General Covariance & the Problem of Time in a Discrete Cosmology

0:00 And having a kind of, in some way, convincing derivation from fundamental principles that seem to be very natural, which are versions of general covariance and of causality. That is, causality here means the idea that not only do we call the order relation causal, but it really is causal in the physical sense, namely that no influences do propagate, except along the path, so to speak. in them, but the intrinsic structure of the causal steps This family of dynamics represents their, what you might call, what's the opposite of static? Well, in the opposition between being and becoming, they're more like becoming, they're processed to use the word that's been used before in this meeting. At least that's how one can think of them. You could always think of them the other way if you really want to, but they arise most naturally to think of them as a process. So you think of, instead of saying, what is the causal set like, you think, how does it grow? So you think of the dynamics as a process of continual birth of elements of the causal set. In fact, my answer to some of this morning's discussion is that the present is nothing but the birth of a new element of the causal set. the process that's going on. What we experience as the passage of time is ultimately dispersed of the elements of the causal set. Once the element is born, then there's a sort of record or a sort of, I'm looking for some word I can't think of at the moment, but like a fossil record of what it's done, and that's the causal set. That's, someone said that once you say Now, it's already past, in other words, anything in the structure is already part of the past. Anyway, that's just an intuitive feeling that can maybe help you appreciate what the dynamics is like. Mathematically, it's a stochastic process. It is, however, a classical stochastic process. So why are we considering a classical process and not going immediately to the quantum one? Well, of course, the classic one is easier. and because of that we hope that it will play some roles for us in ultimately constructing the quantum theory

2:30 it will help us to clarify the conceptual issues involved in building quantum theory it may suggest new physical effects that could ultimately survive like this cosmic renormalization that I referred to very briefly that may help us to explain why the universe is so flat and big from a quantum gravity point of view and then of course that go into it would be the same ideas that go into the quantum theory and the causality and the general covariance. It may actually play a technical role in leading us to the quantum dynamics. So let me show you before going on. Let me just show you by pictures is what I mean by a process, a growth process. And then I'll write a formula if you want that will precisely define what kind of stochastic process it is. So we may start, since we're doing cosmology, we may start with nothing. And if that's the case, then if an element is born, there's only one possibility, which you expect. And this looks still like nothing. There's only one possibility there. of one element. Now it's going again. Now it's going again. Now it's going again. Maybe I should put this up because otherwise these slides will fall off the screen. However, as soon as the second element is born, there are two possibilities. Either the second element is a child of the first, or it's unrelated to the first. With the third element, they're already tied which you can see here the interesting one is this this could have come about depending on the order in which these were born it could have followed this path or this path but leading to the same result which we wouldn't want to distinguish physically and then so on so here the next level is already too big But I've shown all the possible descendants, so to speak, of this middle one, which is the more interesting one, whether it seeks or whatever possible descendants, depending

5:00 on how the new element is born. So in the one extreme over here, it's born unrelated to any of the previous elements. In the other extreme, it's born to the future of all the previous elements. So basically, this is how the Xandamix works. a continual process like this, each element at birth chooses, so to speak, what would be its ancestor, and that pattern of choices, or this record of those choices, is the causal sense. Is it not correct that, if you are looking at this sort of empirically, that there would be a limit to how back in the past you would assume exactly which causal sense? Yes, maybe. Maybe. There's a past, right? Presumably, well, I don't know if there's an actual fundamental limit, but I mean, obviously harder and harder to say definite things. Well, I think if something else was, you know, light cone separation, you wouldn't be able to have enough information. I think it was a good problem. Well, I'm not sure. I mean, this is a model. Within the model, everything is well-defined. If you ask how would you, in practice, discern the structure, I don't see that there's any fundamental limitation, but I mean, we haven't thought, in any way, thought through experimentally how you would do this, it's almost, so far, we don't have to even space-time as such. Isn't this a picture of an objective universal, isn't it? What we, one of us, know, or have. No, it's meant in the formal way. It's not formal. It's meant in the first way, as an objective universe. It's not anything to do with knowledge, as a sense. So, because it starts at one point, it starts in time at one point. So that point can't be just how well we happen to do work. It's not us now, no, this is like the Big Bang, or the pre-Big Bang, or the very first... Oh yeah, I thought I said that this was a cosmological picture. Yes, yes. So let me just give you three... So when this process runs to completion, so to speak, if we imagine ideally that it runs to completion, then you will get an infinite causal set. And I've just given here three examples of what you might get. The one you see here is a chain. Another one is an anti-chain. You might think of it as kind of dust universe. Neither is very interesting. And here nothing is related to anything. Here everything is related to everything. This one has the virtue of being a kind of one-dimensional space-time.

7:30 Another one which comes from a particular choice of parameters is this tree universe in which there's a first element it has an infinite number a countable infinite number of children each of those has a countable infinite number of children and so on I'll write down the parameters in a second these each come from one particular choice of parameters to my knowledge these are the only three choices that produce a deterministic universe you know where the causal set is guaranteed to be something the others produce a random distribution sets and the measure function or the probability function that tells you the probability of the different possibilities is basically the dynamics. Let me just... And you keep track of these, are you keeping track of left-right order as you do with the line of the plane? No, there's no left-right order. There was an issue of labeling, but that was just to take a representation. So they're just tree structures? They're just tree structures. only the relation itself has physical meaning. If it grows in a particular sequence, then there's a kind of labeling associated with which was born first. That's precisely what we don't want to have physical meaning either. And that's what, that will be the interpretation of General Boeberians in this scheme. Although... So the finite ones are in one-on-one correspondence with parenthesis, parenthesizations where you to keep track of order? I'm not quite sure because you have some things, you just represent them, yeah, maybe, I'm not, yeah, if you allow, so do you allow something like this? I guess. Yeah. I think it should be something like, yeah. But I mean, if you can represent each one, you can always represent a causal set as a sort of sub-algebra of a Boolean algebra or something like that. And it's only if I end up with it. Yeah, I think that's it. I wanted to give you this formula, so let me just give you it very quickly. But suppose... Is there a working one? Suppose you had at some stage And then there would be Just let's look at two possibilities

10:00 For the next element I meant to dot that also So two possible ways the next element could be more As a formula of the model that give you the relative probability, or the absolute probability of those two possibilities. With each one, there's a certain number which I'll call pi, which is the size of the past, and a number I'll call m, which is the number of parents, or the number of maximal elements of the past. In this case, let's just look at this for definiteness, pi would be, well, whether it's past this, this, and this, this pi would be 3, the number of ancestors, the number of parents, the number of immediate ancestors would be in this case 2. For any possible way of adding a new element, there will be some well-defined pi and m. The probability of that thing is that lambda, that particular This is just a normalization factor, so it's the same time as I mentioned it. And in this case it's 3, which is the number of pre-existing elements. The important thing is the formula for lambda. It's the sum over all k of n to k. Well, I guess I did it again. and n to this k, the determinant of n is zero. So I wasn't going to give you the same time. So I'm going to write it. So lambda phi comma n is the sum over all k divided by the n to the k minus n, t times k. Okay, these t's are the parameters of coupling mechanisms at the time of a particular model. If I had chosen, for example, t0 equals 1, t1 equals 0, and all the rest, I get the dust universe. If I had chosen t0 equals epsilon, t1 equals 1, and all the rest 0, and send epsilon to 0, I'll get the chain universe, the one-dimensional. If I choose T0 equals T1 equals 1, and all the rest 0, I'll get that tree universe.

12:30 Don't you get anything in between, like with two source elements? You can get it. So this one actually would have an infinite number of first elements because it would give actually a forest of these trees. It would give a count of a number of copies of that tree. A quite interesting one is called percolation, which is t sub n, it's t to the n for some fixed t, which can be thought of as p over q, where p is a probability and q is one minus the probability. That one gives something that's very easy to describe that will produce a random result with a certain distribution. let me just say it quickly and later on if you want I can try and describe it more that particular one is the one involved in the cosmological discussion that I mentioned about flatness that's the one in which each element at birth goes through all the previous elements and with probability p selects them as ancestors each one independent of probability p selects an ancestor and then you have to take the transitive closure I'm sorry, could you say again what the omega or pi you called it? That's pi, that's the astrophysicist's pi. So what's the definition of pi? It's the number of elements, the total number of ancestors of the new elements. Total number of ancestors? Yeah, total number of ancestors. So you just walk back to the ancestors? Yeah, so like in this case, this, this and this, anything that you can reach by going down. And the n is those you can reach by going down along links, along links, one step. And n is just the number of the mediators. Yes, that's n. And n is the total number that were there between this one and this one. These laws are actually the generic solution of the constraints coming from general covariance and valipides. There are some exceptional families, but this is the generic one. The others are of negative zeros. Okay, so that's meant to give a sense of what the dynamics is. Within that, I can now raise the question that I wanted to raise, which is the role of general covariance, which really means nothing more than label independence.

15:00 So in the continuum, by covariance we mean, for example, we mean generally that no choice of coordinates has any physical meaning, but in particular, no choice of the t-coordinate has any physical meaning, that is no particular slicing of the space-time has physical meaning. in particular, slicing gives you produces an order on the element, tells you which ones were born before, although it doesn't totally answer it in the case of a causal set, a label would actually be a complete answer which one was born first and since we don't intend that to have any meaning the only temporal relations that are supposed to have real meaning are those put into the structure of the causal set itself, any labeling that's compatible with that structure are thought to be regarded as physically equivalent. Since we implicitly introduce such a labeling by our definition of the sequential growth dynamics, we have to demand that it drop out of the physics. And that's what we mean by general covariance. Now, there are two aspects to that. I claim they're always there. But maybe we don't think of the second one so much. The first one is that in our dynamics it affects the probabilities It means that the transition probabilities by which this is built up which I wrote down to you very quickly here must satisfy certain identities that mean that when we relabel the elements then that probability doesn't change That's a condition that was crucial in deriving the dynamics and it's very much an analog of something like in a gauge theory, let's say thought of it in a continuum way, you'd have an action, and you'd demand that the action be diffeomorphism invariant. That would be a general covariance in that sense. There's also another sense which has to do with the interpretation of this theory. That is, it affects not the probabilities, but it affects the questions we can put to the causal set for which we can hope to get probabilistic answers. So, for example, in the continuum, it's not kosher to ask what was the volume of the universe when t equals 32 because who's to say which hypersurface corresponds to t equals 32

17:30 it depends how you chose your coordinate system it's for this, just parenthetically I think it's because of this problem that canonical quantum gravity is in some way doomed because it cannot But the starting point doesn't allow it to express in a useful way the kind of questions that would be generally covariant in this manner. This is what is often called the problem of time. In that sense, the problem of time, I claim, is absent in causal set theory or any theory that works with something that's more like a space-time than just like a space-like hypersurface. However, there's certainly an analogous thing that you might worry about is that what questions are meaningful to ask about the following sets? Just like there, in canonical gravity, those meaningful ones are those that commute through the physical constraints, for those who know what I'm talking about. You can formally define it. That's no problem. The problem is to give the physical meaning of these formally defined objects. Well, I should say, you could formally define it if you knew in the Hamiltonian zone. People haven't got that far yet. But if they get that far, they'll be able to formally define the meaningful questions. to give any physical meaning. So in the same way, we can easily define the physical questions, and I'll tell you what the definition is in a second in terms of a sigma algebra and so on. But the question, what I'm calling here the quote-unquote problem of time is can we give the physical meaning of those questions? Can we express them in such a way that we understand what they're talking about? And it seems that we can, and that's a very recent result, so I'll just finish with that, I hope, by quoting the theorem. I'll just give an example of a question that's not meaningful, it's not coherent, is what was the causal set just after the birth of the nth element? That's not meaningful because by different labelings you could make different elements to the nth element, and that has no physical meaning. So although the stochastic process would provide a probabilistic answer, it might say, well, it was 35% to be a chain or something, that answer would not have physical meaning. The question is, what does have physical meaning? Let me give you the answer in the formal sense that I just said, and then try and get a hold

20:00 of that formal answer and what it really means, in terms of the idea of partial stem, which I'll mention in a second. So the answer formula is very easy to state, although there's a lot of sort of technical definitions going in here, which if you're not familiar already with stochastic processes, maybe you don't have to worry about the details here. To define a stochastic process, we really mathematically define a measure on some space of histories that allows us to answer questions like, what's the probability that the history has this or that property? So for example, I could The question Arletta asked distinguishes what we might call an originary causal set from a non-originary one. An originary one would be one in which everything is a descendant of a single element, which the ones I had drawn had that property. A non-originary one would be one like this, for example, in which there are, in this case, three minimal homometers. So a question you could ask is, what's the probability that the actual cos that I grow by with this process for a particular choice of the parameters t, what's the probability that I get an originary one? That would be a typical question. In that case, we would understand immediately what it meant as well. But the point is that the answer to that is just a probability. and a probability is mathematically a measure on a space and any stochastic process is technically really expressed as a measure in that way so if I give you the probabilistic answers to all such questions I've told you the dynamics formally to define the measure we introduced the space omega tilde which is the space of all labeled causal sets because the process until you impose general covariance sets. There's another space which corresponds to, which is the physical one, the space of all unlabeled completed causal sets, ones with the infinite number of elements that stopped being born, they're all finished. There's a measure, this measure is the precise realization of all these transition probabilities, and the theory of stochastic processes gives you that measure once you specify the transition probabilities in a consistent way. What it is formally is a map of some sigma algebra r tilde into the positive reals this r tilde represents the questions that you could ask

22:30 about each element of r tilde is a question that you could ask about the causal set if you were allowed to refer to the labels and you know the yes it's just a sub what it is a subset of the history it's the subset to which the answer is yes the usual correspondence between the properties Is it important that you should have the reals now, the rational ones? No, I don't think it's important that it should be the reals. If you chose these to be rational, everything would be rational. The physical questions, and this is at the formal level, is a sub-algebra of the algebra R tilde. Not any set of causal sets will do, which which was an R tilde, but only those which can't contain any labeling of the causal set without containing all other labelings as well. That's a way of making the question labeling dependent. So the important thing is there's a formal process of passing from this labeled algebra to the unlabeled subalgebra, or what you might call the covariance subalgebra, and all the physically meaningful questions are in here. So I could just stop and say I've told you how to do the meaningful questions to the causal sets, give me an example of what does this mean so the point of the rest is to say what they mean so we conjecture and I think actually now it's a theorem that this can be translated in terms of a set of elementary questions such that all the things of I and R are logical combinations of these elementary questions so this is really all that we can ever know or care and be told by the dynamics about a causal set And the question is, pick your favorite finite causal set, call it S. Does it occur in the actual causal set, the full-completed one, as what I'll call a partial stem? What's a partial stem? It's a subset that contains all of its ancestors. So in other words, there's nothing underneath it in the picture. This is clearly a covariant question. So if I wanted to ask, for example, is it originary or not Well, maybe that's not so easy I can't convert that to a single question, I think But for example, I could ask, does it contain this as a partial step?

25:00 That means, does it have something like this? If I enumerate those which have more than one minimal element and the answer is no, no, no, no to all of them originary plants. So, this is something that has immediate physical meaning, and if we can prove that everything in here is a logical combination of those, then in a certain sense, we've pinned down the physical meaning of the generally covarian questions. So, in some sense, no general covariance question, so I'll ask them to be. Yeah, well, that's it. That's sort of built into the, that would be true even in the larger, they're not past, well, they're not past oriented, but they don't refer to the infinite future as the whole in a certain way that I could make more precise. So, let me point out that if this causal set had a certain property, then the conjecture would trivially be true. That is, if knowing the set of stems of a causal set told you what causal set you had, then everything you could ever ask about a causal set would be re-expressed as questions about the stems that it has. If that were to be true, then the conjecture would follow trivially. Technically, in saying that it follows, I have to use the fact that there are only a countable number of partial stems because they're by definition finite causal sets however, this is not true here's an example of a causal set that I can't distinguish I can't characterize just by its selection of stems so what is this? this is a union a disjoint union of an infinite number of these countable chains and I chose just, for example, a two chain Now, suppose that this two-chain were not there. Suppose that it were a three-chain, or just a one-chain. I claim you can't distinguish any of those different possibilities from each other just by reference to the partial stems. Because what are the partial stems of this? They're just a finite subset that contains all its ancestors.

27:30 So it's got to be a union of a finite number of chains of any length. So that's the partial stem. partial stent is just a union of a higher number of chains of N and N. But I didn't see in that that there was two, right? I mean, if I take this away, I still have the same set of partial stent. So I can't distinguish. So if these grew with some, I mean, this particular one obviously will have a probability zero of growing any exact positive sign has a zero probability, just like you pick a real number, the probability of getting pi exactly at zero. The real question is whether the whole class of ones that are not distinguished, not characterized by their partial stems, has a non-zero probability, and it seems that we have a proof that it does. So I'll just finish with this slide which sketches how the proof goes. instead of three lemmas the first, remember the question is whether knowing the partial stems of the clausal set, the completed clausal set uniquely determines what clausal set I have well if that's not true we'd like it to be true, if it's not true you can prove that it must contain an infinite number of copies of some fixed partial stems in this case In this case, you can see it easily. There are many examples, but say this particular maverick here, because it's obviously an infinite number of copies of that within the whole causal set. that's the first lemma the second is if that happens, if some partial stem occurs infinitely often in C then some level of C must be infinite what's a level? well a level, I won't define it formally but all the minimal elements form the 0th level all the elements that are just above them the ones that have only one so this is the 0th generation the first generation, so to speak those are the levels some level must be infinite. So this is a particular kind of infinite anti-chain that you think about it. And then the third statement is, this is the one that's only a couple of days old, so it's not guaranteed, is that any causal set that grows by this process will not have an infinite level.

30:00 Or more precisely, the probability that it will have an infinite level is zero. almost surely has no angle in bubbles, with some exceptions, which is when this parameter T2, if the parameter T2 or any higher one is greater than 0, then this holds. If they all vanish, like the one I mentioned originally, if only T0 and T1 are 0, you get exceptions, which are the ones I showed you before. But they turn out not to violate this condition of being characterized by the stems. So that would be, that would be the proof. I should mention this proof, this work, the work I'm talking about is done foremost by Joe Henson as a student of Fay Dougher at Queen Mary College and together also with Fay, with me, with Graham Brightwell who's a mathematician at LSE and with Raquel Garcia. This has very much the smell about it of Koenig's Lemma or of Brouwer's Spreithium, they all have somewhat, I mean I'm not saying they're the same, but it's the same kind, it's the same ballpark. Yeah, maybe. That kind of, yeah, I mean that was one thing that we, those things came up, at least I don't actually Brouwer's bread term, but Koenig's lemma came up in our discussion. The one that this is most close to, actually, is the, someone told me it was called the Schroeder-Bernstein theorem or something, which is used in defining cardinality. It turns out that the, it turns out that the Schroeder, the analog of the Schroeder-Bernstein theorem for post sets is actually false in full generality, but enough of it survives to give us this. so that's really the end this now gives us the physical meaning to our formal sigma algebra R and allows us to state questions in a way that can make contact with reality hopefully there was a related issue but since I'm a little over time I'll just leave it there for you to stare at Can I just ask a question of your intended interpretation?

32:30 Are you saying that this is how the world is? So the world has a certain discreetness. Absolutely. that discreteness, then you're going to tie to a quantum discreteness, aren't you? Well, in a certain sense, in a sense that the discreteness scale will be something in order to get to the black hole M3 right, or presumably, although we haven't got this far, presumably in order to get the magnitude of the distance constant right, the scale of the discreteness would have to be around the Planck scale. Of course, the Planck scale, which is if we set C to 1, which is the G-H-bar, has H-bar in it. In that sense, there's a connection to quantum. But it is not the goal to derive things like the characteristic effects of quantum mechanics, like interference and so on, from this discreetness. This discreetness is playing another role. It's giving us the notion of length. It's presumably during the divergences of the continuum and so on. You're going to say that general relativity, as we know it as a student, is an approximation to reality. Approximation to reality. Rather than the other way, right? That's right. So the idea that space-time, as we learned it, continuum, manifold, smooth gourd functions, etc., all that is an emergent structure that in certain circumstances would approximate the closet set, and in other circumstances, it would just be a terrible approximation and wouldn't be used, and those would presumably be, well, if you think about it, what most people would think, for example, inside a black hole, and you fall by a single narrative, and then we know the continuum brings your base down, you would have a closet set, but you no longer have the space-time description of that closet set. Another would be, in the very early universe, if it's like I was saying, it's actually a be approximated in any useful way by a manifold of any dimension. So that would be another important place. And presumably also very likely at very small scale you won't. What do you think a satisfactory constant rapid sphere would look like? Oh, I can tell you exactly. Look out, I'll finish your gunsling.

35:00 Would it, for example, generate these causal sets as an eigen spectra of some sort? No, I don't see it as coming from an operator in Henry's space formulation at all, except maybe some data would be derived from it in some way. The way that I see it being most basically formulated is in terms of histories or something like a sum over histories formulation, and it would be very close to what I told you here. What this classical dynamics is, it's actually a measure, and we call it a classical measure, for reasons that it'll become clear, it's a classical measure on the space of all completed causal sets, on the space of all histories, if you will. That we have already. What I would say is quantum dynamics for causal sets will be a quantum measure on the space And I couldn't define for you a quantum measure, but if you're familiar with the term decoherence function, then it's equivalent to a decoherence function on the space of our history. If you're not familiar with it, then it's a generalization of a classic... A classical measure assigns to every subset a positive real number, in fact, between 0 and 1, if it's the ability of action. This will assign to every measurable subset a positive real number, not necessarily between and not obeying the axiom of additivity, the homogodal axiom. So the union of two disjoint sets of histories will not have a maturation of some of the two. That's interfering, that's what quantum interference is. But obeying another identity involved in three, disjoint unions of three, which I could write down for you, which is a characteristic signature of quantum events that's actually worth looking to experimentally on. Your explanation seems to be to be saying that at least the present familiar versions of the context that there are various versions of, would not necessarily be sovereign in this description. Not sovereign, yes. Not sovereign. But one of them is sort of, The sub-over-history formulation is basically what I'm talking about, or some mild generalization

37:30 that I'm talking about. Well, out of that a crude level, where do, I mean, appreciating your whole structure is discrete, where do particular discrete things like H come from? How do these values arise? You mean Planck's constant? Yeah. The kink's? Well, in the same way that it does in any quantum theory, I think. Oh, yes, that's the great problem. You have a discrete series, so presumably you'll have access to them in a much better way than a quantum series. Well, we'll have a much better defined action. The sum over histories, which is just a formal sum in almost any realization of ordinary quantum mechanics, will here be literally a sum of completely well-defined things. And those will just be numbers, maybe even rational numbers. And the units of those numbers are h, so to speak. But as usual, when you have that situation, you use units in which H is one and they just become pure numbers. So in that way, yes. But from the point of view of quantum mechanics, H is a mystery. I mean, that's... It's what, sorry? It's a mystery. It's a mystery. Oh, yes. It's currently a mystery. I thought if you were having a discrete theory, you would expect to do something about it. It's not great. It's a mystery when the theory is complete. I'm not sure why it's a mystery now I don't understand the question is the mystery is why do the dynamics of the quantum system take the form they do in terms of a decoherence function or in terms of what I would call a quantum measure maybe I can just write down this identity because it's kind of interesting can I say there's a lot of education now that mathematics is a single body of rational thought and that all elements of mathematics can be related one to other. So if you start at any point with a primitive notion which can build a whole structure, then, inevitably, if nature follows mathematical laws, you will get the whole universe. I don't see anybody else here who thinks that quantum mechanics is a rational system. I mean, mate. He's talking about mathematics, not quantum theory. Oh, he did, mate. He did, too. By that token, h-bar should come out like pi, so you never will just come out of it. That's what I mean. Well, we haven't seen it yet. No, we haven't seen it yet, but I fully expect to see it. There's some dimensionless group containing h will come out.

40:00 Yeah, h is a dimension from number. Yeah. But here's the identity, by the way. So, in classical measure theory, you have this equality which is not an equality, not because of interference, but the measure of the discrete union of two separate alternatives, combined alternatives to discrete alternatives. Does anybody know what's in the middle? This is not true, but this is exemplified by the two-slit experiment, in which you have two positives, and when you add them up, you don't get the sum. You could have done a three-slit experiment, or the not-so-famous three-slit experiment. If you did, you would find that when these things are adiapropriate, you do get zero. And this is built into quadratic character of quantum mechanics. Any three alternatives in quantum mechanics that you can define that are described in this way measures corresponding to the beta's identity. So if you did a three-slit experiment with good slits, bad slits, polarized light, unpolarized, so on, always come out that if you superimpose the images you made when you kept an odd number of slits open, these and these, and then you superimpose the images that you made by getting the even number of slits open, they'll agree exactly. This is actually, I think, an interesting For some reason they haven't, but I wish, yeah. Can I help my hand up for quite a while? This may relate a little to what Clive asked and what Peter just said. You know you said that this schema is an objective system. Now, what do you mean by subjective? Do you mean to say that it's mathematical ergo objective? Do you mean that once you've got, and perhaps people would like to answer that, once you've got a mathematical schema, which is repeat, let us say, that's it, that's objective? No, I'm not saying that. I mean, I think I'm saying more than that, or different, something different from that, which is, maybe I should say to preface my answer that I believe that something really exists and it's the nature of a set of particle world lines or a field configuration or something that

42:30 the same sort of thing that would have really existed classically maybe this would be in the terminology of this morning maybe I'm a Bohmian or something like that I'm not sure if that's what I've worked but I believe that something really exists and then the purpose of physics is to tell us what that is purposes. It was in that context, by calling it subjective, I mean that what really exists is one of these causal sets, one and only one of these causal sets. Or it's not yet finished. I mean, it's in the process of coming into being. But that's what exists in theory. So where did the word subjective come in? I don't know, but I hope that would answer the question. It doesn't answer. This causal set doesn't represent our view of the universe or something like that. Parts of that evolving causal set are observers, so there's subjectivity there. Subsystems, structure of subsystems. Subjectivity is a real part of nature. It has to be in this as well. What you're saying is it doesn't create it. There's something there without it, even if you neglect certain of the observers or they go under the soil. Right, they might not have been there, they might have got the chain universe or something like that, or the true universe, and then there wouldn't be an observes, but there would still be this causal set. Do you think it's incompatible with the other position? Sorry? Do you think it's incompatible with the other position? That this is just a subjective? I think it's, well I should say psychologically, yes. I mean, in order to come to this interpretation of causal set, this way of thinking and taking it seriously, I had to free myself from the idea that these elements represented something like approximate measurement determinations or something like that, which is how I had once thought of it. And that was just too far from experiment, and I just couldn't take it seriously that way. Only if I think of them as things in themselves could I take it seriously and put in the effort that it took to work on it. That's just a psychological statement for me, right? It makes you welcome here. I can, I got it, that's right. Am I the chair? Yeah. As you know, the word causality is really loaded. And when I think causality, you can think the word action and you think the word event along with it.

45:00 I'm just wondering, and then you said that what happens is what, that's to be drawn from the set of what can happen, which, of course, I agree with this. I'm just, and so that's a little nexus of associations causing me to ask, which may not be a welcome question. I'm just curious for your gut reaction. causal to me is a word that in view of the fact that it's sort of based on what could happen you don't have bigger balls anymore do you think it might be something to thinking of cause as being some sort of differential relationship to action they're more than just two different words Is there some sort of a level-type relationship, for example, between what we think is events and what we think of as causes? Well, I'm not sure how to answer the question, except that in this mathematical structure, the events correspond to, although they couldn't be thought of as events in the everyday sense, but they're sort of idealization or distillation. Those are the elements, and the causes is the relationship, so there's... Ah, okay. In that sense, you put... Okay, so there's a relational relationship, not a differential relationship. Yeah. I don't know that there's a third thing, though. Okay. To your basic... I would say that the core of your presentation, at least the way you symbolically said that a number plus or there was geometry. Perhaps one should mention two names. Remarkably one person was, I don't know if you know of the work of Rome, 1911. Oh, of course, yes, I don't very much of it. Already, already, when he modelled causality after a partial order, he determined again the Laurentian signature. Even 1914, that's remarkable, I think, this one year before the other. Perhaps you would like to comment on it. We had an email exchange about Lorentz's invariance. The other celebrated result of Christopher Zeeman. Zeeman's result is just like a two-page version of Raab's book, really. So I should say for those who may not be familiar

47:30 with what Johannes is talking about, there's a very old tradition in general relativity. In fact, you pointed out it's a little bit older than general relativity. So there's to regard the causal structure as the most basic structure that space-time has. So I would think that anyone who's not trying to breathe in a vacuum, to use another one of these quotes, in trying to construct a theory of quantum gravity that's not based on the continuum of space-time probably has to select something, one of the structures of the continuum, and use that as a kind of bridge between the microscopic structure and the continuum that has to be recovered as an effective description. And depending on where you start, you'll get different results. You could take the topology, that's a very attractive view. You could take the metric itself as a kind of distance function. Chris, for example, played around with that. You could take different things, but certainly one of the ones with the longest historical pedigree is to take the causal structure as the basic one. And that's the one out of which the causal set idea grows. And you could say, in a sense, that what's happened is that you've taken... It's within that tradition, but the idea of discreteness has come to play a crucial role. It's an added information in some way that allows you to get the full geometry. That if you have... Well, I'm just repeating what I said before. There's a theorem by David Malmöhmann, which is a sort of refinement of earlier theorem. that tells you that in any past and future distinguishing space-time, which is some technical conditions of Laurentian manipulates, that if you know the causal order, that is, you know for every pair of elements of the space-time, which precedes which and which doesn't, for which one can I draw a future-directed timeline for causal curves to reach the other. If you know that, you know everything about the space-time, with the exception of the measure. You know its topology, you know its differentiable structure, you know its conformal metric, So you wouldn't have life. You'd be living in a world, a formerly brilliant world, if that were right. But the missing ingredient, I think, that gets you back away is the disobedience. Then you can bring in Riemann's insight about metriculation, feminine, and counseling. That's where causes that to start. The point being laid also, I would like to remind you, that when I quote and refer to Zeeman,

50:00 Xeon also, you know, as you know, we showed that even the transformation theory on the special relativism is encoded in a partial order, namely all the causal locomorphisms. So that follows? That follows. That follows. Once you have the metric, then the axiomotry should be done exactly. I like your ideas, but could I play devil's advocate? You always do things in my head. Because, as I heard at a conference this week, so I'm sort of repeating the argument, a piece of mathematics is really just, when you write down an equation, you're writing down an invariant from some more general conceptual environment. And if you're writing down an invariant, and a variant has no time associated with it. And therefore, are we injecting into this discussion our own personalised idea that of course there is such a thing of time. We're all hooked on it because we've been educated to believe that there is such a thing. But is there really? You mean, why do I work with something... No, I can understand... I don't understand what the question means, though. I can imagine two different meanings for it. I think that you mean... One thing you could be asking me is, could I reformulate this in the manner of quote-unquote being as opposed to becoming? In other words, could I give a kind of timeless formulation? yes, you could. It just wouldn't be, in my mind, as congenial, but that's exactly what the measure does in a sense. Or you might be asking, do I want anything like physical time at all? I mean, even if I do that, it's still physical time, it's just time subspecie eternitatis or something like that. But you could also be asking, do I want time at all? Do I not want only space, as you have in some sense in canonical quantum gravity? If it was your second question. So if you're not asking the second question, I won't feel that I should answer it. Well, no, it just seemed to be, in a certain sense, that you've taken

52:30 this ordering as implicitly concerned with time. Well, in order to recover space-time at all, that's the only way. We cannot recover just space or something like that. If that order structure has anything to do with reality... Well, no, but there is a sense in which you want to build something utterly from nothing I mean that's that's the that's that's the ultimate that's the ultimate problem that this group seeks seeks in some sense to solve how can we how can we start from utterly from nothing and come up with essentially the whole universe and it's the same dilemma that occurs in the comics or the hierarchy that Clive has struggled with, Clive and Ted have struggled with for years, how do we, in a sense, start with nothing and sort of bootstrap our universe of meaning into existence? And as Clive and Ted have always pointed out, It's very difficult to do this because you do tend to attach meanings from our everyday experience onto the mathematics when perhaps you shouldn't do that because you're putting things into the pot which, in a sense, aren't there at the time. Let me just say one thing in answer to... I don't know if this will make you happier or unhappy, but a finite coset has another interpretation which has nothing to do with causality. Just mathematically, it admits of the interpretation of a finite topology, which Ioannis can tell you any number of things about. This would be another way to go. You would still have the same mathematical structure In fact, in my own mind that preceded this interpretation and I found myself blocked. The one that led to something fruitful seemed to be the causal interpretation. So I don't know if that... Yes, no, because in quantum mechanics you can't have an evolution that has no dynamic origin and only has a geometric topological origin. So maybe if you really were hoping for something like that, maybe this other interpretation would work better? Well, we're talking about

55:00 philosophy now. Earlier you said something to the effect, oh yes, well this means there's no present. It's illusory. Philosophically, that doesn't disturb you? Next question. Really, I don't have anything else to say. If you'd like to talk to me, it probably should disturb me. Going back to this difference about whether there is really time, I mean, in the traditional general theory of relativity, we write down relations and consider a manifold, four-dimensional manifold, and really there isn't any time anywhere in it. It's all geometry, and yet one knows that if you do it on a cosmological scale, in a certain sense, time is the only thing there is, because distance turns into time, and time turns into distance, and you're faced with this ordered set that you've got to set up in order to decide who was before what. but it's a little better in quantum theory isn't it there does seem to be a bit of time but there's certainly in our lives we can't, as Peter says, we can't escape it it seems to me that there is a problem of time in that our perceptions and our lives don't seem to square with our physics I know people talk about the arrow of time I think you presupposed in stating your question that it's not possible to have a dynamic viewpoint on general relativity, and I'm not so sure that that's true, but I do agree that in some way it tends to suggest the static or what you like to call it. But I think when you look at it in the context of the causal set, which is really not that different except that things have become discrete, somehow the table seemed to be turned. It looks to me, certainly this particular family of dynamics that I tried to describe for you, looks much more easily intuited as a kind of dynamic or a kind of changing, whatever

57:30 word, process kind of thing, rather than some completed causal set. The completed causal set, which would be the analog of the static spacetime and the subspecie eternitatis spacetime, is really only some ideal limit that you could imagine, which you introduce to give the measure, to give a less clean definition of the measure. But it's really important if these transitions are moving. It gives you the illusion that you can go backwards and forwards, doesn't it? No, I don't think so. Because going backwards would mean elements to disappear, to die. I'm not talking about this, I'm talking about the traditional... Oh, the traditional one, exactly. It gives you the illusion that you could move in time just as well as you could move in space. But if it's a succession of bursts, I think that illusion goes away. Just to reinforce another point you made in the preamble to your question, the situation with spatial distance in curved space is really very, very bad. I should say, I didn't say anything on the history of this idea, but one of the people who independently proposed this idea was Herak Tuch. And he, in that work, he proposed a definition for spatial distance between points in something that's Minkowski space like. Now, first of all, we could ask about time distance. So given two related elements, one descendant of the other, if that causal set was well well represented approximative by Minkowski's space, how would we read out the time distance between those two elements? Well, if you think just a little bit, there'll be a natural guess, and that turns out to be right, which is that the length of the longest chain, the longest causal chain that you can interpolate into those two elements gives the distance, the time interval. And it's proven to converge very well with small fluctuations. In fact, dimensions, it's sub-square root of n's fluctuation, so it's an amazing thing. Anyway, this all can be made into a precise theorem that has been proved. However, if you ask about spatial distance, the definition that Herod proposed was later shown quite remarkably not to work and to give always an answer zero. There seems, in fact, no simple way

1:00:00 to write down anything that you would call a spatial distance independent of a choice of reference frame or something like that. And this seems to be an aspect of a more general that bothered me for a while, but I think I now believe that that's natural, because spatial distance and relativity in general has no particular meaning. And a good illustration is the Decider space. There exist pairs of points in the Decider universe between which one can pass no geodesic. Well, given that, how would you even think of starting to find space-like distance? So I completely agree with you that the time-like distance is by the way. I have to learn where the psychology of my question came from. I used to work for a radar company. The only way you can get distance with a radar is to use time. But of course, if you wanted to do that on a cosmological scale, neither the observers nor the radar would last long enough. one of the problems i think when you talk about time in these contexts is that questions about time they seem to meet their own tail you know uh you like to say when time began i didn't say it off one sentence i know that you are not going into that but whenever these these uh these questions about time right you get into a language that doesn't make Yeah, you know, that's right, I mean, time is, when you start talking about time, you destroy the concept, it's, what time did time begin, 8 o'clock in the morning? Zero, it begins. That's a good question. Well, I mean, I would just, I don't want to be profoundly correct, but I would just say, what time, the passage of time is what seems to have some little place in the series. That's a good time that the process is approved to run into the project. I think it's tea time. Another quote. Renee Holmes said, we're missing some people are you coming there's a lot of people they're working they're gone

1:02:30 I say I lost it's a shame because I told them I'm going to tell them something they wanted to know Well, okay. As most of you know, I'm a computer scientist by trade, and unless you're a computer science theoretician, you are by definition then a mechanist, which means that these things run on computers, there's obviously a mechanism involved here. And this is the field that I know best, and so the challenge for me some 20 years ago was, can you find a computational way to display what quantum mechanics, how quantum mechanical systems behave? And of course, this would be a neo-mechanist position where the mechanism is an information mechanism. And then I try to produce a model which, in some limit, will hopefully approach QM, and over the last couple of years as I've been working on my house and not only thinking about this sort of thing in little patches, I've come to realize how incredibly inherent I am of physics. And so I've really scaled back my ambitions in terms of if I can build enough bridges so that people whose business it is to do this sort of thing, whose profession it is, then I will consider it to a benefit of success. And so if you combine the neo-mechanist position that I take and the desire of a quantum mechanical model, then that is why I can hope to talk about a particular mechanism which has baffled for some 70 or 100 years, and that is what is it that distinguishes the classical boundary from the quantum mechanical world and what exactly constitutes that. I know there's been a lot of philosophical discussion about this and of course, well I guess I don't know whether physicists agree that the sort of theoretical or formal reason

1:05:00 is that you have a non-unitary operation and that's what breaks through the wall. common wisdom in traditional quantum mechanical thinking, but it certainly is the position that is being taken in the quantum computation community, and so I will adopt that. And before I launch into my talk, I just would like to offer special thanks to Clive and Keith, who over the many years, and not to mention the membership of AMPA, who supplied me with the crucial mathematical tools that you'll see on display here, Clyde for helping you with Clifford algebra, there isn't much to do there with, and Keith for supplying me with the homology, cohomology structure that I have taken over wholeheartedly. Okay, my table of contents, I think I'll write it over here, just study it. How do you reconstruct the world as an observer? what corresponds to psi, that being sort of a shorthand for the quantum mechanical world, what corresponds to classical. This is all from a computational point of view now, and then the QM classical transition. for those of you who haven't seen it and I've learned this is popular anyway the world that I work in

1:07:30 first it consists In other words, my original desire and motivation to all this was artificial intelligence, and in particular that which is called planning, and within planning, sort of thinking about a robot that has to learn about the world and operate in it. These sensors have the property that they either have the value plus one, which means that whatever it is they sense is currently present or they have the value minus one which I often write as one bar which means that whatever it is that it senses is not currently present in other words the opposition is not black versus white but black versus not black and white versus not quite. The next one is the one I'm going to show you now. The demonstration is normally called the coin demonstration, but I don't have the coins so I'll use these addictive instruments here. I have in my hand now, you have to imagine that the palm of each of my hands is one of these sensors. And in this hand then, and that's all I have, I have a magic blindfold in there, have two sensors, one on each hand. And so you look as if you want to smoke it. So I show you this cigarette here. And then I show you this cigarette here. And now I ask you the following question. I ask you how many cigarettes do I have? Now we will stipulate In fact, the two cigarettes that you have seen, one at a time, are in every respect identical, except for the time of their appearance. If you don't like that, we can say they're electrons, or we can say they're coins and spaces. They're totally indistinguishable. So the thing about this question is, you know I have one cigarette, and the question is, do I have more than one or not? So that's a question that has two, the answer is dichotomous, I either have one or I have more than one. And there are two possibilities, they don't overlap, and they expand the space of outcomes. So in traditional, from a traditional Shannon

1:10:00 point of view, there's one bit of information coded in the question. And when you receive the answer to the question, you will perforce have received one bit of information. I either have one or I have more than one. And then I show you this, and you say, uh-huh, he has more than one cigarette. And so the next question I asked me was, from where did the bit come? Where did you get that information from? And the answer to that question is, you saw, you got it only because you saw them at the same time. That is to say that if I look at the world through sensors, there is information about the world to be gained which is not available sequentially. It's not available sequentially. Another way to look at this is that if you have a sensor, it would either be this way or that way. Let's say it's in this state. And I ask you, what's the next state? Well, you can say with certainty, it's this, and vice versa. So there's no information contained in the flipping of one sensor. You can only get information if you see the flipping relative to another one. That's another way to understand where this bit of information comes from. Well, we could have received the information sequentially. You could have dropped the cigarette down at two, and then dropped another one down at two. would have seen two flashes. Yeah, well, no, not really. I mean, you don't know that I don't have some magical apparatus that recycles the ceiling. Oh, yeah. We'd have to know that you didn't have such... Right, right. You need other information in order to protect the non-existence of the concurrence. Getting information out of the sensors always depends on what you're a priori... You don't know a question. But in this case, you had no other a priori information. And there's one other. Oh, yes. And this actually is, I spent years trying to figure out what was slipping through the cracks in computer science relative to claims about Turing machines and, I mean, the dismal history of AI and a lot of other things. And I just had this strong intuition all through the years that something was slipping through the cracks. And this is finally the argument that I developed over the years to beat my colleagues over the head with to show them that there is something that you can't do sequentially.

1:12:30 that there is more going on than sequential computation. And this includes parallelism. And this is for the post-it business. But suddenly I'm thinking, is it true what you say? Because it is, you see, it is looking. To see two cigarettes, first I have to look at one, then move my side to the other. That's not true. They're both inching simultaneously on your retina. It's a sequential. You see, first I move in space. It takes some time. I'm not going to buy that. Actually, no, actually, she's right. You're right. What you're doing is you can do sequentially to do asynchronous, and what you do is you hit one sequentially and you store it and then compare it. Right, but you're doing the memory as to what's going on to show me. Well, actually, the kind of thing that Brian just mentioned, you go back and forth and so on, that actually is the basis of something which I may or may not have time to just mention, it produced the latest of the three patents that I've taken out on this stuff. Do I get royalties? It's already a patent. Get a free stick. Okay. There you go. So that is what I call co-occurrence. Okay. And if you're worried about sort of Einstein's comments about simultaneously, Maybe this is always local, with reference to wherever you are in space and time. One more comment, a little more normal nature. I think that this is the real, really fundamental difference between the way human beings do mathematics and the way Turing would caricature us to do mathematics for the Turingians. Because it's all about co-occurrence. That's right, it's all about co-occurrence. You betcha, you betcha. Okay. Now, the next part is then to figure out, I mean, there's, if we had, in a sense, there's S sub i, then I would write a co-occurrence with a plus, because plus is commutative. That says ordering is just not relevant here. That's why I use a commutative operation here. you can say it's a formal plus but it's not in a sense doing anything, it just is

1:15:00 it means a new sensor you form from the two sensors by asking the question no, it's just pointing out that if you're going to essentially what I'm doing is the structure that I'm going to build is the harvest of the information I gain by looking at co-occurrences and they can be of erity greater than two just easiest to write a move to. So there's no real action involved here. And the next issue then was how to turn this into some, promote this into some concept of action. Now we imagine I've got both hands and a sensor in each one. And now that we know that there's information in co-occurrences, then I notice in this particular co-occurrence here that there's something here and there's not something here at the same time. and the bit of information i get is that this state is not this this state is possible in the world okay if i never experienced this then i don't know but having experienced that this is a fact is an incontrovertible absolute fact and this is also sort of the ontological uh island of stability that i found that i that i needed to continue okay so i have this co-occurrence and I get that bit out of it and then at some other instant in time I experience this co-occurrence which is complementary to the first one using the logic that these are binary sensors and these are plus one or minus one that this state excludes this state and this state excludes this state therefore this state excludes this state there's also a complementary one where it's this state versus mine And that, of course, gives me four points on the plane, where this is 1-1, in other words I've got them both in one hand, and they're both there, and neither one is there, and here, I'm sorry, yeah, that's right. And so I call this co-exclusion. Now, this idea of exclusion, what's called mutual exclusion in computer science, is critical for creating operating systems, and that was indeed where I spent most of my time in computer science was in the design of operating systems.

1:17:30 And it shows up there because you can only allow one process at a time to say to use the printer. I mean, theoretically speaking, you could let them use it whenever they wanted, but then the output would be interleaved and be useless. Or you could not enforce mutual exclusion on a disk block, and that would work just fine, except the users would be very unhappy. And what mutual exclusion does, all it does is it enforces an ordering. You don't really care which of the two processes uses the parent first. They don't care either, basically, in a lot of quantity. But you do have to enforce an ordering, and this is computational simplification, and all it does is it enforces an ordering, that's it, and it's a local order. Okay, I think I'll just mention, okay, and I write that as a product, S1, S2. And in fact, this is, what I actually have done is I'm going to import myself into a clip of algebra here and so I view the sensors not as the sensor values since it's not as pretty scalar values but as vectors and this is the clip of product there and if you had I had a co-exclusion of erity three then I would write S1 S2 S3 and that fits very nicely if you wanted to with this with these four positions here and if you wanted to model a block being moved from one place to the other then you would go from this state say here through this state where the block is up someplace in a hand and then the block is put down here so you wrote you would rotate from here to here to here but actually because uh these things you could actually go the other way if i wanted to move the block from here to here, I could do it the following way. I have a block here, put another block here, and then take this one away. So you then go the other way. And so this spinorial property of the product is just fine.

1:20:00 Notification is commutative as well as your addition? No, it's anticommunitive. And that's because it's picking up action, you see. When you have action, you have time. What does minus s1 s2 mean? It just means you're going the other way. In other words, if I wrote s2 s1, which is minus s1 s2, okay, then in one sense, if one of these I'm going around the one way, and the other one I'm going around the other way. In fact, what I am in, I'm in plus 1 and minus 1, and I'm actually in Z2, the Z2 field, which is minus 1, 0, and 1. I know, but it's Z3, I'm sorry. and so actually then because of that i have a way to talk about exclusion because then if i can say that s uh a plus s b equals zero then i know i'm talking about two states that exclude each other so zero anything that adds to zero when i'm writing down big expressions uh it's an immediate tip-off that there's an exclusionary operation there's exclusionary situation And everything is associated. And it's associated. And distributed. So are you saying you have quaternions over Z3 here? I have what? Quaternions. Quaternions over the field. Yeah, yeah, right. I have cliffordialogers over Z3. Yeah, discrete. That's not that exclusion was represented by a product. But now you're saying it also can be a sound. There are two different kinds. Well, see now, this is the next part, okay? Yes, there's two different things going on, there's plus and there's times, okay? And the structure that I mentioned, the homological structure that I got to keep, looks like this. We'll imagine that I have my sensors down here, and let's just put two of them there. There's S1, there's S2. When I do the co-exclusion, then, I will get a new entity up here, which is S1, S2.

1:22:30 That, too, behaves like a sensor because it has two orientations. So I can do the same trick and build structure. And this operation here is actually the co-boundary operator. And similarly, over here, I have effectors. And effector is defined as S going to S bar. So I don't have to introduce any new notions to get the concept of something that affects. And then there are isomorphisms that go back and forth between here, which I'm allowed to just say they're on S1 for reasons of my own. And so I have S1, S2 over here, too. And so when I hear information bubbles up, and using these co-exclusions and the aforementioned invention, you can very cheaply build a structure. And notice that this structure is always based on observed fact. Observed fact. Absolute fact. Can I ask maybe a naive question here? You say Clifford Algebra there. Is there any particular point of departure where you're faced with a choice of different algebra? What happened was, I don't know if you, you probably would remember this set of four books which were called the Men of Mathematics or something like that, even though there was some women in there. B.T. Bell, wasn't it? Yeah, and one day, I guess it was late 80s. What do you mean the World of Mathematics? World of Mathematics. World of Mathematics. And I was just, and I had gotten a discarded copy from the library. I never had the money to own a set. And I was just sort of idly paging through, and there was an article by this guy called Eddington, whose name I sort of heard vaguely, but I didn't really know much about him then. And lo and behold, I mean, I've been staring at this sort of thing for several years, and I really missed, you know, when I went through undergraduate school as a math major, that was completely wasted on me, but the physics wasn't. I really learned to like physics, but by the time I learned that, it was too late, and I wouldn't have been any good at it anyway, I don't think. But I learned two things from that. I learned about this idea of describing reality with mathematics, which I hadn't really sunk in

1:25:00 in high school at all, which I just really liked that. And the other thing I found was that it had immense power. And I really missed it, because you look at the mathematics. I got into computer science in the late 60s. is completely different, and it's really not very interesting. It all got taken over by the numerical analysts on one side and the logicians on the other, and it's just really the worst shit. And, you know, I really missed having some sort of mathematics. I could do that a lot, and so it was like going home to mama. You were just in the wrong country. I guess so. British computer science theory is completely different. Yeah, well, be that as a man, I just felt like I was going to use one machine. I wanted to go home and I, and so I spent basically, mostly in the early 90s, searching, I knew the mathematicians had this on the shelf, you know, because I had got this stuff on the shelf. And I just knew the padding was something somewhere. So I read this article by Edison, and he's got these little things that are changing in pairs. I said whoopee and I came to Ampa a few months later and at the banquet I went over to Clive after the banquet and I wrote on a piece of paper I wrote down I showed him this article and I said what is this and he wrote clicker and so I was home that's how it happened because it it automatically does these double flips or multiple flips that was my choice that was long answer here. I've always wanted to tell some people. Okay, I just wanted to say that, so you're running the zero business, and on the one side here, we've got the plus, this is the plus side, and this is the times sign, okay? And so over here, what we're doing is we're bubbling up state as it's changing out in the environment. And over here, we're trickling down what in computer science you call goals. Now there's no teleological content in that word. What a goal-oriented computation is, is a computation whose state includes a state that it is trying to reach. That's all that in the word goal. So for example, over here

1:27:30 I might write S1 plus S2. Over here, there's a goal. You have a goal to produce S1 to S1 bar. That would just be something sitting on a memory that anybody who knew about S1 could find and say, oh, I know how to do that. And you arrange the way you name these things that everybody can always find very quickly what it is they're interested in now if you look at plus then it plus in Z3 if you if you write down what is one plus one you get minus one and minus one plus minus one is plus one but if you write one plus one bar you get zero so let's say if they're the same you take a little absolute value here you get a non-zero value whereas if they're different you get a zero so you have this property under plus and you have the same thing under multiplication. If you multiply plus one times plus one or minus one times minus one, you get one. If you multiply opposite signs, you get minus one. So again, you have an exclusive or-like behavior. It's got the fuzz on. I mean, it really is you really do have these two exclusive or-things going on. But they are not the same and they're kind of twisted. I've acquired a PhD student who was interested confrontation, and he complained about this a long time. He's been doing circuit design for years and years, and it really bothered him. He couldn't get nice, clean XORs and man's out of this. But nevertheless, there really is an XOR flavor on both sides. And if you think about the fact that it is action that produces change, and it is change that we associate with time, therefore, time is, has, if you think about the nature of time. If you're in this instant, then you cannot absolutely be in any other instant. And every instant has that same relationship. So there's this exclusive, exclusionary quality to time. Whereas space, namely, has this co-occurring quality. And that's actually exactly what Leibniz said. He said that about indistinguishables. He pointed out someplace, not that I'm any student of Leibniz, but I've read somebody who read somebody who read somebody who said that in distinguished tools you have the germ of the concept of space. And then I'm just

1:30:00 adding to that saying that in exclusion you have the germ of time. Well, I just want to make sure I've got something that's simple, kind of straight. So, if you have a bunch of things, then a co-occurrence of them is represented by Clifford Products, that's A, B, C, D. A different sum or a different problem? The co-occurrence is a sum. Because it's a problem. Because co-occurrence is commutative. They're neither one or the other... One thing has been co-occurring like one in one hand and one in the other hand. That's right. That's a plus. It's only when you... That's a plus. Yeah. Multiplication is action. So what does AB mean? AB means and. You would write that as AB on A plus B. What does it mean physically? It means that this would rotate at 90 degrees. In this direction. Or it means you do that action. Right. This is an algebraic description of the way the program would behave. If you issued it a particular goal and the action a, b was in an appropriate state. Yeah, you could say that, but again, it's not... I think even AND is not completely clean, but you can get exclusive OR is buried in there in NAND and OR is something like A plus B plus AB. In that respect, it's apparently very identical to a Gawang. What does it mean to multiply this plane A by A plus B? I don't know, the program isn't constructed to do things like that because all actions are necessarily at least at area 2. So what you would get here, you would get a plus b, I take the squares of all my senses to be plus 1, so you get 1 plus b out of that. I'm sorry, that's a, b. So this looks more like a lint co-boundary operator. What does it mean to add two axes, like a, b plus c, b? Well, s is the co-occurring. So here I have s1, s2. Maybe here I've got a, b because I have an a, b down here.

1:32:30 S1 is 2 plus AB, then I'm saying these two actions, or these two metasensors, over on this side they're sensors with their orientations. And so I would say those two metasensors co-occur. Over on the other side, I would say those two actions occur at the same time. And in fact, you can see that if you are playing quantum computation and you want to multiply by a catamart transform, if this is number one here, and this is number two, and you want to do them both, then you can multiply them sequentially like that, and you get one result, and if you multiply them like that, you get another one, and this turns out to be the one that goes someplace in terms of talking about teleportation. So it rotates them both at the same time, and that's exactly what happens. To introduce a note of levitate, your remark about reading somebody who read somebody reminds me of Tom Lehrer's famous song about Lovachewski Tom Lehrer's song about Lovachewski now I realize if you were in Grossman rather than so that AA was zero then writing a product of things as a co-occurrence would make some sense because if the same thing occurred in two different places, the whole thing goes away. Writing a product of co-occurrence, I don't understand those words. You want to allow co-occurrences without having duplications of the same thing. So, x squared is zero. Well, the reason I like the Clifford product and not the Grassman product In computer science, we're not really able at all to do engineering-type calculations like we do with bridges. How much can this bridge hold? We're not able to do anything even remotely resembling that in computer science because we don't really have that kind of mathematics that can produce numbers. That's because we had all these logicians. Whereas here, you have physics-type mathematics. for computation and the hope is to I mean the program came first I got a program that works

1:35:00 this way and the code came first and the mathematics came afterwards and as far as I'm and the code the way the program design of the the code that does this it's it's really one of the prettiest things I've ever seen and it's it just clings as a hound's tooth and so the mathematics will fit, okay, not the other way around. Yeah? Maybe I foresee a bit of an interpretation of problem here with this. Are you going to read the plus sign as the, that connecting as quantum superposition? Not yet. No, no, no, no, just co-occurring, okay? Superposition states are, for example, this one, if I, you see, remember the signature of my census is plus one, okay, but if I decide to write S1 as S1A plus S1B, okay, and then do a co-exclusion, so now I got the product S1A of S1B, now that's got square minus one, and the superposition of up and down would be this state for this guy here. So that's how you capture a superposition state. You would agree with me that that scheme would have... I mean, you think of space-likeness in conjunction with quantum paradigms. For instance, what is the product here? Okay, we have two operations. Okay, let's be simple. We have the addition and the product. Okay, the product catches the sequential application. Yeah, it captures exclusion, which is like time. Yes, the change. And while the plus captures the processes happening in parallel. No, well, yeah, it has this concurrence. It has this idea of going on events occurring at the same time, right? Processes are, you can start talking about various processes now, right? You can talk about the process of some particular thing bubbling up through here, following a certain path, certain things, certain of these meta-meta-meta-meta-meta centers changing their orientation depending on their own. And likewise, you can talk about goals trickling down here

1:37:30 and certain paths being available because that is the state and certain paths not being available because that's not the state certain paths exclude each other you can't only have one or the other you could have paths that are superposed and so on it just depends on how everything has been built up and what the current state is I might add that we really don't know about what's going on out there a persistent problem in computer science has been to get state and action on the same piece of paper You know, and it's, I mean, here I'm separating out the state part from the action part. I mean, we don't know what Mother Nature's got behind the screen there. And I've never been satisfied with the way computer scientists have tried to separate the two elements so you can look at them and nevertheless have them intertwined. And what this says here is that for every state, there's an operation, an exact anti-gizmo that will change that state. For every single state that you have over here, there is something on the other side that will change it. And it's local. It's local. Which I've just learned kind of is the gauge-like aspect. But don't crucify me if that's what's wrong. But isn't the test going to be that your system sounds, if it will, physically realize concurrency in the sense of concurrence. But I'm not sure that any of the computational theories that claim they apply concurrency do actually achieve. I don't understand your question, Peter. What is called parallelism has got nothing to do with this at all. Parallelism gets the same answer, but by the law clock it's faster. There is no semantic addition by computing certain things. No, no, I think you'll think it's right. But I'm not sure, as you say, what the computer scientists are doing. Well, I'm a computer scientist. They're not doing this. That's one of the reasons I quit. I just got tired of pissing against the wind. Okay. I'm going to pass on the patentable thing. We've got time to hand off. I'll come back to that. It has to do with how do you discover these things efficiently, the co-exclusions.

1:40:00 So you go up with the co-boundary operator and you go down with the boundary operator. This whole structure is isomorphic to differentiation and integration in the world of linear vector spaces. okay so you have oh i just want to say you know when you're trickling down here you look at sort of the causality situation right you trickle something down but nothing and all you're doing on the way down at least as i currently have a program all you're doing is sort of each one the goal is coming down each one of these guys is looking up just as each of these is looking down and see what's coming up and so he sees oh you want to do something that's one and i'm s1 that's No, you want to do something with S1, S2, this guy says. He says, well, I'm S1, S2. I know how to do that. In order for me to flip, well, then S1 and S2 separately. Something's got to happen to them. So as the gold trickles down, it fans out. And this is where you get the multiple computational paths in a quantum computation, in my view. And then finally it gets down here, and if you do something, the hand grasps or something, and you pick up the block or whatever. but you think about it now you sort of set everything up on the way down but when the change occurs what's the vertical axis well this is the cobalt going up it's bubbling up and it's trickling down is this in time there's no time only time that is the thing This is sort of the quantum mechanical world, if you will. Things are happening, but there's no time. There's order. You do have space-like and time-like. There's space-like and time-like, but nothing has happened yet. But co-occurrence is space-like. It doesn't happen until you actually change the guy on the boundary. Then something happens. This is something that's happening on something like a simplicity or quantifier. Yeah. I think that's the first thing. Thich just said, when you said there's order, he said you mean sequential order. Do you mean sequential order? No, I don't mean that. I just mean there's... Partial order? Well, you've got to be sequenced to distinguish them, surely.

1:42:30 No, you don't have to distinguish them. They're just there. Each one of these things knows what it is and knows what to do. Not called Clarence's. The left-hand side. The left-hand side. The trickling down. These aren't changing any because we're inside the boundary. So all that's happening when the goal trickles down is it gets fanned out into subgoals. Actually, the reason I asked you about simplicial complexes is... Well, listen, let me finish that question. One of the things Yanis was showing us recently, which was new to me, is that simplicial complexes are, in fact, special cases of cosites. Well, I'll tell you this about simplicial complexes. I tried for a whole year with an American mathematician to, this was many years ago, back in the 70s, and I tried as I might, I couldn't get simplicial complexes to harmonize with my concept of computation. So, I don't know. Mike, what is the difference between a co-occurrence and a failure to co-occur? You didn't get the information. I mean, what sort of co-occurrence imply that they happen at the same time? That's right. They do. Yeah. And the failure to get a co-occurrence means that they're at a different time. Right. So we're discussing a manifold which is sequentially ordered. I don't see that. Well, no, sequentially later. This is entirely local. Every one of these guys is doing his own little thing, right? together. You're saying that there is a clock which is telling us. Well, I don't want to talk about finding this stuff, okay? That's not what I'm talking about today. We can do it afterwards. That's the pattern. There is something that's finding co-occurrence. Yeah, that's in our world. That's not in some of the machines. I mean, I really don't want to spend time talking about that mechanism, nifty as it is, okay? We can do it afterwards. You say co-occurrence, you're It's within a delta T. Co-occurrence within a delta T. Yes. The word declines time. You're saying the difference. Well, I just really don't want to spend time talking about that, even though it's very, very many. In that case, you cut it off. We have nothing about that. Okay. How are you going to detect co-occurrences? That's the problem. Okay? You can detect co-occurrences by having a little clock, right? And it's going to go back and forth.

1:45:00 But actually, you don't have to do that, okay? Because you can use a wave interference kind of a thing and find out whether, if you imagine a wave coming to a wall, and you just have two wires out to do the currents at them, you really don't have to have a little clock for that, for the co-occurrences. But co-occurrences is the easy part, co-exclusions is the hard part. And if you think about the way you would find a co-exclusion, well, first you've got to find, say, A plus B. And then you've got to find A bar plus B bar. And only when you've seen both of them are you allowed to infer the B. So how do you do this? Well, what you would do is you would set up some process in the machine, right, and it's going to sit there and wait, maybe using a delta-t, depending on what your co-occurrence detector is, okay, it's going to sit there and wait around until it sees this. And when it sees that, then it'll go looking for this one, or you can split this into two and have them handshake and exit from the whole thing and say, I found a pair of, kind Co-occurrences. Okay, wonderful. Okay, supposing that I have any sensors, okay now in a real system is gonna be a lot of hundreds thousands, right? Okay, so I want all the two co-exclusions and I want all the three two co-occurrences and three co-occurrences and however high anerity I want and this basically goes with an extra factor of two in there because you've got plus and minus. So that means you're going to have two to the n processes out there. That doesn't go. That just doesn't go. If I had a thousand senses, I'd need two to a thousandth processes inside the machine. It just doesn't fly. So here comes the nifty part. How do you find these things then without spending exponential time and space on them? Well, what you do, what you do is you make a little buffer, we love buffers in computer science, and what I will do is, as every sensor changes, before I pass it on to the rest of the system, I drop it into this buffer, and the buffer is delta T long.

1:47:30 Here you do need a delta t. That buffer is delta t long and when a given sensor change has been in their delta t time gets thrown away. Delta t is the granularity of my co-occurrence detection. Right. Okay? So every time, so A went from A bar to A, I'd say. So I put A in the buffer. And within delta T, B bar goes to B. So I put B in the buffer. Now by definition, everything in the buffer has, the module of delta T has occurred at the same time. Right? So I have the co-occurrence A plus B, but I know that they change, quote unquote, at the same time. they arrive in the buffer, it was a bar for q bar. So I can infer the existence of the other one by means of this mechanism. Really what I'm looking is I'm looking at the differential. I'm looking at the time derivative of this. And so instead of exponential, I got linear. It's linear in the length of the buffer. So that's how you find these things efficiently. It's only the fact that you have 1,000 things side by side. That's right. Does that answer the questions that we're... And actually, if you think about it, there are basically two steps involved. The two steps are the one pair arriving, and then the deduction, let's say, is free, and then the conclusion. So you have A plus B plus A bar plus B bar, and then the conclusion, which is AB. And that, this is a time-like development. You can't say this until this is done. And I think, therefore, that the measure of that time is H. So you're talking about logical sequentiality, not 10? Yeah, I understand. I have one question. What do you mean by the conclusion is A-B? Well, going way back, if I see a co-occurrence and its complement, I'm allowed to conclude A-B. I can conclude the co-exclusion, which is represented algebraically as the problem.

1:50:00 Why do you conclude the co-exclusion? Because the code came first, so I know my initial little Gedanken model was this block So if I experienced a block here and out there, and in some other instant I experienced a blocked here and out there, then I can conclude, because of the information I've harvested from the respective co-occurrences, I can conclude something has changed. Something happened. An action. An AB is an action. An AB is an action. How do you conclude it's that action? Well, because it fits. It does what I want. It does what? It models. if I then take this action AB and represent it as a software entity, then the goal to do something would be represented basically as AB to AB bar. So the goal comes down and says, you've got to change something because somebody up on high decided something had to change. So the goal comes down, this guy up above is Bill Hyatt AB and somebody else. But this guy only takes goals for himself. So he says, I gotta make something happen. So he takes the goal for himself to change and translates that into a goal for A to change to A-bar and or B to change to D-bar. It depends how you define things, okay? And when that happens then, then one of these things will change, it'll bubble up, come across, and he says, ah, I'm done, okay? So you're not interliving the action A and B as multiplication by A and B, but rather as the action of complementing A and B. and then the complementation of A, B to A, B bar contains within it the complementation of A and the complementation of B and thereby complements the code. I have the operation of this, I have something, I have this operating on that state. So that's the sequential description. That's what happens when you code, yeah. It happens sequentially. It's just you packed it up unsequentially in your mind and we've seen a sequential Well, yeah it's hard and it's like everything else

1:52:30 you don't know what it is, people don't understand until they ask, so thank you for your questions Any others? How are we doing on time? Actually, we're almost there I'm going to sort of ask you to take with however much faith in me that this is fundamentally a quantum mechanical situation because nothing actually all this stuff is happening but but it's not really happening until something happens out here. So if you imagine this represents some physical system, and you poke it, you prepare it, you poke it, and this stuff circulates around, finally something drops out. But notice that what drops out doesn't have any sort of obvious functional relationship to what you put in. What comes out is a side effect of all this structure, all this sort of interlocked, intercorrelated structure. just as the fact that an elephant walks is a side effect of all the stuff that's in the elephant. That's again about what you mean by macroscopic chronology. Right, there's no difference. There's no difference. From this point of view, see, because this hierarchy is made from the same materials at every level. Now, let's wait for that whole exploration to my talk on Monday or Tuesday. So a really interesting question now is to ask, well, what does a sequential computation look like? What does a sequential computation look like? Well, it's going to look something like this. Here's the two pieces, A and B. And I have another action over here, which is only to after this has occurred, so I can imagine my sensors A, B, and C down here, and so going through the bubble up and map across and all that stuff, eventually this guy says, ah, I'll do my thing, and the result of that is that let's say B changes to B bar. So this action here, the next one, knows that it can start only when that one is finished, and then it's also built out of Now, if we go back to what I said before that multiplication is action, let's just look at a few things. If I just take AB and imagine the way this is working here, either I have AB as an action and I can ask, what happens if it acts on itself?

1:55:00 Well, that's AB squared and that's one. It's reversible. And then if I took a, b plus its constituents, a plus b plus a, b, and ask what happens when that operates on itself, again, I get one or minus one, forget it. So I read this as these things are reversible, but notice that you have to always take the whole clump. You just can't take part of it. And that's basically what's going on over here. We're just taking certain pieces. We're not taking any sort of natural integral pieces. We're taking unnatural pieces. And so if we multiply that together, we have A plus B being operated on by B by plus C. And that is not equal to 1. That's equal to B, C plus A, B and so on. And that's not reversible. And so, I claim that I have delivered on my promissory note to explain to you the mechanism for how you make the transition from the quantum mechanical world to the classical world. Namely, you perform non-unitary, non-reversible, sequential, Newtonian, functional, as in LISP, and computation. It's not reversible either, is it? It's the arrow and the arrow is in one direction. So which transition is that, then, at that hour, is that from quantum-mechanical to quantum-mechanical? Yeah, yeah, when you, yeah, at this point, at this point, when you did this, this, okay, everything was still ducking, basically, because that was A plus B plus AB. Okay, so that was okay, okay? But as soon as you conditioned, as soon as you tried to enforce a time-like progression, an ordinary timeline, this is what you're doing, and it's not reversible anymore. If you look at the bits, what you will probably find is like an and. Someplace two bits went to one. You threw some information. You mean something in addition to sequential? No, to me, sequential, time-like, and if you will, list-like, and classical. They're all synonyms. It's all the same worldview, and ego consciousness.

1:57:30 All those things, from my point of view, are identical in their foundations. It's this world over here. You can say this is self, but I'm going to be talking about that, I'm going to be putting that set of eyeglasses on in the talk. Are there any other questions? and these times are implicitly proper times you can't actually talk about that that's an interesting question it turns out that if you look at the combinatorics of this straight flat buildup then you get the combinatorial hierarchy you get onesies, twosies, threesies and so on did you have any answer to the question about the cutoff? anyway if we sort of believe the combinatorial hierarchy When you get to A, B, C, D, as it were, you get at four levels, then you are in the realm of gravitational stuff in space and time. And the idea of proper time and all that only can be stated at this point. And my musings are that, I mean, the basic idea is you don't have enough information carrying capacity in the structure to represent what goes on at this level until you get there. So that's to say at levels one, two, and three, you have no space-time structure. And I'm used that that is the inflationary period. But I don't know if that's right. At least, you know, a lot can happen before space and time show up. I must say I have the intuition that that sequence must be going on everywhere. I know. Well, see, that event window I told you, I showed you, but that's actually the mechanism in the void. Because really what you're doing is you're taking zero, a plus b plus a bar plus b bar, and translating that, it's essentially a co-boundary output, and you're getting a, b out of it. So that's really, you know, that's how structure is coming out of the void. And that's why I impute Planck's constant at that point. It doesn't get out of the void from our sense until it's done four levels. No, no, it's out of the void in terms of where is this structure coming from, but it's not able to talk about itself in terms of space and time, because it doesn't have enough

2:00:00 information carrying capacity. I can say this, that if you think about the co-exclusion inference, the fundamental syllogism in Buddhist logic is A and B implies C, implies not A and not B implies not C. That's the basic syllogism of this logic. You haven't got C? I'm sorry, C. That's the basic, I know, it's not like they discovered something Aristotle didn't know, but in Aristotle's formulation hidden away down in some little on and or not, I believe. By the way, the book that you want to get a hold of, about this is a really wonderful book, but I need Theodore Chervatsky. Do you want to reverse the second arrow? Yeah, yeah. This one here? Yeah, that's right. Theodore to the right. To the right? No, no, no, no. A and B implies C implies not A and not B implies not C. Here's the alphabet. Here's the alphabet. And then you're thinking X implies Y implies not Y implies not X. If you make A equals B. Yeah. Yeah. A and B implies C implies not A and not A, implies not C. Mike, at least in my kind of notation, there should be other brackets. If you remember, I tried to do a few steps. You don't need any more than that. That's the whole thing. Why are you trying to truth table? Well, I have, too, and you must have made a mistake. That's all I can say, because it comes out all truths. No, Mike, you don't need one. Yeah, you've got an arrow. Which one? The last arrow. You've turned it on as the corner. The last arrow, turn it around, change the hand to R. Change the hand to R. Yeah. Well, we'll do truth tables. We can do it at the farm. Mike, when I do the shame, watch. When I tried to do that, I couldn't get it right in my nomenclature without the brackets.

2:02:30 There are brackets missing. When you put the brackets in, I got it right. I don't know which ones you're missing. This? You have this? You need to have that. I think Viv wants to put brackets around the A and B. Well, but some people work on the basis of... That's okay. Thank you. A distinction between... Okay. The book about Buddhist logic is in two volumes. The second volume... The second volume is a translation of the... sort of the Aristotle of the Euclid, or whatever you'll call him, of that school of life and the name of Dharmakirti. And the first volume is Chabatsky's commentary on that. I much prefer the volume one, because I don't really care to read in detail what Dharmakirti had to say, but I'm very interested in what Chabatsky had to say. This book, by the way, was published in 1930 in St. Petersburg. I don't think it was called Leningrad quite yet, and that's how Cummings was able to write and publish this book. I actually have a book in my room. It's related to your shell game, because they talk about existence and nonexistence, and neither existing nor non-existence. In fact, their censors are exactly like mine. So, Mike, I'm supposed to say, then, that, okay, it doesn't work out in, say, the sort of logic that I'm used to, but it's okay No, no, no. I can't believe I've written this down wrong. I've actually done this to him several times. I just can't believe that I've made some mistake three times, so there must be some trivial problem. Well, what's the problem of a mistake? It works out when I put certain brackets out there. If you put a square bracket... You can actually write if and only if you just reverse the knots. Can I squeeze in one question about physical interpretation? Just one small one. A lot of people were talking about co-occurrence as though co-occurrence could give you time straight away. Would you like to comment about that with respect to relativity and the relativity of time?

2:05:00 a two-ary co-occurrence as a photon. I should also just mention that Dharmakini had a really great definition of what a syllogism is. A syllogism is the means of conveying an indication to another person. This can be done on the napkin. Yeah, that's right. as a photon? That's my interpretation. My interpretation is this is the simplest co-occurrence. It has all sort of like the red properties. It's timeless. You can sigh, you know, and all that. And it's the simplest. That's okay, but what I want to say is that this, interpreting the tree of co-occurrence, I mean, on the ladder of co-occurrence. On the ladder, yeah. Okay, the bubble up part of the ladder. That does not work. What doesn't? In relativity, because there's a relativity of simultaneously. I know that, but I heard that comment in the room, and I want to get clarification on this. What you have is proper time for the observers. Yeah, you are the observer. You are the observer of the co-occurrence, and in a sense, you are living in the time frame of the co-occurrence. You're living in proper time. I think you're doing it right, Mike. Yeah, I think so. I may foresee a problem here. If you want to retain finite dimensional algorithms, say finite inputs and whatever they generate, and then make connections, if you want to retain finite dimensional algorithms, unitality, and then you want such ideas to have say to have some space-time implication we may apply some ideas from relativity then you are bound to run to the notorious problem that we have had in the past in uniting quantum mechanics with relativity namely on the fact finite-dimensional unitary representations of the Lorentz group. Well, see, I'm puzzling, I mean, I sort of feel like...

2:07:30 I'm just saying... I'm just saying... I'm starting to puzzle over relativity now. I haven't worried a bit about it. I would say, though, that the basic layout of the whole thing is fundamentally relative. I mean, the values of one entity to another are only, you know, they're relative. everybody's their own observer and nothing happens until you get some sort of influence and so on. There's no coordinate system built into this except the coordinate systems of their values of their states. It's got a lot of room doing geometry and so on. I would say that you really can't even start talking about relativity until you get to the fourth level. You just don't have enough guts. That means, really, that you shouldn't be trying to apply those kinds of things to things that are going on at the lower levels. Because, OK, it's just not, it doesn't make any sense. And you can think of EPR in the same way. The EPR influences, you could think, I don't actually like this way, but you could think that they are propagating at these lower levels, but you can't see them except from up above. And the system is self-consistent, so you'll never see anything that violates looking down from above, but there's actually stuff going on underneath that's passing, as it were, through space and time. I think I have a crude question. Where does three-dimensionality begin to come in? I think that it comes up when you first get to level four. I don't think that... I don't know whether you could argue there's a two-dimensionality at level two. I don't know. I just don't know. It comes up co-occurrence with that dimensionality. You have geometry already here. You have a seed of geometry. Your co-occurrence implies dimensionality. You can't have it without it. It's at least two-dimensionality to get co-occurrence. Yeah, right. Each of these things is its own dimension. And the size of the space is at least two to the n. There's no such thing as a line without two-dimensionality. I don't buy that. I think that's a false statement. Well, anyway, I don't know. That's too hard a question for my poor mathematics. No, no, no. No, co-occurrence would appear on distinguishable photons. you can't begin to talk about I'm inclined to believe that's sort of my inclination because we're talking about space-like dimensions this is not space-like

2:10:00 these are dimensions of distinction those are the space that all this is in is the space of distinctions do you think I'm going to say something because I think Lou took Clifford Algebras and more or less got a special relativity out of it in a paper. Well, that's true. That's the... So it certainly looks as if Mike's formulation will end up by being obeying the rates of variance of that sort of thing. Well, let me explain that. Perhaps. Yeah, I don't know. That's what I say. I start getting real humble about my physics. Youngs, I want to make sure that you are not thinking of trying to apply... No, no, this is the first... It's up here, so we're across. There's another thing. What is the constitution of a property that makes a level? And what it is is that, from your point of view, they're all there at the same time. That's it. Okay? So, essentially, all the mathematics of general relativity is going on here and with this neighbor over here and up, starting, say, from level four. Down below it has, it just kind of talks. So you say the EPR, for instance, gives us glimpses or shadows of the lower. That's one way to look at it. My preferred way of thinking about EPR, and this is something I thought of before I did this, and actually I got a paper in essays about it. It's called Synchronization, the Mechanism of Conservation Loss. Essentially, if you look at these synchronization signals, I was talking about mutual exclusion, you've got to wait for the printer and so on. You draw a synchronizer like this, and one guy comes in, he does a wait. You've got one secret bit in that that says whether you let somebody through or not. When you get done with the printer, you come around and you signal. now that's all it's really doing is if you think again of the of you know this playing with four points essentially what you're doing is you're deciding am i going to go this way around or that way around that's who goes first who got to go first okay and uh so all you're doing all this

2:12:30 synchronization signal is doing is adjusting the phase of user one user two system okay and phase Phase differences are not information-bearing, and therefore they can exceed the speed of line. So that's the way I have always thought of the EPR experiment, and if you think about the EPR experiment, you can't actually make the detections of whether of the two correlations. You have to see both ends. Otherwise you don't know which, you don't know where this photon is you got at the one end, You don't know if it came from each hyperspace or compared with another. You have to have both of the photons. Essentially, the logic of the EPR experiment rests on the fact that, say, spin is being conserved. So if this one's up, that one's going to be down. So that means you've got to see them from both ends. What is seeing from both ends? You have to actually see both photons. We talk about doing EPR electrons or whatever it is you're sending off. the logic and the reasoning of the EPR experiment rests on the fact that there's a conservation law binding the two together observing, measuring, whatever you want to call it I'm just saying that you have to have you have to actually the logic of the EPR thing is based on the fact that you have a conservation law, so if something happens over here it's bound to be the opposite over there in order to preserve the conservation now if you look at this photon You grab it, you get a hold of it, and you measure some other photon, not its mate. Then you might see them both up and say, hey, what happened here? So that means that if you're really going to see the effect on an individual basis, you have to actually have both of them. And all you'll see is randomly that one's up and the other one's down. And the reason... They're correlated. Right, they correlated, but they're correlated in this particular way binding them, and the conservation law really is the fact that from an old-fashioned operating system resource invariant point of view, there's always one printer in the system, and the question is, who's got it, and so the synchronization signals are determining who has the up, because the other guy's got to have the down, and so when you reach into the system and you measure one, it's a random question, who had the stick, who had the synchronization token, you find

2:15:00 and not the other. So there's no mystery. I think we should stop now. I think we should stop now. Yeah. Thank you. The sort of general methodology that we were using. And I'll say something. What do all the letters mean in this conference? Oh, this is the conference for computing anticipatory systems, which is the field in fact that Daniel Dubois works in. And you'll just get a brief glimpse of that as I hear. Out of interest, what does chaos stand for? Well, no, it does stand for something, but it's meant to be what it, it's just an acronym for itself in a certain sense. Well, I want to try and do the thing that, I know Ted, which can't be done, and that is to say, I'm going to assume the undue effective mathematics of mathematics in physics, and I'm going to assume that quantum mechanics can be made and consistently explained. And so I begin by saying that the formalism of quantum mechanics is ambiguous. State vectors are arbitrary up to a constant phase factor. Yet, as Berry showed, gauge invariant relative phases of the state vector constitutes a new class of quantum observables called the geometric or Berry phase. This must therefore be the starting point to any valid interpretation of quantum mechanics. For otherwise, the ambiguity with regard to phase means that any interpretation consistent with the formalism is a valid one in some domain. So you can act yourself blue in the face about which interpretation is correct, but in some ways each of you, the view that you hold about it is valid in some domain of the physical

2:17:30 world. Aspects. In some aspect, yes. So how do we resolve this? We say, well, so I say there's a large body of often heated argument of the last century today as to which interpretation is to be preferred. And there's really no way of telling that from the formalism. From the formalism alone. So what we need is a boundary condition. Because boundary conditions are always essential to correct solutions. So if we really want to, the question is what is the boundary condition? And it turns out that the boundary condition I'm going to show you is remarkably simple and has absolutely fascinating implications and corresponds with the belief that I think is held by many people here that essentially start from that. Okay, I'll show you that later. Now, since I think that the very phase is fundamental, we start off with considerations of quantum chaos, which must arise from boundary conditions, geometric phase and quantum holography, And then we identify self-referential in the main paper, quantum mechanical models, which explain this ambiguity. Now, it turns out that another set of models which fit into this framework with the ones that I think I've been working on, Schweppes have been working on, and DuPais have been working on, and I, in my opinion, the combinatorial hierarchy have been working on, for example, concern Kenneth Wilson's renormalisation group methodology for the universal determination of critical, of stable and unstable points, which are attractors, for which he received the 1982 Nobel Prize. These points define material phase transitions, such as those from liquid to solid, or from crystal to solid, or gelatinous, paramagnetic, ferromagnetic, i.e. they are material properties, which in general, and they are quantum mechanical rather than classical.

2:20:00 Now, I think the correct philosophical word for this is that it's noumena. Now, I think if in fact you label noumena, you give noumena a label, then that corresponds to what we would call quality. That will say more about it. So, Wilson proved very nicely that while the corresponding classical field equations to their quantum mechanical counterparts can usually be applied on scales large compared to the plankton, care is essential. As this cannot always be said to the case, as it's often assumed, because, for example, the person who was working about the same time as him was Landau, and Landau said quantum mechanical effects don't percolate up into the map of Scotland. But if you look at the Eisen model, but Wilton actually does, in his theory, he shows that these quantum mechanical effects can affect anything in up to three and four dimensions. But in five dimensions, no. Well, yes, I mean, in a more practical sense, Chandra Staker showed that quantum mechanical effects can hold up white laws. Yeah, yeah, well, but I mean, the reason I put this in is that, you know, if you tell physicists that you're going to have a quantum mechanical explanation of a biological phenomena, they say, well, first of all, it's noisy and it's in the macroscopic world. and there's some recent papers by Tegmark who has conclusively proved that you can't do this. But I don't know why they would do that because, in fact, this work concludes and shows that under the particular circumstances in a phase transition... Now, what happens at a phase transition? Let's take the simplest phase transition, one of the things you can think of, which is water. Liquid water to steam. At that particular critical temperature then you actually have drops of water containing bubbles of steam,

2:22:30 and bubbles of steam containing drops of water on all scales, from the microscopic right up to whatever scale. So it's fractal? It's essentially fractal or self-similar, and since these things can only be predicted quantum mechanically, you can perhaps infer that in fact there's some influence at this particular point in relation to quantum coherence. I don't see that this has anything to do with quantum mechanics because the using models... Well, you don't that's fine, that's fine can I continue and I will tell you why I'm sure it's got something to do with quantum mechanics I'm just saying it arises in a classical version of the theory as well. Well, of course, because classical physics is just invariant properties of the quantum mechanical domain. Well, that says everything is quantum mechanics. Yeah, everything is quantum mechanics. Okay? And that's my assumption anyway, that the universe is in fact totally quantum mechanical. and I intend to show you why. So that's a sort of example of this. It also turns out that in Wilson's theory he had a mathematical method of doing these calculations which was essentially to impose a three-dimensional lattice structure on the Hamiltonian and by this method to map the Hamiltonian on itself repeatedly until all you were left with was essentially the data which was these critical points which in some sense are the centres of attractors. Now, in this particular approach, not only does quantum theory contain, quantum theory but it essentially contains the whole of Kells. And now I'll try and tell you why

2:25:00 how all these theories come together. It's really to do with the fact that Now, there are two ways of solving this problem. You can solve it in terms of what you might call the dynamical variables which are x, or you could have many dynamical variables or you can solve it in the parameter space by looking at the base of these. Now, the A way of looking at the parameter space, the methodology is less well known, it's concerned it was worked out, or started to be worked out by ServiceLean, and it concerns Li transformational systems, and in general Li showed that more or less any problem, whatever kind it is, can be solved through the parameter space by essentially tuning these parameters. and then these parameters are in fact invariants of the solution space. Now, if we do this in quantum mechanics then we know in the dynamic space that the solutions concern the eigenvalues of operators which are often the Hound terms. But we now know from Berry that in the parameter space, where in this case it's of the phase, in this particular case, that in fact there are gauge invariant phase, observable gauge invariant phases of the state vector. So you've got a second class of observables. from playing with these and others which come essentially. Is that meant to be the characteristic equation? No, no, it's just an illustration of this idea of the parameter space. It's not meant to be anything. It's just to illustrate the concept.

2:27:30 So, now it turns out that this space is really continuous, and it has a geometric quantization. And this space is in some sense highly, the space associated with this equatorial galaxy is the discrete space. So, in this formulation of quantum mechanics, you have both continuous properties and discrete properties just because of the very fundamental nature of mathematics itself. I'm finding it difficult to know what this polynomial is. Well, it's just made to illustrate what the parameter space concerns and what the dynamical space concerns. The A is the parameters on the X. There's some sort of variable. There's some sort of variable. It doesn't even have to be a polynomial. We need to be talking about... It doesn't even have to be a polynomial. No, no, it doesn't. Yes, yes, yes. It's just an illustration of the I. But all modern quantum mechanics mixes up variables with continuous spectra and discrete spectra. Well, let me go on and tell you how the work of the various people in this paper sort of hangs together within this framework. Dubois began his work into competing anticipatory systems by looking at cellular automata. But instead of taking logical rules as the things that change the state of the cellular automata, he took the canonical equation of chaos, and what he discovered was that you could essentially turn the butterfly effect on its head. What do you mean by canonical equation of chaos? Well, you just need a quadratic... You mean an example of a chaotic process?

2:30:00 An example of a chaotic process. He took the simplest one, which seems to more or less include everything. That, in general, can be chaos, or factually put. It's general, but usually it has a chaotic process. So you iterate on, in some sense, what you do is you tune these parameters, okay, until the chaos in the automata comes very, very close to the behavior which you want to simulate. that thing generates new values of the variable and you tune these values you tune these values so that the chaotic effects exponentially fast you mean x equals x you tune this you tune this thing so that you tune the thing in the parameter space so that very rapidly the thing get there much more quickly, you get there exponential facts. So, effectively, you've used the butterfly effect. I think maybe it's not clear on what you're doing with it. You're writing f of x equals that, and then you're turning with x. f of x and n equals. Sort of, yes. The Julia set is something like z nu equals a constant plus z on squared. it's a complex number in that in that particular case because you're on the other side of the ground but anyway that's what he did and he did lots of work with this and he called this fractal intelligence because in fact usually it ended up by giving you a fractal to assimilate so that was A false pendulum can do this. Yes, yes, yes. But this is to illustrate the fact that he, in order to get the solution, he was working in the parameter space, okay? Now, when it comes to quantum holography, we discover that Schwemp is also working in the quantum mechanical parameter space now. And in the quantum mechanical parameter space, he chose the Heisenberg group. Now, the Heisenberg group has three parameters,

2:32:30 and you can show that in relation to the Heisenberg group these parameters correspond to actually spatial measures in order to get in order to simulate a holographic process you essentially you essentially you essentially tune tune these parameters. And it turns out that in relation to the Heisenberg group, that the Lie algebra of the Heisenberg group, because this is a Lie group and it's nil potent, so it's always solid. And it turns out that nil potent is a very, very important property. I'll try I'd say something going on a bit later on, that the algebra models the Heisenberg commutation relations in this particular problem. So what you're doing is you're saying that you've got this group G, which is the dupe of these parameters. This group has a dual, which is related to, which is related to, this one has an exponential mapping, okay, this one has an exponential mapping, okay, and there's a Lie different, because it's a Lie group, there's a Lie different morphism, and so there's a differentiable inverse to this group, and so for every mapping that goes this way, which is an exponential mapping, you can find a logarithmic mapping, which takes you back again in this case you have turned the Heisenberg uncertainty principle you use the Heisenberg uncertainty to compute and so it turns out it turns out as somebody actually proved with the Confidence Act that quantum holography is the optimisation process is the optimisation process for quantum mechanics. And it so happens since quantum holography describes how magnetic resonance imaging machines

2:35:00 actually work as they are in probably all the hospitals in Oxford and all around the world. And how they get their images out is because essentially one of these maps concerns the holographic encoding, and the other one corresponds to the holographic decoding of the information to get out the image. Excuse me, Peter, when you say holographic, you use holographic connection with... Well, this is generalized syllogary of any... Right, right. The pictures you usually get from NMR are planar. Oh, planar, yes. But using this technology, you could get holographic images out. Well, yes. I mean, if you go to the Duke University website, they do magnetic resonance microscopy, and they have three-dimensional pictures. Which are obtained in that way. Which are obtained as a holographic. Yeah, yeah. And not by building up a bunch of planar pictures. Yeah, yeah, yeah. Yeah, okay. So we know that this technique works, okay? And the approximation process to do this optimization is called spin echo. Okay. Now spin echo means that what in these brain and body slices that people usually want to look at, they usually choose to look at the protons, let's say, of the oxygen that are actually spinning, because water is very characteristic of structures in the brain, because water is almost heavy. and now we can go back to Wilson. You say, so the structures in the body on the whole are stable. In a stable configuration of, you've got all the little protons spinning and they've all got their characteristic, they've all, to some axis, they've all got some characteristic angle at which they're actually spinning it. And what the machine does is It pulls the spins all into a single spin state and then that spin state decays. It's released. And then this information from the release to a certain state is collected in a coil.

2:37:30 And then you actually feed back what you received back into this system, and you keep pulsing it. The info comes out as radio waves, doesn't it? Yeah, it comes out as radio waves. And so, if you do this spin echo technique, you can make better and better images. There's a lot of... So, it's actually been known that you can control these quantum mechanical processes in relation to nuclei, the spins of nuclei, since about 1950. First of all, it was done in magnetic spectroscopy. And then somebody got the idea that, in fact, you could use it to construct these imaging machines. And these imaging machines are getting better all the time. And Schrenk, who I work with, is, in fact, a specialist in this. places like Stanford and Harvard and all around the world working with the medical doctors and they want better images. They don't really care how the machine is like, they want better images. And he advises them on how to tune the machines in a better way to get out the kind of images that they would like. Can we just go back to the phase changes of materials like water? So you can think... Well, the connection is that in the material, first of all, the spins are in a big stable configuration, okay? You perturb them into an unstable configuration and then they decay back into the space. So the nice thing about this is that, in fact, the decay concerns entropy, but on the other hand, you're getting a nice image out. So you can consider the entropy not as a measure of disorder, but equally as an information metric which gives you the image so what you're saying is common is that both phenomena the boiling of water and this are sort of catastrophes in the common sense so this is this now when I earlier when I did some work with Jessel Ross-Kennedy and

2:40:00 We first began by looking at a machine which was constructed at Imperial College by Dennis Gabor. Now this was a non-linear, this was a non-linear computer which essentially could employ filtering, prediction and simulation in terms of a learning process, okay, in order to solve problems, okay. It was a mapping machine. It was a mapping machine. It was a mapping machine. And we sort of looked at this, but linear, it was a non-linear mapping machine and apparently it looked like, you know, the computers that they looked at that time, it occupied a whole room and filled boxes, but it didn't work according to the same principles. And because computers at that time, the digital ones were in a sense, they were just as big but they were somewhat simpler in their construction and giving their results, nobody really took Gabor's machine seriously anymore, except Fatmi. Fatmi was a student of Gabor's and he kept up interest in that machine which led to a lot of rather unusual research at King's College. Okay? Now we discovered that in fact because Rosconi had a Rosconi had a theory which was about Huygens principle of secondary sources and you could, it was discovered that we could write down a new theory of machines based on