The Internal Logic of Causal Sets

0:00 Thank you. Thank you. Right. So I've given today's talk about the internal logic of causal sets, what the universe looks like from the inside. And basically that's what I'm trying to answer. And my universe happens to be a causal set, so the answer turns out to have to do with its internal logic. And I hope to explain what that is throughout. Here's my outline, which might not all make sense to you because several of these things I need to give the talk before I can tell you what they are. But basically, I'll first give you the motivation of why do I care about an internal viewpoint in a causal set, and the motivation comes from causal spinetworks and trying to build a quantum theory that has causality in it. Then I'll journal description is, and it involves what I call evolving past, how the past of a given event evolves throughout the causal set. And to give them, I need to generalize set theory and specify them by truth values, which will be time till truth values, and these are related to seed sense, and I need to go ahead and tell you what these are. Now, the result of doing this is that we have a way to express causality algebraically so the causal set is being transformed into a lattice you have concepts like the complement of the evolving past you have an algebra of such sets and the corresponding logic is this out of focus or okay and now the thing is that I'm I'm just going to talk about causal sets today, basically just a causal structure. And this technique I'm going to use is actually very powerful. For the causal set, they are about the simplest thing that you can use that explains to you what the basic structures are. But in many ways, it's too simple.

2:30 What I care about is the quantum theory eventually. So what will come up next is in some sense an overkill of the problem using this technique. So you should keep your mind open throughout what I say and try to think what might happen in the quantum versions of such things. I have a couple of transparencies in the end where I'm saying what we're looking at now, which is the quantum version, but it's all in progress and basically I don't even have very much time to go through a lot of that because the material is quite a lot. But what I'll try to mention about the use of this in the quantum theory is generalizing the eigenvalues of a projection operator to give a time-dependent spectrum. What does it mean to have evolving spaces of states? And an application of the same technique to relational quantum theory. So, as I said, this came out of looking at causal spin networks. and I guess you probably almost all have heard about what they are but just quickly it's a combination of spinetworks with causal sets I assume you all know what spinetworks are and anyway I don't use them in that talk what causal sets are is a discrete version of the causal structure of the Lorentzian space time so you have events P, Q and the causal relationship between the two and the structure of this thing is that it's a partially ordered set of events and the before Q and Q is before R, then P is before R. There are no time-like loops. You can go up and back to the same point. And it's locally finite, so between two events I only have a finite number of events. So what we do is that we combine these two things. We assume that spin networks in some sense represent the quantum tree geometry and causal sets the causal histories. And we get histories of spin networks where the evolution is given of graphs, of the SPI network graphs and their labels, and these local changes are subject to the causal fed ordering. And an overview of what is going on with the causal SPI networks and where the internal viewpoint I'm going to talk about today comes in is, well, we have the first idea of making causal histories out of spin networks. Then we found that histories allow bubble evolution,

5:00 that their generator moves, and that causality implies that there is no cross symmetry, as it happens with Euclidean spin networks. We've looked at the state space, we've defined that, and local operators on it. Samir and Ruman have looked at and got results for scaling of 2 plus 1 causal speed networks. There are discussions on how to get the continuum limit using non-equilibrium phenomena by Stuck-Aufman and Lee. And basically these two are part of a rather complicated issue which is how to get a continuum limit from this Planckian theory. There are stringian supersymmetric versions of it, little. And there is some related work which is one 1 plus 1 causal evolution, the stuff done by Jan Amjorn and Renat and all. And the main directions that we're, like all of these people or some of these people or together with doing other stuff, are looking at is one is building that causal quantum theory, and the other is getting a classical limit, because you have a theory defined at the Planck level, and the question is whether that would give you continuous space time, and so on. And both of these meet the internal viewpoint, which is what I'm going to talk about today. So how did I start thinking about this, which might not be the best way to go about it, So that's, anyway, the historical motivation, is that you usually have a space-time, and, canonically, you have a spatial slice in that space-time. And the idea is that in a quantum version of this, you might have a state representing the three geometry of the spatial slice. But then, who can possibly see that spatial slice? By construction, the answer is either an observer of the infinite future, so that all the information from the end can come up to him, whatever that means, or an observer outside the universe. The standard way we do that is actually that case, you assume there's an observer outside of the universe, and basically you treat the universe as a quantum mechanical system. So you might think that this is not totally satisfactory. And the physical situation you might have is finite time, physical observers, inside the universe, which have an inside viewpoint, what I call an internal viewpoint.

7:30 And unavoidably, they will only see a part of the spatial slice, what lies inside their causal path. So any physical observer, if you wanted to have such a description, would have a partial knowledge of the universe, and that would depend on the event that he is sitting on. And you don't really want to think about an observer like a human or anything like that. It doesn't matter. The point is that you would really eventually need to have a description of whatever quantum mechanical theory, whatever is supposed to represent your universe, where it's going to be with respect to some event inside it. Sort of. Now you might say that quantum mechanically, sorry, classically, this is fine, it's been accommodated by the theory that you have. So why should I care about whether the description is from inside or outside the universe? And basically the point is that I want to get a quantum theory that has causality, and not only for that reason, but basically whatever I try to do, I get stuck with the fact that the quantum theory makes certain assumptions. And roughly speaking, it only makes sense when the split between the system and the environment is fixed, and when the time is fixed. And I tend to use the word time stage because my time is not, of course, a parameter T in an R line. It's an event in the causal set which has some meaning with respect to the rest of the events of the causal set. So I use the word stage for giving a time in that way. And, I mean, you might also say that since there are such assumptions here, it might mean that you can't really use quantum theory or whatever, but my attitude at this talk is that this is too far-fetched, and I should just say that the problem is not that there are assumptions, it's that they do not appear explicitly in your theory. Generally, if assumptions in whatever physical theory you're making appear explicitly, you don't end up in trouble. You can tell what is going on. And basically what we're trying to do here is put them in the theory in an explicit manner

10:00 and hope that this will get us out of several problems that one encounters. And because of the technique I will use to do this, I will name the assumptions context. So here's the transparency that is supposed to tell you roughly what the material, the mathematical stuff that's following is trying to do. It might not all make sense right now, but we can come back in the end after I've done it and think about it. is a technique from categorical math to which I can give a physical interpretation of context dependence. So, what I wanted to get out of is having fixed assumptions and physical statements that are given only make sense for these fixed assumptions. So, in order to be able to accommodate all possible such assumptions for context, I have a category where the context sleeve, and a functor that goes into it and gives me the physical statements for these assumptions. It's the whole talk as an example. I mean, really. So an example is this. So here I care about statements that are event-dependent, they're time-dependent. said the only statement you can make, the only physical information encoded is whether something is in your past, in your future, or in either. Right? So a physical statement here will just be that, whether something is in my past or not. And the context will be at which event do I say this. Right? Make transparency, I say exactly what that is. Yeah. I mean, whether something is in my past or not depends on which event do I see when I think about it, about the question. Yeah? So what will end up being the case is that the context category will be the causal set itself of the collection of possible events where I can make such a statement. And the physical statement I'm looking at is a very simple one, which is just whether a point is in my, another event is in my past or not.

12:30 Now that other examples of such a thing is the application of the relational quantum theory that we're supposed to be working on with Carlo Rovelli, where the context category is all possible splits of a system and its environment. I give you the split and you tell me the quantum theory that makes some sense for that. Anyway, just the next transparency for what I just told Jorge. I learned about this sort of math from Chris Itham who first used this for consistent histories and his context was consistent set of histories or course gradings. So certain statements made sense at a certain level of course grading and not in another one. So here's the setup and what we're going to analyze for the rest of the talk. So a functor is a generalization of the content of a function. It is between two categories, and the idea is that it should preserve the structure of the first when you're going to the second. The categories in question is the causal set, which itself is a category. The objects are the events, P, Q, and R, and the errors are the cause of relations, whether P is before Q, and so on. The background to this talk is categorical, and I don't really think that you're supposed to know what categories are, but I'm going to use such basic things it doesn't really matter very much. Basically, what you have in a category is an object, is arrow, A, B, between two objects, A and B, in such a way that this arrow is associative, and there is a unit arrow from A to A. And you don't really care to think too much about what these things are. You can just think about what they are in my physical case that I apply them to. So the category that my founder will take me to is set. That's because the causal path of each event is the events in the causal set that is in its path. Yes, I have a given event P, and as it's passed, I've got all sorts of events, and this is a set.

15:00 So, I'm mapping from the causal set to the category of sets. And I use a function that I call past from C, the causal set to category of sets that does the following. It takes an event P into past of P, which is the events of the past of P, rather simple. Every time I have a causal relation, I can go from P to Q. up there, I get the path of Q and I have an inclusion map of the path of P into the path of Q, right? So every time P and Q are causally related, I get an inclusion of the smaller path into the bigger path. Fixed causal set, yes. In the causal set? P is before Q, causal relation. I mean, I'll have it, I think. It doesn't, not every two elements have an arrow between them. So the relation of the past should preserve is associativity, so I can go from P to Q to R to S, let's say, and it doesn't matter which order I go from the two end events in the causal set, this is preserved in the set that past is mapping to, since whether I go from the first little past to the last one, it doesn't matter in which order I take the middle ones out. And, obviously, there is a unit one, since for any event, its past is the same as its past. It's a nice, trivial mathematical statement. So, in all what these do, is here is a little causal set. And this function of past gives me a collection of sets of the events in the past of a given one. So here is six, and here is everything in the path of six, including six. And for each causal relation, I have one of these green arrows that are supposed to represent an inclusion. Now, the point is that the way to think about this functor is that you can think of the functor itself as a generalization of the concept of set. What it is, is a set that varies over the entire causal set.

17:30 Yes, given a point, it spits out a set. So there is a variation of sets throughout the causal set. And mathematicians tend to call these factors varying sets. Since my category here that I start from is something that represents causality, I call them evolving sets. And that is basically the key to how to treat these factors. need is a generalization of basic set theory in order to give such a factor like paths. So, how can I specify paths? Yeah, I hate to do these things. Oops. Now, let me think of the causal path of a single event. So here's a causal set. It's got a few points. And here is event P. And how do I give the set that represents the path of P? I do it by asking the question, is some event R, say here, in the path of P? The answer to this is either yes or no. Some mathematicians do the same thing by saying that the answer yes or no is isomorphic to 1 and 0. And what you really mean is that if an event is at the path of P, you stick 1 to it, and if it's not, you stick 0. So what you do by asking this question is that you get a map from the causal set to the set 1, 0. And this map is called the characteristic function. So the path of this little t being a subset of c is equivalently described by the characteristic function chi from the causal set to the set 1, 0. And the good thing about this is that given chi, I can find the causal path of P by taking the inverse image of one under chi. So basically just picking up the events that have the label one.

20:00 And now because one might think of one representing it is true that R is before P and zero representing it is false that R is before P, there is a name truth values to 1 and 0, and so the set 1, 0 is a set of truth values for the path of a single event. So this is standard set theory. Now the question is, does the same characteristic function and set of truth values work for the function of past? And no, it misses the point. The point is that is an evolving set, not just a collection of plain sets. It has a structure of the causal set inside it. So what you want is a characteristic function and a set of truth values that capture this. And here we go to the truth values for past. So the right question to ask for an evolving set is not whether R is or is not before P, will ever be passed and when this will happen. That can give meaning to R being outside the past of P now. So what you can say is that at stage P, here, when I'm here, R might or might not have occurred. If it hasn't occurred, there could be a later event, Q, after P, in which R has occurred. So one proposition is to assign this causal relation from P to Q as the truth value at P for R being passed. That would be pretty much like saying I can have a very simple causal set, which is just a line. Yes, 1, 2, 3, 4, 5, 6, and so on. And say I'm at 2. So at 2, I can ask you a question about whether 5 has happened or not. So is 5 passed? And instead of saying no, I will say 2 to 3, which means that it will be passed after 1, sorry, 2 to 5.

22:30 it will be passed after 3 time steps. Yes? So the good thing about this is that I can tell things that are partially true. I mean, we'll get back to that later. So the point is that P goes to Q is the time we need to wait, which could be forever, before R is included in the functor passed. So P goes to Q instead of it being called truth value for this obvious reason it's called the time till truth value. But you have to be careful. Now if R is passed after that time, it is also passed after any later time. That is, If this is a good R, so is this one that goes 2 to 6. And not only that, but it's not necessary that the set of events Q, which have in their past both P and R, might not start with a single event. So I might have the situation where P is here, R is there, but there are two events, q1 and q2, that are the first ones that see both. bit more careful about what the time-till-truth value is and we do this in the following way we define the set of all causal relations that start at p r of p these are all causal relations in the causal set that start at p then we define what is called a sieve on the event p which is a subset of all the possible causal relations, such that if P in Q is in the C's, and Q prime is a later event, then P to Q prime is also in the C's. So that means that just if 2 to 5 is a truth value and there is a 6 after 5, then 2 to 6 is also a truth value. And another way to say the same thing is that sp is a subset of rp that is less extendable since p to q prime is the composition of the original truth value with the extension from

25:00 q to q prime. So here is, here I've drawn the C for p and r. So you have to wait till Q that has both of them in the past, then P to Q is a true value, but so is everything else after it. And here's another Q that has both in their past. So all the red arrows are all the events in the future of P that know about R. Is that okay? I mean, I hate saying this. just too little. So the point is that C is a way to say the fact that what becomes true stays true. And as truth values, C can accommodate partial truths, point being that R being passed is partially true as P. We'll get back to that in the next transparency. So the set of truth values for past as P is a set of C's on P. Okay? That is basically the main point of using these things. A nice fact of life is that you can also have a map that given And the set of truth values at P, you can get the ones for any later event, which means that omega, this thing here, is also an evolving set, is also a functor from C to set. And the set of truth values at P is its component at P. This is essential for the whole construction to work, that the set of truth values is of the same nature as the functor you're trying to describe. this is when you were trying to just say that the set A is a subset of x, and instead you had the characteristic function from this to 1, 0, it's essential that the set of truth values is also a set, like what you're trying to describe. And it also works out here. The the truest values is also a functor, like the other that you're trying to describe. Now, there's an important sieve that lives in here, which is the largest one, rp itself.

27:30 Yes? Because all the causal relations that do start at p is also satisfied as properties. It's just the larger subset, rp itself. Yes? So there is a largest sieve in here, is totally true at P, which just means the following. Here we're about to use omega to find the path of P. And as I was saying before, the whole machinery, it is quite an overkill of this problem here yes so you're supposed to try to understand how the mechanism works so in the example i had the previous page that r is passed at p here r is not before p but it becomes passed later so that situation gets a safe truth value that says that R is past is partially true at P. Now, past P itself contains all the events that R before P. For them, S is past for these events is totally true at P, in the sense that it is true at P and at all other events in the causal set later than P. And these are obtained by the errors in the largest Cs, the whole causal relations. That simply says that all these events S here are true for any event later than P. So past P is defined using the truth value RP, which is the largest Cs, and that generally, even if you generalize and you go outside the causal set that I'm treating here, is always the case. What is true now is the largest Cs. So, where we had this example here for set, and we could say that A contains those elements in X that map to 1 under the characteristic function, the past P can be derived by all the elements in C with the characteristic function for past P now being the maximal C's, the largest C's. So, we do have the characteristic function for past P.

30:00 Here I'm doing something which is rather funny, which is I'm trying to tell you how the whole thing works without showing the diagrams that are supposed to explain exactly how this goes. Because I mean I've got like two transparencies here if you think that it gets too much if I'm just trying to explain how it works, but I assume that you don't really want to see categorical diagrams right now and you'd rather know what it all means. So if it really gets problematic, I can just tell you exactly how it works. Okay, whatever that was, now we get into something more interesting. We've said when R is in path P. The question is, when is R not in path P? Now, for a set, you have A subset of X, something is not innate when it's in the complement of A. So what we are really asking is, what is the complement of past P? Now, but this does not work in the way you would expect. For the evolving set complement, the point with evolving set is that you ask when something will become true. So, the events that are not in the past of P are those that will never join the past of the future of P. Right. So, this is not the complement. It will eventually become past. So, the complement of past P is the events in the causal set such that R is not in the past q for any q later than p. And here I have an example of the causal set I had before. I start at event seven and I can calculate what is the complement of seven. So what I need to do is go all the way to the future, like six, find what is in the past of six. All these things are not in the past of 7. What is there is 3, 4, 11, 10, and 12, and it's the stuff that will never have an arrow from 8 to the future of 7. Yes? So 7 will never receive information from events in its complement. And I might say that this means that the events in the complement of a given event are beyond its causal horizon and I can try to do

32:30 things with it which I'm just speculating since I haven't fully worked it out and say that the complement of the path of a given event is events beyond its causal horizon now note that this is an event dependent horizon and not a global But what I really care about is a global property of the causal set. So could I use this complement to find such a thing, for example, a black hole? You could imagine that a black hole is a collection of events that never go into path, so they would, in some sense, be in the complement. Now, this is tricky because I need to know how to distinguish between branching and singularities because even in the continuum case, I can be here, and I will never get information from there unless there's a hole, and it happens up there. But this is, so all these things are events that will not send information to me, but that's not the same thing as a singularity. So, there is some problem here that we are trying to solve with Samir. the point is that you expect that such a global property of a causal set would come when there is a part of a complement that a large class of events in C will agree on. The point is that you can't have all events agreeing because the causal set has, you allow highly non-trivial causal structure, you're not seeing as a causal set that corresponds just to Minkowski's based on. And the good thing about this is that computationally this is much better than a row causal set. And I have to admit that it doesn't exactly mean that this is easy, it just means that the row causal set is pretty impossible to use. So using sets instead is much better. And now if the case with the row causal set is when I would just give you bag of events and causal relations, and I tell you, can you tell me if there's a black hole there? And we've tried to see whether there's any criterion of any kind that you could use that, but, I mean, it's pretty impossible. If it's any large number of causal relations, then it's a mess.

35:00 Now, what is better is when I'm instead looking for past or not past and so on, and then I can look for all the laps of these things. Basically, the point is that I'm turning the causal set into a lattice this way. And I can treat these things in a much easier fashion, hopefully. Well, in some sense you have it here because you need to have the entire causal set to answer this question. Yeah, no, I understand what you mean and that is in some sense what we are quite confused about. Well, no, in some sense you do have the black hole from infinity because the point is that to ask what is not in your past, you need to go all the way up to the end to see what did not join your past. And that seems to be an equivalent. You can't get anything better than that in this. If I give you a space frame, it's a space frame. Right. Yeah. Just with a congenial. Can you tell me what it's like for you? Well, I've got... No, I can't say that many things. I can say a different, well, somewhat different thing, which is the following, that if I'm If I have a causal set which is a lattice, in that I have an initial event and a final event, then the complement is empty for all events in it, yes. Eventually you will see everything if you wait long enough. So but that doesn't see anything of what you mentioned. This just sees the fact that there is a big bang and a final crunch, yes. The rest I just don't seem to be able to see. But it's a unique big bang. Yes, a unique big bang. Yeah, if you have two or three of them, then you might not see things. But that's what corresponds to your branches. Right, sure. So, I mean, if I understand what you're saying, you're saying you're not interested in the case where there is a tribe, that corresponds to something, having to plan it. Yeah. So, if you were interested in that, then you might be interested in the context in which the universe is something like space is closed. Is there a notion of some region becoming causally isolated from other regions?

37:30 Right, but, yeah. But, I mean, to be completely honest, I'm only interested in that because that's the only thing I could possibly do. Well, I'm not so sure what I would mean by the universe being spatially compact in that in my causal set, that's my problem. I mean basically I only have one criterion, which which is where, if I wait long enough, I see things. So, I think that's an equivalent situation. You have no preferred set of events that you call, which you call a space for a 50-year-old. Okay. Now, you may have a sort of a problem, you may have a, you know, you may have a black point, which is playing something like the role of it. Okay. But that's something like your timeline. Well, I mean, I don't know. That's the answer. I'm not saying that it's not possible to solve it. Yeah. Yeah. I mean, basically, the problem is that if it is a black hole as opposed to branching, there should be something about it that makes it a black hole, so stuff goes there. So, basically, there is a form of asymmetry between a branching and a black hole. Yes, because the branching here, I could just look at the same thing from the other side and symmetric as a problem. The black hole would not be like that. And I hope that I can say that better, but I can't right now. So what you can do is you can tell when there's a branching. I haven't done it, so I'm not going to claim I can do anything. I'm just saying what I would like to do. You can tell when there's not a branching. No, I can tell that I need to be able to tell when there's a branching and when there's not a branching, that's all. No, no, no. It's...

40:00 I mean, I... Yeah. If there's a single first event, whatever that means for you. I mean, to be able to say more things, basically a causal set does not, cannot incur very much. Okay? I mean, I need properties about the geometry and stuff like that to say more. And maybe the point is that I can't really answer this unless I stick three networks or whatever else you want spaces in here so that I can tell about the geometry and the features. This is very likely. As it is, that's all it can do. But of course, in a sense, when you have the whole space with all the points around the file tensor is in the positive structure. Yeah. If you see a single function is the class performance, the class is a big guy, that's quite different. Yeah, yeah, I mean, I can't... Sure. Well, yeah, yeah, sure. Yeah. I mean, I'm really calling them infimum and supremum in the sense of a lattice. Basically, my entire point here is that a causal set as it stands is not great. You can't do very much. And in this causal speed network approach, we are supposed to be able to use it. So that's the main thing that we're trying to address. Now, when you go from the causal set to the collection of paths at every point, you have transformed from the causal set to a lattice. A lattice is a much better thing to use. You can think about intersections, union, and so on, as we are going to do right after this. And that might be better. Can you think of the word lattice number is like that? Can you think of the lattice? that for that given P and Q, I've got a least upper bound, I've got a greatest lower bound, which basically there is a last event in their common past and a first event in their common future and I assume that there is also a supremum and an infimum in that there is an event here that is the least upper bound of all others and it's an event at the bottom that is the greatest

42:30 lower bound of all others. So that says certain things about the causal structure that I have and with using this machinery that turns out to be basically trivial causal structure I just can't about this thing. So I can, yes? Right, so I can tell when my space time is a lattice, yes, because all the complements come out to be trivial. So that's one thing I can say. Now, I mean basically the whole, supposed to work for space times or causal structures that are more complicated than the basic ones that we usually use that are lattices, okay? It just can't tell between one lattice and another lattice because they're all trivial for it. Yes? But there is, I mean the point is that the standard space time is a continuum space time and it's what I'm supposed to get at the continuum level when I've got there. I want to use this causal structure microscopically so there's just no reason it should look like this. That's the assumption. Okay. Now, all this stuff about the complement and so on fits into the fact that evolving sets without complement is part of an algebra and hence an underlying logic of such evolving sets. And that's a generalization of what happens for standard set which are those that have two values one zero and you have in the standard way if x is in x is also in a is one if x in x is not in a zero and all the possible subsets of x obey the standard set operations such as union when something is both intersection one or the other intersection is both implication is those that if they are in A they should also be in B and the complement in the standard way so the subset, the power set of a given set obeys the Boolean algebra now the characteristic of the Boolean algebra

45:00 is the complement not A intersection A is empty, and NOT A union A is the whole set that you started with. And thus NOT NOT A, the complement of the complement, is the original set. Here. Now that means that underlying this set algebra, you have a boolean logic. And for two statements, the negation of your statement and the statement statement is false, and negation of your statement or your original one is true, and so not-not-something is the same as saying the original thing. Yes, that I assume that we all know since that's what we always use. Now, in the evolving set, this is different, and I guess here I have a transparency for what we've already said, which is the case where the causal set is a lattice, and then the past of a given event is smaller than the causal set, but its complement is empty. So while the past intersectional with its complement is empty, the past union with its complement is not the original past set. And therefore, the complement of the complement is smaller, is bigger than the original one. Yes? So this is part of the fact that evolving sets, which are founders from the causal set to set, form a category which is usually called set C, all possible founders from C to set, and given some factor in set C, all its subfunctors, which is the equivalent of its power set, obey a hiding algebra. Now here I've got what is a subfunctor, but I might just as well skip it since I want to say different things. And see that the hiding algebra also has these four operations, the union, the intersection, the implication, and the negation. And the first two are the same as in the Boolean case. But the implication and the negation have evolving versions. And we have seen the case of the complement.

47:30 So what we have done is to change from the causal set into a lattice, and when you have a lattice, as happens with a Boolean in the hiding case, you can get an algebra, and that means that you have an underlying logic. So you must include the following from this, that the standard way is to have a physical theory that is based on set theory, and so the underlying logic of any physical statements you might make on something that you have used set theory for is Boolean. Now, if you want to have this internal description in a causal set, which would let me to use this evolving set, then I get a hiding algebra, and interference means that when I want to take the temporal dependence into account by this method, that is, using such functions, then the underlying logic will not be Boolean, but intuitionistic, which is, yes? You thought that claiming that not X is not equal to X is an intuitive thing. Yeah, right. That's the same question I always get. Yeah, right. So this is a very counterintuitive intuitionistic logic that we seem to encounter in trying to use the temporal context in this way. So infusionistic logic is a bizarre part of mathematics. It's not so bizarre, but I certainly do not want to advocate that. But basically what it says is that not, not X is not the same as X. And it comes from constructive mathematics, where there you can understand what this word means. Constructive means that you should construct whatever proof you have. So you do not allow any proof by contradiction. So you can't say for all X, either X or not X, which is a standard way of proving things. prove or disprove x because x may be true tomorrow or false tomorrow, which is basically the same idea we're using with whether r is in the path of p or not. And as far as I can tell, that makes the life of a mathematician pretty difficult, but it has been very fruitfully used outside this original context for in the applications like I did here to specify functors of certain types.

50:00 And that really does not relate to the original motivation, it's just a good method. Anyway, that's where it comes from. Now what I want to do here is give a quick summary of what we're supposed to have understood at this stage. And then I've got, I guess, three transparencies on quantum versions of stuff, which I hope to mention. Classical. Oh, classical, yes. So, summarizing, the internal point that Kozl says, which is what I started trying to say that I wanted to have, which is the end that a physical observer might have, can be is this functor from the causal set to set. The point about using a functor is a basic, I'll just return to what we were saying earlier, where I had my context and my physical statement, and the deal is that I can have P and parts of P if I've got this functor passed, and And then I can go from P to Q, and this tells me exactly how to go to past Q, yes? So this was my context part, and the point is that I can change context, and my funter makes sure that my statements are changed accordingly. This is really a very simple application of it. You're supposed to do much better things with it. Anyway, the internal viewpoints, that is the causal past, are described in terms of time-till-truth values, which are Cs. The point here is that you want, at each event, past P is a subset of C. And the good way, the solid way to say that past P is a subset of C is using the characteristic function. To do this, you need truth values, and we calculated what these were. Now, thieves is a way to specify time that can be used in quantum theory. And that's a very interesting thing, that the point is that once you try to think about the causal set and so on,

52:30 you've lost t as a parameter in an hour line and so on. And now you have a problem of how am I supposed to tell you that something has happened at time of t and so on. So, and just giving you an event, it's not so great. Basically, it turns out that Cs are much better because they give you the event and everything else in its future. And because of the way that this is part of the entire causal set, it turns out to be much more solid in what I'm trying to do next. Anyway, I'm just saying that Cs are great things and I really like them. The causal paths of all events in C form a lattice even if the underlying causal set is not one. And that's where we hope that maybe we can have better control of the causal set. And evolving sets also form a lattice between themselves. So I could have this functor past and I can also have a functor future. Yes, with all the events of the future at each point. Or events with special properties like the events in my past that made me happy or something like that. a factor, and all these, the one with the other, form a hiding algebra. Well, sub-whatever is of the original thing, power set. So a hiding algebra has a non-standard complement we looked at, and apparently it caused much of the causal structure of C in that it misses completely the difference between one lattice and another and so on, so it depends whether you like this or not, I do. And so if the causal set is a lattice and the complement is empty, as we said, and importantly, this is done for a fixed causal set. Yes? So I probably have to say that this is a kinematical description of what I'm trying to say. It's an internal viewpoint description that is kinematical. A dynamical version would need to change this. I need a causal set that is not fixed, presumably, and this is not done this technique. Okay. Now, what should I try to say? So there are basically a couple of things, well, more than a couple, that we're trying to do in the quantum version of this. It is all in progress, so I'm not really going to report concrete results. I'm just going to tell you roughly what are you supposed to understand that you can do next and hope

55:00 ideas, or tell me that what I'm doing is stupid, or try to work it out, or whatever. So, the causal quantum theory version would be where you would like to have quantum states, or space to spin networks, or whatever, in the past of the events in the causal set. So you don't just want to have bare, empty, no-information-containing events in the causal So what you might try to do, and it has been discussed occasionally by people without, Hilbert spaces to each event, and then you would have a functor, instead of going from C to Z, it would be going to Hilbert spaces, in such a way that if you go to a later event, then you might have a linear map from the Hilbert space of P to the Hilbert space of Q, and you might think that if P and R are space-like separated and they're both going to Q, what you need is to go from the tensor product of the 2 to the new, later, Hilbert space. Now, that's problematic. The main problem is that the whole philosophy behind what we're trying to use is that we need the concept of an object and a sub-object, which is just the generalization in categorical terms of a subset and a set. And the problem here, of course, is that HP is not in any natural way subspace of HP tensor HR. And basically this naive version of sticking Hilda spaces on each event does not work. That is not exactly a problem, and you might argue that it shouldn't work from the start. And basically there are loads of ways to go around it. I'm currently, and a couple of other more here, trying to sort out which one should you use for certain things and what do you want to do with it. So I suggested that instead of mapping to Hilbert's spaces, you should map the C-star algebras and say that you have your observable paths and algebras of observables for each event, which is a nice thing, and he's supposed to tell me how to do that. Another possibility, and it generally tends to work much better, is to work with projection operators. It depends on what you want to observe. It depends on what you want to observe. Yeah, it depends on what you want to observe. So maybe you have some quantum fields living on the cargo set,

57:30 and so at each point you have the algebra generated by creation and annihilation operators at that event. This tends to do a quality of theory for existence. Okay, another thing that is also not fully done, so I'll just tell you how, what the idea is, is to relate projection operators and C's. And the idea is that the standard spectrum of a projection operator is 1 and 0. And, of course, you get this because you say that the projection operator is characterized by if you measure again, you get the same value. But then that's the same thing that we got with thieves. Once something happened, it kept being true. So we went, like, from one zero true values by what we understand as relativity, which is just the causal third, to thieves on events, the internal viewpoint of causal set, I should say, here. Now, 1 and 0 in quantum theory appears as a spectrum of a projection operator. So what you would like to think about is whether you could have projection operators that are now internal in the functorial way we had before, whose eigenvalues are C's. Well, this is supposed to be an argument, while something like that might make sense, hardly make sense. So, basically, roughly the point is that you could imagine if P and Q is the complete path of R, then you could have an evolution operator, E, from the tensor products of the two to the last one. Now, if you project down to a subspace of HP by some projection operator like P here, so you go from HP tensor HQ to SP tensor HQ, and then you evolve this, then what you expect is there should be a projection operator from HR to SR going from this one to a linear subspace, and that you could find this from an equation of that type. Basically, the idea would be that projecting down to a subspace persists in time.

1:00:00 Now, while this might be a reasonable thing to expect, that a projection is essentially something that you could evolve in a sense, this way to do it turns out to be too naive. You can't really push this diagram too much. So there's a way to do this, which is suggested by the, basically by the math I'm using. And it's along those lines that the 1 and 0 eigenvalues come from the fact that projection operators satisfy P squared equals P. And so if the eigenvalue of P is A, then you have this equation to solve. And clearly, this is, oops, it's missing something. It's supposed to say here that A is 0 and 1 when A is either a real or a complex number. But now what I want to do is an internal version of P. And when I, to do that, I would go from set, which is what we usually use here, to set C category. And then what is a real or apparently, changes when you change category. And you do get a different, a larger spectrum. And in very simple case, I can check that this is a C, but I do need to do a full causal set, and basically I'm sure that it looks completely nuts to you, so I have to come back, hopefully next semester, and tell you how it really works and what the infused interpretation is. That will accomplish the number of changes. What on first does it change to? Well, you have an internal definition of what a real number or a complex number is, that, I mean, as most things... I'm just going to go back to hear that question again. A real complex number of changes. What on earth does it change to? Well, you have an internal definition of what a real number or a complex number is But, I mean, as most things in categories, it's defined in the most abstract way you could imagine. But basically, what, the point is that when you go into a different category, you should

1:02:30 say what a real number is there. It could change to things like continuum functions and so on. So, for example, could become, well, because real number of how they mean there's some field of numbers we can add in multiple. Right. So, is it still even a field or, like, functions that you can do? Yeah, sort of. I mean, I really need to work this out to come back to you with whatever. I mean, they really do it in a very different way, which eventually you can understand, like, with what else I said here, but right now it's not totally transparent to me. I can tell it changes, and I've seen examples where it changes into continuing functions, but then you do things with. Or in lots of curves, I mean, I can get 1-0 to turn into 1-0, cross 1-0, and so on. But, I mean, I really need to finish this. Yes, exactly. Shall I just tell one last thing? Okay. The last thing is applying the same idea to something where my context is not causality anymore, but Carla's relational quantum theory, which we're supposed to be doing with Carla, And I don't want here to advocate that you should believe in relational quantum theory, which I should tell you in a second what it's supposed to say very roughly, but I'm just saying that you can just apply this technique and turn it into something that you can use and decide then whether to believe it or not. And what he says is that you can have an electron, an apparatus looking at the electron, and then assuming that the initial state of the apparatus is on, operator in the John Hilbert space that would take me from the initial state of the spin and that of the apparatus to some later state. In the standard way if I have a superposition of possible spins I will get a superposition of up electron spin and the apparatus seeing up plus down electron spin and the apparatus seeing down and if the electron happens to be in the definite state then I should and yet my apparatus is giving me a good measurement. And it's important here to say that I assume that A is a small system, so it should have a small number of states. So I'm not considering something where A is very big and you should be treated as classical and so on.

1:05:00 Now, you could imagine somebody like me looking at the whole thing, and if I'm going to consistently say that A, the apparatus, measured a certain spin of the electron, then I did a new interaction operator in the Hilbert space of all three of us. So you had better not give me one of the collapsed states because I won't be able to use it. So the question that is raised in the relational quantum theory is when are you going to go from generally a state in the tensor product to a particular one, and when does it make sense? So what do you need to do here? the idea about how to apply the previous stuff here is that all this makes sense if you tell me which system is supposed to say what about which other system so basically i have all possible splits between me and the two things or the two things between themselves or you looking at me looking at them and so on and so my context here is the human spaces of all possible systems and subsystems so the apparatus looking at the electron me looking at the two of them or each one And this is, just because of systems and subsystems, basically it's a partially ordered set. So I'm back to the same deal, where I would go from a partially ordered set of all possible systems and subsystems to measurements that I can possibly make. And when you want to check whether this technique of using these factors would work, what you can do, the first thing that you do to see if you have a hope, is to see where the seeds will come from. Because what's characteristic is seeds, that something is true and then it stays true. And here if you think about it, where they come from, is that if you here see me saying that the electron is up, then you know that the apparatus showed up and that the electron is up. Yes? So if you go inside the subsystem, it stays in what you initially so basically you can if you have projected a particular state then you can imagine that everybody else down has projected a particular one so that's where the thesis first come from and i'm also supposed to actually finish doing this but basically i'm just trying to say that you can this is a rather powerful technique that once you understand how to do it which takes you ages then you can do all sorts of things with it i should also say that i'm not saying that

1:07:30 this is going to solve the problems of quantum theory for you. What I'm saying is that there are often confusions and ambiguities and stuff like that that come from the fact that you have not said when you do certain things, what your assumptions were. And this is a way to accommodate them automatically and not having to worry about them. And I would hope that this can solve things for you or make you understand what's going on. Thank When you did your quantum mechanics, you were right, so using the Hilbert spaces or algebra observables. Right. And the other stuff you were doing was, you know, more lines of logic and synthesis. Did you think of trying to use sort of algebra?