by just multiplication by some function. So the two spacetimes are the same up to a conformal sort of mid-seventies, and it's got no kind of conditions missing, you know, well, yeah, there's an extra condition, which is that two spacetimes have to satisfy a slight strengthening of the condition that there are no closed causal curves. So in fact, they have to be. If they have to be past or future distinguished that's the condition. It's just a slightly stronger condition than the they don't contain causal codes but otherwise there's no conditions you don't put any condition on this function n so it doesn't even have to be continuous so the causal structure of the space-time fixes everything about it apart from the overall conformal factor

17:30 Or, in other words, apart from information about the volume of space-time regions. So, here's a space-time. Given that, what could be a discrete underpinning of a space-time that only sees the structure of the length? The idea is that you choose some points in the space-time, somehow uniformly distributed, and choose one per-plank volume. Then endow them with the partial order inherited from the spacetime causal order. That will give you a causal set. It will be locally finite and it will satisfy the conditions of being a partial order. So, for example, if you choose this point A and this point B, then A will precede B because A is to the causal class of B in space. You can discretize this space, this continuous space-time, by choosing these points and giving them the partial order that is induced by the space-time part of it. And that will be this discrete underlying substructure. Note that this description introduces volume information via the correspondence that the number of points corresponds to the volume of the space-time region in the Planck unit. So because we're choosing one point per count volume, it introduces its corresponding number and number to volume. So in fact with this prescription we have a hope of not only recovering the conformal, the space-time up to conformal factor, but actually fixes conformal factor also. So you can ask any of these questions, but is that, I mean are there sort of background assumptions way back, like is this all 3 plus 1 for example, or is it anything plus 1? No. It has to be anything plus one because it's only in anything plus one that you have a causal structure. I know, but the anything is anything. Okay, so that was the general idea, but how are we to choose these points in a way that respects local Lorentz invariants?

20:00 Well, the answer is that we choose them at random by a Poisson process of sprinkling. So literally, imagine sprinkling with a sulfur points down into the space-time. So that the number of points in a particular space-time region R is a random variable, and it has an expectation value which is given by the volume of that region in time, in time units. So, let me repeat it again. The suggestion is that a space-time Mg is a good approximation to, or emerges from, a causal step P, when P could have been made by sprinkling into that space-time. And Raphael has conjectured that this is, in fact, the only way to discretize a space-time in a locality of the rectum-vary mechanism. What does locality of the rectum-vary mechanism mean? It means that, precisely, this process of sprinkling, of this cross-on process, will give you a probability distribution on the set of all causal sets, in fact. And locally Lorentz invariant means that that probability distribution doesn't depend in any way on any coordinate choice that you make in actually doing the sprinkling. So the sprinkling, this random process, doesn't depend on any coordinate choice. I mean to do it on a computer you'd have to choose some coordinates to do it. So it doesn't mean the local Lorentz group sets the tension space? I mean, of course, there's no such thing in the causal set. It just means that the sort of these causal sets, P, are themselves and depend only on the space time. So is that to say that the other stuff isn't Oedipium orchids? Well, there isn't Oedipium orchids in the Oedipus forest, is there? No, there might be. Let me give you an example in which it's not locally directed. So if you think about two-dimensional microscopy space, one way to discretize it is to make a null lattice like this.

22:30 And that looks quite uniform. It looks like you've got one point per bank volume. But you've only got one point per bank volume if the volumes... All of these are matched to the coordinates that I've chosen to do this latticing. If I boost this lattice, I get something like this, and if I boost it very far, by something close to this beam of light, the distance between these groups of lattice points gets very, very large, and these ones get squashed up together. So you can see in this coordinate system, this lattice looks very un-uniform. It doesn't look like this at all. So the notion of... but if you were to do a sprinkling into this two-dimensional encaustic, then any boost would make the sprinkling look exactly the same. You wouldn't get big voids and gaps and the conjecture is that any way that's not random of discretizing the same encaustic is going to have this problem that there'll be some coordinates in which it looks... If you haven't sampled it properly, if you haven't got a roughly one point, how about something like a Penrose tower? Well, I mean, the conjecture is that that won't work, so that anything that's specific like that that's deterministic isn't going to work. I mean, I haven't proved anything. Just to clarify, presumably that rather innocent phrase could have been used sort of close to the maximum likelihood, I mean it always could have been used, so you have to sharpen it up. Yes, that's right, you have to sharpen it up, all of that needs to be sharpened up and made mathematically more precise. Okay, so what about, so that was just kinematics, that was just a description of what the underlying... What the underlying reality of the universe could be, that is a causal set, but of course we need more than that, we need to know how it behaves, we need to know what the dynamics is, and we believe that the universe is fundamentally quantum mechanical, so it has to be a quantum dynamics, so I'm going to move on now to describing these two different routes, pathways to finding a quantum theory.

25:00 The first is quantum measure theory, which is Raphael's. And the second is discrete collapse models, work building on the inspiration of the work by Girard and Lee. Before I tell you what they are, I want to stress that both are what I call history-based. That is, there's a sample space. They have this character. There's a sample space. Of possible histories of the universe, namely, in both cases, the set of all causal sets, omegas, one of which, one of these causes, one element of omega of which happens here, I acknowledge that I'm using a naughty word as far as Chris is concerned, one of which happens, it's realized, it occurs, it is the real substance of the universe. And it's the job of the dynamics to predict the properties of this actually occurring history, objectively, without recourse to notions of observers, observables, and... Moreover, root one, this is this quantum measure theory, if applied to quantum mechanics, has the character that, as you'll see, it leaves the Schrodinger equation intact. The dynamics of ordinary quantum mechanics. But root two, if you applied it to quantum mechanics and not to quantum physics, it does mess with the Schrodinger equation. It postulates fundamental violations of Schrodinger evolution. So the two roots differ and you may prefer one over the other according to this difference. Presumably it's not the job of the dynamics. To sort of over predict the properties of the history, what I mean is to do better than the uncertainty principle you get with the Schrodinger query, I mean, you know, it's okay. Okay, quantum measure theory. So the idea is that we seek, on our sample space omega of all possible causal sets, a so-called quantum measure.

27:30 And before I say what that is, let's remember what a standard classical measure is, or a probability measure. That is... First of all, a collection R of subsets of over, which are closed, a collection which is closed under counter-receptor operation, that is, counter the union's intersectional time. And this set of subsets of over, this collection of subsets of over, is known as the measurable set. So these are, that means that if you take two elements of R, and you take their union, Then that's also in R. And if you take a countable union of elements of R, that will also be in R, et cetera, and intersections and complements. Then, B, there's a real non-negative function, mu, from this collection of measurable sets to the reals, such that the measure mu of, which is called the measure of the disjoint union of two and the measure of the whole sample space is one. And you can see that in this case you're going to be able to interpret mu of a particular set A, say, a probability, it is indeed interpreted in that way, you can interpret mu of the subset A as the probability, the history that occurs, belongs to that. Can I just ask, when you say disjoint union, do you mean the full blown thing in set theory where effectively you sort of label them, or do you just mean union if it's disjoint? I mean that A and B have no intersection and go back to their union. Yeah, you don't need the thing called this time. Now, a quantum measure from Omega will have A, so it will have a collection of measurable sets, but not B.

30:00 So this thing doesn't hold. Instead, you have a real non-negative function, mu. Which doesn't satisfy this, so the measure of a disjoint union b is not equal to the sum of the measure of a union b. But, something involving three sets, which is this condition holds, and that's the measure of the disjoint union of three sets a, b, and c, disjoint set a, b, and c, is given by the measure of a union b plus the measure of b union c plus the measure of c union a minus the measure of a minus b. This condition here is the first in a hierarchy of generalizations of the classical question. Yeah, that's the first time one quantizes and quotes by turning a rule with two things into a rule with three things. I mean, what's quantum about that, if you know what I mean? And if you go to four things and five things, is it sort of second quantized? Is that good? Is it something we've never heard of yet? No, it's something we've never heard of yet. Something we've never heard of yet, okay. There are no theories, as far as we know, which satisfy the higher generalizations. So consider non-relativistic, so let's ground ourselves a bit, though it's rather abstract, let's consider non-relativistic single-particle quantum mechanics. There, the sample space is the set of all possible trajectories. We don't have a mathematically well-defined measure yet, or even ever, but roughly speaking, the quantum measure of a set of particle trajectories A is given by D of AA, where D is the so-called decoherence function. And I'll define it, define what D is. So A is some subset of the set of all possible trajectories, D is this decoherence function. Then, the measure of The difference between the measure of the disjoint union of A and B and the sum of the measures of A and B separately is given by twice the real part, the decoherence functional in A and B. Now, as far as quantum measure theory is concerned, we don't require that this off-diagonal term of the decoherence functional, that is D, A, B, vanish.

32:30 We don't require that for A and what's equal to B. And it, well, just to comment here, that, in this case, Calling it the decoherence functional is rather confusing. It rather ought to be called the interference functional or even coherence functional, but let's not confuse matters and we'll just stick to the familiar terminology because it's quite embedded now in the literature. So what is this decoherence functional for particle quantum mechanics? All of this is one of the paths in the set A, and you multiply it by e to the minus i s, where s is the action of y, one of the paths in the set B, take that thing, you sum it, sorry, you weight it by the wave function of the initial point of the path x, the complex conjugate of the wave function of the initial point of the path y, and then you sum that over all paths x, y, and z. And I put that in inverted commas because it's like... There's a continuous space of paths, so these sums are actually integrals. I've replaced psi, psi star by a more general thing which is the density matrix, but it doesn't matter, you can just think of it as psi, psi star. The slight detail here is that you have to wait. Again, you have to weight the path by adding this delta function between the final point of x and the final point of y. Slightly complicated technical reasons for that. So at some final point, tf, you have to force the two paths to come together, where tf is some truncation time, and it turns out that the value of this decoherence functional does not depend on the truncation time, you can take tf to be whatever you like, so long as it's to the future that any of the conditions relevant for defining these sets can be. So this suggests that causal sets are way forward for defining a quantum measure for causal sets.

35:00 We could try to guess what S of a causal set, and then use exactly this kind of expression for the decoherence functional, and then by this equation the decoherence functional would give us the quantum measure. It turns out, and I'm not going to say anything, give you any details of this, but for now it seems actually... It might be more fruitful to seek the quantum measure directly rather than going through this route of looking for the action of the causal set, but actually look for the quantum measure more directly. Let me say a bit about the interpretation of this. There's no deep coherence here. Mu doesn't behave like a probability distribution. How then are we to interpret it? And Raphael's suggestion is to use the concepts of propensity and preclusion. The conclusion is from Gorosh in an article that he wrote about Newton. So we say that mu A, this quantum measure of this set of histories A, is the propensity for the realized history to belong to that set A. And if mu A is small enough, then we say it's precluded that the realized history should belong to A. That is, we predict that the realized history will not belong to A. You make that prediction. Whatever the realized history is, it's not going to belong to A if mu A is very small. Now, there are various problems with that, but one I'll just highlight here. There's a conjecture, I believe it, although it hasn't been proved, even in ordinary quantum mechanics, that given any history H, there will exist some measurable set A. And so on and so forth, with H being in that set, but such that the measure of that set is zero. So, give me a history, any history, I'll produce a measurable set which has zero measure, and therefore I'll say, okay, the realized history can't be in that set, and therefore it can't be H. And that's true for every history.

37:30 And therefore... So this history would be the one to do the course. Fine-grained histories, yeah. Well, isn't the obvious itself the set H itself? I mean, a single place, doesn't it? So that, I mean, that, yeah. Even, right, even aside from that problem, the sort of the, what would you, what's that called? The, um, the lottery problem. So even aside from that problem, I mean, this is a, I mean, this expands, this is an empty statement, almost. It must be noted that the set contains an open set, or something like that. Something like that. Okay, yes. Right. So give it some width, then. So this, the proof of this, if you can come up with one, is going to be related to low local hidden variable theory, a common set of theories, probably, in a passage for all time. But the conclusion of it is that nothing ever happens. Because the realized history can't be any of these, any given history age. And that's a problem, it's a problem for me, not for Chris. And to solve it, Rathod proposed that there should be a restriction on allowed preclusion. So this problem arises because you say any set with measure zero gives rise to a preclusion. So if you restrict the possible sets which give you an allowed preclusion, you might evade this. And that's exactly what he did, he proposed a restriction. And the condition that restricts This is an abstraction of the notion of measurement. But unlike the usual notion of measurement in the Copenhagen interpretation, this condition is actually very simple. It's completely concrete and perfectly objective. I'm not going to say what that condition is. Roughly, it is that the situation must be that you have three space-time regions, A and B space-like to each other, and C in the common future. Conclusion is predicted when variables in A and B become correlated with each other and also with variables in C. And if one of those correlated possibilities has negligible measure. So this is a sort of, it's a mock-up of a measurement type situation.

40:00 Sorry, correlated means what? It means sort of in the little set that you're choosing, taken as a probability distribution in some sense, and then the correlations? Correlated means that the histories in which the variables are not correlated together form a set with measure zero. Okay, so let me pause here and compare and contrast this with the decoherent histories. I noticed also that that rule is sort of patently kin asymmetric in the sense that it talks about the future and, you know, there isn't a big bubble down below and blah blah blah. That's by design, is it? Yeah, I suppose so, yeah. Is it equivalent to its upside-down version or is that half-none? I don't think so, no. So, let's just think about decoherent histories then. In the decoherent histories approach, the sample space is not fixed. So we take omega, this is the set of what Jim Hartle would call the all of the fine-grained histories of the system, and consider all possible partitions of it. So take omega and divide it into a whole bunch of subsets, A1, A2, A3, etc. So that omega is the disjoint union of all those subsets. And of course there are many different partitions, there's another one, so consider them all, all possible partitions. If a given partition, say this first one here, satisfies the condition that the decoherence functional of a i a j is zero when i is not equal to j, we call that a decoherence set of coarse-grained histories. The a i, these subsets of, these sets of fine-grained histories are The coarse-grained histories say each set AI of fine-grained histories is itself called a history. It's a coarse-grained history, but it's a history. And we say that this partition A1, A2, A3, A4, that is a physically possible sample space. So let me stress, omega is not the sample space in this case. The sample space is this set of There are sets of fine-grained histories. Each set of fine-grained histories, AI, is a coarse-grained history.

42:30 Given such a decoherent partition, so an AIA, A1, A2, etc., it has a probability measure. Notice probability measure here, not quantum measures. Probability measure satisfies all the usual rules of standard measure theory. And the probability measure is given by probability of the coarse-grained history A.I. is the diagonal term of D.A.I. We interpret this P.A.I. as the probability that A.I. happens, not that a fine-grained history H is, not that the fine-grained history that occurs is one that's contained in A.I., That's the ontology, not the fine-grained histories within it, but the coarse-grained history itself is now to be thought of as one big, thick, unique, indivisible thing, the coarse-grained history. Because there's a probability measure, there's no need to introduce this notion of conclusion. We can talk directly about probabilities and predictions, probabilistic, stochastic predictions. There's no need to introduce them. So that's all I've got to say about decoherence in history. So there's similarities but also differences with this quantum measure theory. Now, will quantum measure theory work? I have very mixed feelings about it. In particular, I just bring up these two issues or problems. Is it true that we can turn this notion of preclusion into the usual probabilistic predictions of quantum theory? And you need approximate preclusion. In order to address real-life situations, we're going to need to make preclusive predictions when the quantum measure of a set is not just zero, but only approximately zero.

45:00 For you to be able to make those predictions. But we want the predictions to be objective and not depend on any subjective smallnesses of terms. So the second route to a possible quantum theory for causal sense. I was motivated to think about these things because I had a lot of trouble with this notion of preclusion and quantum measure. In this way of thinking about things, you get an ordinary common law garden probability measure on causal sets, or if we could extend it to causal sets, that's what we would get, not a quantum measure. So before I say, in fact, before I get on to discrete collapse models, let's think about collapse models and quantum mechanics to begin with. And this is a description of the very simplest GRW, Girardi-Riemann-Weber, collapse model. We have a wave function, psi, just a single particle wave function, psi of x and t, which evolves according to the Schrodinger equation almost all the time, except that spontaneously and at random, and on average once every length of time, tau, this wave function collapses. ...spontaneously, just by itself. And the way that it collapses is that the wave function, which was this at some time, becomes psi-primed, which you get from multiplying psi by a function j, j of x, and this parameter position x-bar, which is called a jump factor, and then dividing by n of x-bar, which is the normalization function.

47:30 J, the jump factor, is normalized with a 1, and GRW suggests a particular form for J, which is an exponential. So you're multiplying the wave function by a Gaussian, which is peaked at a particular x-bar with a particular width. And x here is configuration space x, isn't it? And finally, the collapse centre, x bar, is chosen at random with the probability distribution given by n, the normalisation factor. So the collapse centre is not fixed, when the collapse occurs, it's chosen at random, and it's more likely to occur where the wave function is peaked. And GRW suggests these values for those constants of nature so as not to violate any known experimental results. So there's some discussion about how to interpret this formula. The interpretation I want to focus on is due to John Bell and also stressed by Adrian Kent. We take the be-ables or the elements of reality or the things in the theory which are supposed to correspond with what's real. To be the collapse centers for events, that is the sequence ordered in time of the X bars that are the centers of the collapses as they occur. Of course those are random, you don't know where those are going to be or even when they occur. But once they have occurred, those are the vehicles, those are the real things in theory, the only real things. The wave function is not to be thought of as real, but the wave function is simply an auxiliary piece of the machinery that you use to generate these collapse centers. The GRW prescription then gives you, for each history, each sequence of collapse centers that you can think of as a possible history for the system,

50:00 Now everybody knows that it's hard to make realistic versions of quantities real, relativistic. This is not a relativistic model, it's non-relativistic. One attempt by Trevor Samuels deserved, one attempt to make a realistic version, relativistic version by Trevor Samuels deserved attention. We know that if you go to the... If you want to incorporate relativity you have to move from quantum mechanics to quantum field theory. You can't maintain single particle quantum mechanics in the face of special relativity. So we need to do quantum field theory. And Trevor's model is a stochastic model of a quantum field theory on a 1 plus 1 neural lattice. I won't give all details of it here, but we'll use the relevant bits to build our collapse model of the quantum field theory. Null lattice constant field theory. So here's a 1 plus 1 null lattice. This is 1 plus 1 in the cosine space. And our space time is given by these, with null links between them, between the vertices and the links. The field values, so there's a field... Just to note, that's exactly the lattice that is the sort of problem, you know, in terms of if you give it a boost, it would look well, you know, blah, blah, blah. So, the field values alpha are, we're going to take the simplest possible field, they take values either 0 or 1, and they live on the links, not on the vertices, and the fields live on the links. And we can call them optication numbers. So you can think of a field value 0 here as being, as saying that a fermion isn't there, or a field value 1 here as saying that there is a fermion there. Now a wave-functional psi is a wave-functional of the field configuration cell, so it's a wave-functional psi on some space like hypersurface, on a space like hypersurface sigma, is a wave-functional of the values of alpha on that hypersurface, and I should tell you what hypersurfaces look like. Hypersurfaces are things like .

52:30 So this is a constant time hypersurface. It cuts these links. And this is a general, a general, some general space like hypersurface, just cutting links, so that the hypersurface is cut through the links and not through the bones. So on any hypersurface, there will be field values sitting on the links that the hypersurface cuts through. And the wave functional is a functional object of those field combinations. Does that Wrigley one count as space-like as well? Are you allowed to be light-like? Yes, that's right, you're allowed to be light-like. Now, you can evolve psi, the wave functional, if you know it on one space-like hyper-surface, to any other one, by evolving it through a sequence of so-called elementary moves. And here's an example of an elementary move. You can evolve from a sigma, which is this green surface, To sigma prime, which is the green one, but then goes through the blue one, blue bit here, by making the elementary move, which flips this bit up to that bit across that rhombus, through the vertex v, and here I've made the two links in the original hypercircus, one and two, which are replaced by the two links, one prime and two primes, in the final hypercircus. And you can evolve The way that you evolve psi through, and you can see that you can change any hypersurface, say for example sigma t, to any other one, just by going through a sequence of these different, of these elementary moves sometimes. So all we need to do to evolve psi from any hypersurface to any other one is to know how to evolve it through a single elementary move. And we do that using the so-called R matrices, which are unitary matrices which live at the vertices.

55:00 So for the elementary move that I showed there, you evolve the wave function on sigma to the wave function on sigma prime by evolving it using these unitary matrices, the R matrices. So in general you can choose the R matrices to be different for the different vertices, but for a usual quantum field theory you want it to be homogeneous, so that the R matrices are the same for all the vertices, and here's a particular example. You can take the amplitude, the non-zero amplitudes, to correspond to the following configurations. When alpha 1 is here, these are the original two links and these are the final two links. And take an empty link to mean field value 0 and an arrow to mean field value 1. And the final two links have field value zero. Then there's the non-zero amplitude. And similarly, 1, 0, 0, 1, etc. These are the non-zero ones. And all the other ones are zero. And you can see that this somehow embodies conservation of fermions. If there's one fermion here, there's one fermion here. You don't mean it to be two the same, do you? It's just a mistake. You mean the bottom right one to be sort of south-east to north-east. Oh, yeah, right. Yeah, there are six mountains. And finally, we take the lattice to be, just for ease of normalisation, we take the lattice to be periodic in space of, say, width 2m. So field configuration on a space-like surface is from sequence of 2m 1 to 0. Okay, so let's make a lattice model, a collapse model of this quantum field theory. In quantum mechanics, what collapses are positions, and the be-ables, or the elements of reality, are the collapse centres which are the positions of the quantum field. In quantum field theory, it's the field values which are going to collapse, the be-ables will be collapse centres which are values of the field. So this is what we do. These are the rules for this collapse model of quantum field theory. We start with the wave function on some hypersurface sigma. We choose some elementary move at random and evolve sigma through the move to some new hypersurface sigma and evolve the wave function to sigma prime using the r-matrix of that vector.

57:30 So there'll be a collapse on the left one, the new left one, and with columnarity p there'll be a collapse on the right one. In the case that there's a collapse on the left link, the wave functional becomes the following. Psi goes to Psi' which is the original Psi multiplied by this jug factor and then divided by the normalisation factor just as we had in the GRW model. I'll just talk about these details because I'm running out of time, but of course you can't have a Gaussian in this case because you don't have continuous values of the variables, the only possibilities are values 0 and 1, so it's much simpler than having a Gaussian factor. So again, n was that normalisation factor, and the value, the field value that you collapse to is randomly chosen according to the probability distribution of n. The variables are these values of the field that you collapse to. If you collapse, then you say that there is a realised field value, whatever it is, on that link. Now, this algorithm will then generate a probability distribution on the set of histories, which are assignments on this lattice of one of zero, field value zero, field value one, or no field value at all, because it may not collapse. It's random whether or not the weight function collapses on that little piece.

1:00:00 Signs of either zero, one, or no value to every link in the lattice, to the future of the initial hypersurface on which you're giving your wave function. This probability distribution is independent of the sequence of hypersurfaces which a given run of the algorithm happens to generate. So if you start with some initial hypersurface, you choose at random an elementary move, you do that. You choose at random another elementary move, you do that. You gradually creep your way across the whole of the lattice, and that generates a particular set of hypersurfaces on any given run, but the probability distribution that you build for these histories is independent of what the sequence of hypersurfaces happens to be. The sequence of hypersurfaces is not a foliation, because there are too many, so after an elementary move, you haven't foliated the surface, you've just changed the hypersurface locally, so it's not a foliation, but it doesn't include every hypersurface, so there are some hypersurfaces which your creeping hypersurface doesn't coincide with, and that's it, that's the model, that's the collapse model for quantum field theory, and of course the first thing that you must do, In having built a realistic model of a quantum theory is to show that you preserve relativistic causality. In other words, you have to show that you can't signal that at the moment. We know that it must be a non-local theory because Bell's theorem tells us that any time you try to make, any time you make a realistic model of something which you claim agrees with the predictions of quantum mechanics, Then it must be non-local. So, in fact, you can see that this collapse model is non-local, but we want to show that even though it's non-local, you still can't signal faster than light, and Joe, my student, is showing that. So that's good, that's the first hurdle that you have to open. There are many questions, of course. What is this in the continuum limit? Can you take the continuum limit? Is it one of the known continuous spontaneous localisations, or is it something different?

1:02:30 And finally, can you generalize this to a causal set? Will you be able to generalize it first to having a quantum field on a fixed background causal set? Could you generalize it to pertain to the dynamics of the causal set itself? And my feeling, well, I have various thoughts about this, but my feeling is that these two things might in fact be the same thing. So I've been completely vague about how to generalise to causal sets, which was my motivation. I'm sorry, I apologise, but all work in its infancy, in its first stages. I'll just summarise my talk. The causal set theory is a proposal for a truly bottom-up theory of quantum mechanics, and it's what most people can't just make do with. With trying to follow our noses with continuum of space quantum rules, it uses the fact that the causal structure is, in a very precise sense, nine-tenths of the metric in ordinary continuum of space. So exactly nine-tenths, because if you know the causal structure, you know the metric up to the conformal factor, that is, you know nine of the ten independent functions in the metric. And discreteness, the discreteness of the causal set via this relation that the number of points in the space-time region of the volume package gives you the other tenth, so it defeats the missing side of the information. Quantum measure theory, Hawking's quantum measure theory proposes a measure on the sample space of prime-grade histories, in this case causal sets, which measure includes interference. So it's not a decoherent. It's not a probability lecture. It doesn't satisfy the standard classical rules of probability. And in order to interpret it, we need to make sense of its notion of conclusion. Applied to quantum mechanics, it corresponds to keeping the Schrodinger equation intact. I described a discrete collapse model for a quantum field theory on a null lattice.

1:05:00 And that provides a hint as to how a probability measure could be constructed for these terms. There are no philosophical problems here, or no special ones anyway. No problems with having to interpret this thing called a quantum measure. But applied to quantum mechanics, it's the GRW. This is a model which means that the Schrodinger equation is fundamentally violated, the Grinning equation is a problem which is violated every now and again around it, so something like this is something to be avoided. So finally there's lots to be done. Thank you. In the context of the Schrodinger equation, does this have any... Any close resemblances to Nelson's formulation of quantum mechanics, which I'm surpassed as one can explain, if you haven't derived the true equation so far? I have a question. When you take sigma, the hyper-surface, is there any more reason to take a bounded domain, or only sigma, and look at two different domains? And that, I think, if you want to take a look at it, you know, there's a way to define a random process which takes from one space that happens to happen to the other space that happens to happen to the other space that happens to the other space that happens to the other space that happens to the other space that happens to the other space that happens to the other space that happens to the other space that happens to the other space that happens to the other space that happens to the other space that happens to the other space that happens to the other space that happens to the other space that happens to the other space that happens to the other space that happens to the other space that happens to the other space that happens to the other space that happens to the other space that happens to the other space On the edges? Yes, the point is, if I can construct from the process provided on the exam that you have,

1:07:30 what you want to do is basically, you can construct through the space, and if you can't construct through the space, that doesn't work. So instead, I mean, in a sense, that's what I did, I mean, it was, I mean, it was a, it was periodically identifiable, so, yeah. So one of the things that I guess you want to say is that at some large distances we could do such a thing, so how do you, is there an notion of distance and why should you expect that the dynamics of this would have anything to do with anything? Is there any evidence that this has anything to do with physics of large distances that we need? A continuum of space-time, then you have distance. Is that what you mean? You're saying, for example, that this has one point and volume. So, the physics that we're talking about is at the time scale. A very large distance compared to the time distance which we could use in some sense. In which there is some experiment there. Right. Yes. So, how do you define distance in the first place and why is this important? So, we don't define distance fundamentally, but we define volume. So, what you're saying is that at large volumes, so that means at large numbers, so in terms of the causal set, that means when the causal set has very, very many elements. All right, so when n, when the number of elements of the causal set when n is very large, and you're right, when n is very large, we want to be able to predict that the causal set looks like a continuum spacetime, and the continuum spacetime is a classical one, so it satisfies the Einstein equation. So, right? Yeah. So, okay, so we can't do that because we don't, we need the dynamics to do it, we don't have the dynamics. No, it just has to do with the people's general feeling that the discreteness scale should be the Planck length, so that it's at that length, at that scale, that the continuum description of space-time will break down, and that it's at that scale that it will have to be replaced by something else.

1:10:00 Yes, this has nothing to do with it. One more. So you cited the theorem of the cycle of time. So, when you're saying that two spaces are related by the dichromorphism, what does dichromorphism mean, since you don't have any form of it? Oh, so that, right, so the... The theorem of all things. Right, the theorem... ...is to do with continuous spacetimes, not causal sets, right? So, the theorem is that if you have two Lorentzian manifolds, so they each have... So, M, G, and M prime, G primed. So, M is a manifold, M primed is a manifold, with topology differentials. Each one has its topology, each one has its differential structure. Each spacetime has a metric, G and G prime. This function, f, between the two, which is a bijection, so it's one-to-one and onto. But you don't know anything else about that function. You don't need it to be continuous, you don't need anything. It's just a one-to-one and onto map. If that function preserves the causal order, then that function must be a diffeomorphism between M and M'. So M and M' in particular, they're diffeomorphic manifolds. Right, and also it relates the two metrics g and g-prime. So it's not a theorem which applies to a causal set? No, it's a theorem in the continuum. But it's one of the reasons that it's possible even to have any hope that a causal set could look like, could give rise to.

1:12:30 To a continuum of space-time. Because if all the information about, almost all the information about a continuum of space-time is encoded in its causal order, you can hope that by having some discretization of that causal order, that could give rise to all the space-time information that you know. I don't account for it. It might be nice to have nothing but natural numbers in the whole thing. That's right but I don't see any reason why we can't try to try to form I mean what's real is this discrete thing I mean the the the decoherence function itself is not real that's just something in the calculation that we're using to um that's not part that's just part of the the the cogs and the wheels behind the scenes the real thing is the causal set and that's that's discrete You're snorting, for real! We can apply things like, for instance, matrix theories, for instance, like the other types of matrix theories, going back to the early 90s, for instance. There's actually a very, there's an interesting connection between, possible connection between matrix models, random matrices and causal sets, because you can represent a causal set by, What's called an incidence matrix. So the rows and columns of the matrix are labelled by the points of the causal set and you put a 1 in an entry if, say, the row is, the row precedes the point, the point which labels the row precedes the point which...

1:15:00 Which labels the column, for example. So that's called the incidence matrix. Irreducibly or transitively? Just the irreducible ones? No, all the relations. Oh, right. Yeah, all the relations. So a theory which randomly generates... Matrices like that, so a random matrix theory is a random cause-and-effect theory via that correspondence. I don't know whether that has anything to do with matrix models, but at least you can. I mean, there is some connection between matrices and cause-and-effects. So that might be a dynamic review, which, for instance, means that you should wrap up and scatter them out, for instance, rather than in particular matrix models. That's possible. What does quantum relativity basically mean? It's sort of sidestepped the problem of time as it's referred to in the canonical theory There's no 3 plus 1 split, there's no timeless formalism, there's no rule of DeWitt equation, so, because it's history based, you know, in the same way that decoherent history sort of sidesteps the so-called problem of time in the canonical theory, but time, as we experience it, emerges, hopefully, if we can work out what the dynamics is and we can predict the classical space times. The causal sets approximating classical spacetimes are the ones which are most likely. Time just emerges in that sense. It's not there. The only thing that's there fundamentally is the causal order. And time will only be meaningful in the context of the emergent spacetime,

1:17:30 On that note, can you imagine if you write down a causal set sort of at random, I'm using the phrase very loosely, you know, haphazardly, it won't necessarily look particularly 3 plus 1-ish at all, you can imagine bits of it would just look like sort of temporarily 3 plus 0 or temporarily 4 plus 0 in the imaginary Euclidean time sort of thing. As N gets very is-a-nose. The red thing on the wall looks like a bloodstain. Sorry, you just sort of struck me as rather gruesome. Is that what you do to your speakers that you don't like? You talk about realistic theories. Sorry, so you can say more than that. You can say that, in fact, we know that the causal sets which look like continuum space-times These are very very rare in the set of all causal sets. So there's a theorem by Kleitman and Rothschild or Bosman, Pytar, I can't remember. Two mathematicians. And they say if you consider all causal sets with n elements, and let's let n get very, very large, and as n gets large, we know that the proportion of causal sets which have a three-layer structure goes to one. So, causal sets with a three-layer structure of one. The proportion of that would go to zero faster than it goes to one. So, almost all of them. Is that theorem possible to even sketch an intuitive explanation of all of it? No, I don't understand it. So this looks nothing like the continuum of space-time. And those are the vast majority of causal sets as the number of causal sets. So you know that they're very rare and you're going to need some very special dynamics to figure out the space-time. That's a good question. So, it's possible that you don't need to include it fundamentally at all, that you can get everything, so that would be the ultimate unification, right, so then you would unify space-time, matter, metric, differentiable structure, topology, everything in terms of one thing which is the order, so the way that you could, alright, this is, you know...

1:20:00 I'm not suggesting this is true, but it's a possibility. You could get all sorts of, every kind of matter that we see now in causal sets by getting, once you have the continuum, once you're able to predict continuum space times, you could get, you could get gauge fields via Kibbutz-Klein reductions. So you can get gauge fields from pure gravity, and you could also get fermions from pure gravity, so you could get matter from pure gravity via quantizing topological geons. So topological geons are particles made from non-trivial spatial topology, and you can quantize those as fermions. In fact, you can give them all the quantum numbers of the standard model, in fact. I'm not saying that you would be able to derive the standard model with all its 23 parameters and predict that the mass of the electron is half an MeV, but there's a potential there. Or, you may be right, you may have to put extra structure onto the causal set. Fundamentally you may have to put fields on a causal set, I don't know, but it's possible that you may not have to do that. We did that just for flat space yeah so this this is for math that's much more than much much more difficult yeah it's much more significant Well, they don't specify topology, so that's the thing. I mean, you haven't specified what the topology is.

1:22:30 This might be pretty relevant. How would one think about the notion of quantum dominance in a causal set? I think there is no way to think about it at the causal set level. It's only really when you get to the continuum. Because a causal set, it's so fundamentally itself that there's no notion, there's no way in which I could think of wick rotating some time parameter and producing a kind of Euclidean causal set. There are very, very few operations you can do on a relation. You could imagine reversing a relation. That's one thing you could do. You can imagine completing a relation. That means that you add in extra relations to the ones that you already have until all elements are related to all other elements. So you could imagine doing that. That's called a linear completion, I think. And you make the causal set into a totally ordered set. But apart from that, you could take relations out at random. That's another thing that you could do. But apart from that, it's very difficult to think of operations that you can do to a causal set as a whole. In particular, it's very difficult to imagine what you might mean by a Euclidean causal set or an instanton kind of causal set. So it would have to be once you We've gotten a continuum approximation to causal set theory. At that stage, you can maybe think about instantons and tonnels. Is there a version of this which is similar to, for example, what Thorpe did, some theory of the discrete energy, which also has dots and operations where you get a 1? I suppose I would reverse it and I would say, is there any reason why the theory should be deterministic? He's driven by the desire to have a deterministic theory, and my reaction to that is, well, why?

1:25:00 Why? Why do we want a deterministic thing? It seems to me that the lessons of quantum, of everything that we've learned about the quantum world is that it's not deterministic, it's probabilistic, it's random, we can't predict for certain. So that, so yeah, so I would turn the question around to Ertug to say that he, he should justify why he wants the deterministic thing. What we know about the quantum world is again, scales which are extremely important. It's entirely possible that things look so different, completely different from anything I've suggested, anything anyone has suggested at Planck scale. What more can I say? Rather than going towards determinism, we might go to one of those higher, higher academic theories, both knowledge-based. Yeah, that's also possible. Should we say that again? No, but this question was about... The higher generalizations of the measure. Oh yeah, that thing, yes, yes, because after all, if they're totally new, they might be totally new and very flowful. Yeah, that's true. So quantization interests are only a pretty dynamical view, in the sense that there's no interference of what happens in OCR. Right, no. Right, that's right. It's just in the dynamics, yeah. There's only, there's one and only one realized causal set. Would you like to comment on the Hegel? The Hegel? It's not the Hegel. I was thinking, it's not that it's not the Hegel, it's possibly, it's possibly Kip Fergie.

1:27:30 Sorry, why we called it non-Hegelian. So, correct me if I'm wrong, but, yeah, so Hegel says things only have, their properties are only relations to other things. They don't exist in isolation in and of themselves, but they're. So if something has exactly the same relations to everything else that exists in the universe, if A and B have exactly the same relations to everything else in the whole universe, then A must be the same thing as B. So these three points would be non-Hegelian because they have exactly the same relations to all the other points. In the, all the, exactly the same relations to all the, I mean precisely in mathematical terms, the same relation, you know, this relation to all the other points in the course is that, but they're different points. I mean I'd label this one A, this one B, this one C, so they're different. So they're non-Hegelian, they're, they violate whatever, Hegel's philosophy. I don't know, this is what they tell me. But then you get this, you can, we've generated all sorts of hideous terminology. This is non-hegelian, but you can hegelianize it by introducing an equivalence relation, so everything is equivalent which has the same relationship, you know, so that's hegelianization. It's no worse than Euclidean ethics. I would say because Kant asked once if there was only one kind, a single kind, in the world, wouldn't it be meaningful for you to ask whether it is the kind that you like? So I think it's the kind that comes from the nature of how you choose. That's actually a very good non-facetious question, you know, you just have to have some tea.

1:30:00 I'm just a student, very suspicious of you. I'm just a student, but let me just make one comment. One of the things that... I'll come and see you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. More like the second one. Yeah. The first one. The second one. The second one. I see. The second one. The second one. The second one. The second one. The second one. The second one. The second one. The second one. The second one. The second one. The second one. The second one. The second one. The second one. The second one. The second one. The second one. The second one. The second one. The second one. The second one. The second one. The second one. The second one. The second one. The second one. The second one. The second one. The second one. The second one. The second one. So, let's say in the case we have two leaders, and she shows an example, so she gives an example of quantum mechanics with one quantum function, and she only used one. She only used one of these guys. She used two, but one quantum function.