Quantising on a category - new approach to quantum spacetime

0:00 I don't mean by this to talk about problematizing electric tensors or anything like that. I mean at a slightly more fundamental level. People, for example, talk about quantum physics, of course, but I'm not trying to speak for all of you. What does it really mean? What does it really mean? There are several connotations to it. It can of course mean, what does it mean to be a philosopher, i.e. what is the concept, the concept that I'm using? Or it can mean, what is the appropriate theoretical structure? Now, my view on these things has always been that before you can really talk about concepts in philosophy or physics, you really need an underlying theoretical structure to talk about them. I only say this because I realise this is a philosophy or physics meeting, so I apologise anyway. Because, in fact, I'm not going to do a philosophy talk, I'm going to do a physics talk. However, I'm very much aware, a few people are very much aware, of discussions on quantising space, and space is quite a doable way to account for some of the things I've said in my previous lectures. Apologies if you were. So let me start off by giving you a few motivational questions, which I sort of think might not necessarily be interesting. Let's suppose for a moment that we're working for something plausible, so we actually space and time, but where time is that sort of continuous, usual, real parameter, but space is sometimes quantized. I mean, this is actually what is done in the context of all the mathematical questions. But I mean, like I said, what's on the country level? So what is meant by quantized topologies? So people talk about it, they talk about topological positions, one topological space result, and one three-dimensional manifold result in the context of quantum gravity. So what does this really mean? What I mean by this is, what is the reason that prep-mathematical structure is talking about? Now, one of the instincts, at least I think as your physicist, is that when we talk about quantizing something, you tend to think of wave functions defined on them if you're interested in them. If you're interested in actually values of fields, you might decide to try, and indeed you might want to do something like that. So Berghove, if you're interested in the concept of topology, presumably you might have some talk. I mean, Mr. Business Induction is true, in one case, it's always true, it's very useful.

2:30 So it's true that Cybeck said, presumably it's true, it's actually true. So you can try doing that, you might think, what's what you mean? Maybe a weight function, a bunch of topology, in some sense, a bunch of business. It's not really justified. I mean, now from a point of view, is the physics law of induction really describing the internal degrees of association? So if you have a sine Q in general, Q-sum, the configuration itself, whatever it is, has some internal degrees of freedom. It has a frequency of constant. If you represent those with a constant, then you will find that in effect your state vector has to be vector based on that. And those extra degrees of freedom in the state vector are allowing you to describe the conservation of the internal degrees of freedom. Here we have an example that's not passed on my mind, so you see this is actually true of the metric of mathematics. Although we are used to writing psi over 3g and psi over 3geometry, it's quite plausible to argue that actually this vector should have internal indexes, which is associated internally with these three metrics. This is never done. Anyway, that's by the way. OK, so I'll show you that in a second. Now, so to agree with this is really the question, what actually are the basic quantum observables? If you actually ask where does the side x really come from in all the wave mechanics, why do you like the side x actually if you're quantizing the part that works for your mind, there's a dozen ways you can answer that, but one way, one very sophisticated way of answering it is when it comes to representational problems and computationally related to the fact that the operators x and p, when it's like x, p, y, h, and r, mentally it's actually a big thing in the case of non-dimension. So the wave function is the representation. So that basic algebra. Quantum observance is in a way what leads to the most aware functions of all the quantum mechanics.

5:00 Now think about the basic algebra. This is not all possible observance, of course. It's just that it's the key. This is one of the fundamental things about quantization, is if you cannot represent quantum mechanicality, all the things you can represent classically are different. So you have to find some special algebra for observance. So it's a quite a good question to ask, then, what are the basics of quantum observance? And if you knew what they were, if they satisfied some sort of algebra, then you could legitimately say, well, quantization means finding representations by algebra. But it might be that some of those representations are functions of a coordinate. It's not something that can turn superhuman. You know, play it late, it's a software. That's the way to do it, by the way, you know, to get it to the ground. It's something I've been trying to do, which is to try to develop enough out of the structure for doing this. But not just as topology, in fact, but much wider, with lots of things, as you'll see. Here's now an example, physically addressing the software. I talked about the canonical sense of scare quotes there. I was talking about physical space. People these days also, in the context of quantum gravity, often look at causal sense as a model of space-time. Now causal sense is a discrete model of space-time, I should add. Does this thing work? Well, I ask, but it won't be true in college. So, a causal sense is something like this. We have a series of points. Each one of these is supposed to represent a point in space-time, whatever that might be. And the arrow that appears here simply decays for some causal relation from here to here. So if something happens here, it can communicate with that point, maybe it can't communicate with that point. And most of Penrose has been much of this in his career, I must say. So, there's a lot of discussion of things these days, you know, there's context, and that's a very simple problem to set just two points. Here's a simple space-time that can be measured, just a simple point. So, if you have such models of space-time, again, the kind of implication is that you can quantise them. That is to say, you can have whatever you'd like, many quantum fluctuations from this to this. Now, let me be careful about this. This is the model of space-time. So if I have quantum fluctuations, it's not something that happens in time. Each one of these is a model of space-time. So you cannot, in principle, discuss this using anything like mathematical quantisation. As a matter of fact, you can't discuss it at all using conventional quantisation. There's quite a lot that gets overlooked, I think, but it actually doesn't make any sense to talk about it. However, it does make sense in the context of a quantum history theory, and there are such things that can be played, although people argue that they really don't make sense unless they do exist.

7:30 So in the context of a quantum history theory, it is mean to talk about quantizing forces of sex, so that in some way we have quantum superpositions as history states. So you might have a superposition, a supercausal set. What does that mean? Now there you do get deep conceptual quality, and if I tell you the quantum state of the whole universe history-wise, so there's a superposition, say, two wider states, a causality. What does that mean? It's a representation of time. So I can ask the same sorts of questions here, by asking the context of the quantum apology. Now, in the use of history, it's like doing that. So once again, I mean, I can ask, would you write a slide of C? Again, this is induction, please. I think it works as a part of the movement of both sides, and it does work as a part of the set, and of course the meaning of these states is different, I must emphasise, because they're not canonical states, they're history states. So you can think about the history of something like that. So, apart from that, the mathematics of the state is canonical. So, would you write this? Well, you could do, I suppose. It's obviously quite damp, but again, is it correct? What is justification? And again, one might ask, well, should this be better spaced out here? After all, it also does have internal degrees of freedom. I mean, apart from anything else, you can see that, because I can commute some of these things around. For example, I've got this structure. These points are labeled. I can kind of flip it around. There's a group of transformations that acts on this thing. What might go in that? This should represent the consequences. Now, it won't work if you've just got Psi or C. This is unlikely to be a much natural way of doing that, is to have again a thing that takes its value from vectors away. So perhaps you want to do that. And again, you can ask therefore, well, how do we find these slave vectors and so on? And the answer is the same as in the canonical case, that actually, you can find an algebra of what you might call physical observables, or variables, whatever your philosophical budgets are. Then representations of algebra will keep the association constant, and again maybe some of them will be the form psi of c, maybe some of them won't be the form psi of c, let's try to see. Of course in this case, like I said, there is this particularly interesting conceptual question from the class of Picasso. So what do you mean by this? I'm just saying that perhaps you like the states of cause and sex. You'll see later in my talk that I actually did this by myself. What does this mean? I mean, the state that sits in there, what does it mean? What's the motion of time?

10:00 I mean, I'd be much interested in the problem of time, because here you see it absolutely, crashingly clear. Well, it doesn't mean the answer's crashingly clear. What is crashingly clear? Okay, so that's what I want to do. That's my motivation. But now I have to see. Now, let me introduce here a very short discussion of what you do actually do when you do know what's going on. There's a kind of analogue of all of this, and that's where you have a system whose configuration space, q, is not just a real line, it's a manifold, like 2-sphere, or something like that. Now there's a very well-known, well several well-known techniques for contrasting this. So, for example, if q is a two-dimensional sphere, if you have such a system, for some interesting reason, you're just part of a room on the surface of a 2-sphere, once it's moved to the sphere, there's a very well-known answer to that. But it's based on the fact that this is actually a homogeneous space. It's the quotient space of SO3 over SO2. In other words, SO3 acts on the two spheres. It's a group of transformations, but it leaves certain points fixed. Any group in the interaction leaves a certain axis fixed, and that's where the SO2 comes from. So it's a very important idea that you have a group that acts on the two spheres, and this group is the analogue of momentum. In ordinary quantum theory, a whole breed of physics, if you have a powerful movement on a straight line, you know, both classically and fundamentally, you think of momentum as the generator of translation, and momentum takes you to one configuration point from another. That's the key point. Mathematically, I think, momentum does in quantum physics. So the analog of that here, this is the generators of the best-known query, represent momentum, the analog of momentum for all this theory. And there are three of them. So basically there are three fundamentally independent momentum variables on this issue. So this can be generalised to any situation where G is in the form of G is for a page, and you can think of G as the analogue of a document. If, by any chance, you have a manifold which is not of this form, similar to that of the original, it just doesn't have to be in the quotient space of two or three groups, you can always find it in the four-digit report. Now if you do that, the quantization of this system is actually fairly straightforward. What you basically do is, first of all, you say, right, we represent the group G, so it's not unitary.

12:30 Unitary representation is G, so we've got to represent something else. So that's like representation and translation. But of course, always in physics, you have to do that as a momentum, you have configuration variables too, I mean, this is that of the x's, so in ordinary quantum physics, we have x, x squared, 27x cubed, the fourth, 9x, the fourth, et cetera, okay, whatever, functions of x, which is also what you said. Certainly, this applies in general, but if you have any function of q, say it's the root function of the configuration space, that of x is x squared, so I'm going to represent that too, and what you want to do is represent the additive structure. In the case of all the quantum mechanics, what that would actually correspond to is e to the i of a, say, x plus x cubed, and then you get the... So you get that. And then there's the question of how the two subgroups seem to try to be right, because you've got the root of G and you've got the derivative of Q. Now what you find as you work it out is you get this result, that U of G times U of beta is not V of beta times U of G. It's kind of twisted. And this beta is not tau of G, because tau of G is simply the name of the action of G on the Q. So what this actually does is it translates the beta before you actually do the beta, so you get that translated. And the ordinary quantum computation data can be written exactly in this form. If you are e to the i a x, this is a number for b. If e to the i b, p hat, this is a number for b. You can use them to find out something exactly like that. All this corresponds to the integral semi-direct class, I've just recorded. Now, in the case of all quantum mechanics, this would play just fine. And its representation is given uniquely by the famous weight country of representation, quantum mechanics, which we're familiar with. Applied to the case of Q, then you're making a whole host of representations, actually, as algebra. And each one of those is an inequivalent quantization of the system. So you see, because you've got an algebra, you can actually classify all possible quantizations. So you see, the number one problem for me is, well, so where's an analogue of this? The cause and sex, which, apologies, sounds a bit way out at first, but it's not. Well, it's just the risk. Now, a various representation structure is obviously, one obvious representation is the one that some mystics tell us you can just take away functions. So you write the functions of Q and you find immediately there are representations of them. Just like this, it's a very, very familiar, if you've seen it before, it's amazing.

15:00 Again, just what all the way the country looks like, written in a slightly snazzy form, and for a general configuration, it's very cheap. However, it turns out that there are other representations where in fact the weight function is not just confidence value, it's vector space, and this does not happen in the form of the weight of the chemist, but it does happen in general. It happens in two-sphere, it happens in three-sphere form, but you do have a vector space B. And this is carrying the new additional representation of H, so when H comes into the sequence point, let's start with H, it comes in here as the axis of the vector space, which is the target space. And if the representation looks like this, then it's just a couple of individuals, and the only thing that you have, which is different from this, is you have this thing called a modifier, which is an operator, and you can show all representations of this type, all of them including this form. So this is just battery. So what I'm saying is that if your configuration space is a manifold, you know what to do. And that would actually be a history theory, where the history is all the manifold. You could also do the same thing for the history of the system, history of the system, see what else can go on. But of course, these are the same exceptional causal sets of technologies that are far more in control as you can see in mathematics. So this is nothing to be concerned about. Right, so what are the underlying causal sets? Well, what is it? Well, as I say, momentum really can be thought of as generating, well, generating, momentum translates you across the configurations. So, you might think there's an amount here, a lot more precisely, without the question, what is the answer to this? I've got a collection of topologies that some of them are taking from each other. Or, I may be using this up for your course of sex, because I think they're a bit more physical. Why don't you like, I've got a very simple, four possible models for space and time. This is not very well priced, all of a sudden, is it? Nonetheless, four possible, single point, two points in seconds. The configuration space here is certainly these four points, and that's why I'm reading this screen at quite a number of points. What about momentum? How do I get from one of these to another? Now, we're certainly not in the lead group of acts, I'm sure. In fact, we're not a group of acts at all. What does act? How do you get from here to here? What relates this to this? Mathematics. What do we do? What's it done with? Now, I've talked a lot about this, and I've come to the conclusion that there's an obvious answer, which is actually...

17:30 Protector or preserving maps. What relates one causal set to another? It's all the ways you can map from here to here that preserve the causal structure. But this is what I'm postulating. After all, a function is the function between two sections. It looks like a metric. I call it a function. Well, I meant to say it because it's mapping that to that. So, being a physicist, I'm reading that to get started, but it looks like a metric. But of course, there are many ways you can do it. For example, you can put this in there like that, like that, like that, like that, lots of ways you can do it. On the other hand, you can't flip it around, that would violate the causal. So, what I'm going to explain is that in the case of causal sets, one can actually develop a four-pronged scheme, a four-pronged function scheme, which is just as precise as an elementary wave, and it's actually a very particular theory, at least a particular concept, or any vector in mathematics, where what you're focused on is what the function is, which is the main idea. So you transform one corpus set to another by using all-or-none-preserving maps. It's very un-augmented in some way. Because in the topological case also, what you would do, you would say, well, now there is continuous maps. So you have a pair of topological spaces, and you ask, well, how do you get from one to the other? Now, it's the way you think of all the continuous maps between them. In some way, they're telling you how to get from one to the other. Now, I could at this stage simply develop the theory with these two examples, but when I began thinking of this, I realised that actually... This is at least what we have here as a very special case of a much more general structure, which in itself has wide possibilities of applications, which I want to show you as I do it, because it's this. I always like doing this, I must admit. I like to start off with a really concrete formula, so no one can play me too hard because it's too disconnected from reality. And when you've got everybody nice and comfortable, when you have some massive generalisation, it's really abstract. But because we're doing it this way round, you can buy it quickly if you want. So, here's what you're doing. Let me consider a much more general system. Let me consider a system where I've got a general category, q. Now, most people don't know what a category is, but the category is a very general idea in mathematics, a very important idea, really. A category consists of a set of things called objects, A, B, C, and I want to put arrows between them. Now, if these are set, the arrows may actually be functions, but they can be other things as well, so it's a big generalization of the way it's set. Now, a good example of a category is the one at the top there. That actually is a category.

20:00 Those four objects ABCD, those four are called subsets, and the topology is a category, in fact, with the arrows between the two objects being just order-preserving maps. In other words, if you look at the axes of a category, you'll find that thing completely suffocating. So this is an example of a left-forced mischievous idea. I'm sending this to a topological place because of the... That's the key point, yes, that's right. And also, you can compose them. See, that's actually the key thing to category. If you've got a particular example, if Frank goes from here to here, they're not all preserving that. They're all preserving that. If I compose the two, it's also not all preserving that. And that's really the key point, is you stay within the class of things you're looking at. I mean, it's not a requirement, it's just associated with something else. So what I'm going to do is study this much more general question, is supposing that I'm typically given a category, I say this was a small category because the last likely logical thought was here, some categories can be so enormous that the collection of all objects isn't actually a set. Speaking of the set, I'm going to get involved in that sort of thing. So a small category, so the one where everything is what we say. So a small category, where the analog configuration space is a collection of all objects, or a history analog of all objects, and then the arrows between two points, two elements of object symmetry, are thought of as the analog of the group element that takes you from one object to another, in the case of a manifold. So basically they all went momentum, that's the idea, but in some work, the arrows in the category all went momentum, and the objects were like configuration pointers, that's the idea. Now what I'm trying to do is develop a quantum theory for any system, by the way, in general. Medicals have been done, but obviously I can apply it to these particular cases. As a matter of fact, this is quite a broad-ranging idea, because if you think about it, there are actually many, many possible examples. They're actually all connected in all sorts of sense. By connected I mean they're just one tree, it's quite a bit. There are many, many variables like that. You've got quantum space, you've got quantum space, you've got quantum space-time. Again, you have lots of different types of topological space.

22:30 Of course, physics comes at some point, you have to sort of like model, choose the sort of things you're interested in, but mathematics is very, very important. You can be manifolds, you've got quantum manifold theory, if you want it. Quantum differential structure, if you want to do that. And more of that, we need quantumized groups. I thought this was a smart effect, because it's kind of funny, really. So, if you give the new meaning of the word quantum group theory, that's what was being preempted. But here, what would happen is you have quantum fluctuations in a group. Now, you might think that's a bit strange. But if you really weigh out, remember I used to, oh, sorry. Those of you who've known me for a long time, remember I used to work with smart. And as you go, and this refers to me all the time really, in terms of things you think about, there's a really wacky question for you. The internal symmetry group of the grand unified theory, how do you know it's that all the way down to the big bang? Maybe there are quantum fluctuations in the group itself. Can you imagine such a thing? And if so, how would you describe it? I'm talking about the quantum group theory, but you could do it like this. This scheme could be applied, so you can imagine. It's crazy, having lectures of groups who are interested in that show fluctuations over time. That's the great thing, it's quite true. However, there's a problem. Sorry, it's a sonical problem. That's what I was going to say. There's a great art I've learnt from the lecturers. In the middle of their lecture, I only give the sonical problem. I'm used to talking about it, so yeah, it's a very big problem. It's crazy. Part of the lesson. It's a bit useful, I guess. It's always been that way. Here's the problem. I want to think of these arrows, if you like, if you want to think of a concrete example of the measure these are all observing, that's what we'll set, and that's what we'll find. I want to represent these in some way, operators, and we'll let them implement it. So I'm imagining I could operate with d hat, for each arrow, I'll put the d hat of g. Let's take a look at u of g, that's the unity operator for the first layer, so what this is like for the first layer of three nodules. And I have been quite dissatisfied with this. If I have G followed by F followed by G, then the representative of the composition is quite a bit more complicated. That's the analog of saying U of G1 times U of G2, because the group norm in a good natural manifolds is not analog of this composition. Or maybe the other way around, because the representation and the representation is partly analog. This is the analog of this. However, the problem is that in general it doesn't make any sense. Because the composition of plurality, as we like to all deserve to be mapped, is only defined if the range of the first one is the debate of the second, in other words, f has to map a to b, g has to map b to c, if g maps r has to map b to c, of course you can't combine them, so g can't make any sense.

25:00 On the other hand, you can always say the project needs to be operated. So if you simply had an operator d hat of g, or b hat of o, you couldn't, this would actually make sense, it would make sense, there's a contradiction here, you would always get the clients to operate. And that stops you just doing anything sort of naively, if you like, and just say, right, we just write down the hats and keys and see what we get. So what I've done is introduced something called an arrow field, which turns out to work very well. And it irritates most people, if I'm correct. Who is not irritated by me doing this? People are changing. I like this, I'll leave it off. Okay. So, in general, what's an arrow field? And this is very symbolically real. It's that for each object in the cat, an arrow field is something which simply associates an arrow to it. Each object has one of these. Now, there may be identity arrows, because one thing about the cat is it always has an identity arrow. So it could be that, but to each point you have such a point. So to each object you have an arrow. And I've got an arrow field. It's like a vector field. A vector field is a manifold associated with a point. So an arrow field associates an arrow to each object. So let's think of those from the call of death. So, as I say, for each object A, there's an arrow that connects them. This remains the same, so it's not the exact same as that of B. For example, here's a simple example. Suppose we have a category of just five objects. There are lots of arrows in between all the objects, but here's a particular arrow field, associated with first A1, this arrow from A1 to B, the second object, this arrow, that one. You see, the problem, let's say, with the individual algorithms is that, generally speaking, you can't define them as one or two.

27:30 In fact, they call them, I think, what's called partial semantics. The part is only partially defined, and they're very difficult to use, so to speak. The reason is, so now, to perform the genuine semantics. And it's very simple, really. What one does, I wrote down the formal definition here, but here's what it is, a sort of simple term. We're going to play around with it, and we'll be finally quite up top. Like this, which object I act on the first object, much as the object on the first element, it will take me there, and I pick up that point and act on that with the x2 of that. This is always defined, you see, because an arrow is drawn the same time for every object, there's always something here, there's always some x2 of a bit, you may be going, it still exists. So the result of this is this product is always defined. Where the product two arrows, the location is defined. I would feel it's always fun. That's the heart of what I talk about. That's the part where everything's painful and it's fun. Could you just say a little bit again exactly what you mean by that arrow field? I mean, what does this arrow actually do to these ones on the left here? Well, I'll tell you what it does. I'll tell you what it does. These are causal sets. You see, with these arrows, it would be possible a causal order-preserving map should appear here. So, what that arrow field would be is for each causal set, I would associate an order-preserving map with some other causal set. Just with one other causal set? Yeah, just one. Just like a vector field. Like a vector field with a single arrow at each point. But the thing is, you can obviously combine those, you see, because like I said, you've got a pair of these, you go from here to here with the first element, but then there's always something that can happen with the second one, because the second element will be associated with every object, so there's always a composition. So it makes sense. Now, of course, it doesn't make sense, but there's a part of law that's defined in the sense that this is certainly another area of field, but what is also very important is that it's associative. That's very important. I think the group of a times bc is ab times c, and that's absolutely controversial in the group. These things are called semigroups and monomers, and that's the potential of associativity. I mean, again, you can see what made any sense if you didn't, because when we represent these things by operators, then operators are associatives, right? If you have three operators, a times bc is ab times c. So if I can represent a narrative by an operator, then I have an associativity narrative. If you represent the things that operate because they're associated, what they're represented must be associated too, so it's a simple matter of units. So anyway, I'm going to prove that it's a simple matter of units.

30:00 And also, there's not much identity at the bottom, which is the arrow field, which is associated with every object that has the identity arrow. And that's clearly an identity for this structure. So what we actually have is a monoid. So a monoid is a semi-group of units. So any difference between a monoid and a group is that you don't have inverses. And that's obviously true in general. There's not an inverse to that. If you have an area going from here to here, you can't go and see which area is going backwards. That's the big difference. These are semi-groups, they're not groups. Apart from that, this is the same number of groups as it was. In fact, that's why I claim this, that this collection of all arrows are more negative than the analogs for attracting them. And that if you're more than a group, that's the key. It's very important to work on that. So, you're saying that... Well, it may or may not be. You see, I mean... No, semi-groups. Oh, no, I said it's analog. No, it's a physicist. It's the play of an analogous role in a concept. There's a process you can get to if you want to look at a category, because in general, you don't emerge in a category. No, you could, of course, look at the set of convertibles, if I'm not mistaken. You'll see that when I come to consider the actual expanderases. You see, the unit operator, if you do that, it's one. Here you've one. You'll see that when you do that. So this is the idea. Of course, I'm not going to surprise you in any category, but you've got to think about it in terms of causal sense. That's absolutely fine. Now, if this is the algorithm that you've bought... One critical thing about the Lincoln-Watson book is that it acts on a manifold that maps points around. Of course, that's not quite what the Lincoln-Watson is. If you have a manifold Q, then at any point in the Q that you hear, the Lincoln-Watson takes you to another Q up there. And of course, the case of the Lincoln-Watson is emergent, don't you think? It takes what it translates you around. That's why it's a different dimension. It's just a configuration. It's a system. That's why the Lincoln-Watson is really interesting. So if you have an analog here, well of course there is obviously, it's the obvious thing, if you have an arrow field, the obvious way of getting at all the kinds of objects is what you would expect to justify the reactions of Ls or A, just the range of Ls or A. So for example if we come back to this arrow field here, what this arrow field would do in translating objects, it takes this object and that object, that object and that object, that object and that object, takes that object and that object, this object leaves the other.

32:30 It's not like I've got a group of actual manifolds moving points around. That's why it's not like an augmented reality. That's the idea of an augmented reality. Now, what is important is that you can show that this action is advertised as this. It represents one way or the other. So that's just like a relatively lost group of actual manifolds. That's with the representation itself, I think, of the representation of the actual manifolds there. So here's the key point. The last point I'll pass on here is that there's a special time around the field, which is simple, of course not very useful, because if you just have to take a single arrow to mark the single order preserved in that, it's that you pick up the arrow field which is equal to that one, so let's say for example I have a pic with that clear edge, you pick up the arrow field which is associated with that object, H, there's the other object, no density, so there's nothing, so in other words you could represent each individual arrow in this way, so you've not lost anything by doing this. But of course, now, you can't even have clubs still, because even if you have two arrows with a special time, of course, it's just an individual arrow. Well, the product of the arrow's been on this planet, the product of the arrow's been east, that's the key point. Anyway, you can do that, and that's the obvious way. And these things are important, but technical reasons are common. Right, now then. We will use this to discuss quantum theory at some point. So again, I don't think this is causal sex, as I said at the time. I want now to know what I'm going to do. What I would like to know is what is the algebra, the quantum algebra I'm trying to represent. You see, I've got these arrow figures up to here. It's good. That's my momentum. I might well imagine that the analog of a configuration spectrum is the order. Now, do those things with the label vector spectrum. Those things stick together in some sort of study group or whatever, with representations I can then say falls in front of it. I could guess it, actually. I could write down what is analogue, or what works for a manager. But in fact there's another way of motivating it, which is to say, well, let's start off by taking, let's try and guess what the representation might be of this operating representation, or this analogue. Let's try and represent it first, and then see what analysis we get. This is working backwards, but it's perfect. This is persuasion. In fact, this representation of power can be visited for a start. So just work the obvious way. This is the element of this. I mean, the way a group acts in a wave form is u of g of psi, which is a psi of g of q. So that works.

35:00 In the same way, the rise up to a narrow field, an average. So it just does this. And that's the obvious thing to write down. The critical question is, these are operators now. Do they represent that semi-group structure? That's what's important. Now should they do? I'm not going to point it out. I don't think the cataclysm is important. It's just part of the definition. And you try to use the fact that this is a real action system, so you preserve these monolingual acts of knowledge. I think it's a fact. So you can find this is true. It's actually an anti-representation. That's unavoidable. It doesn't matter. Matter of fact, it's still regrettable that you couldn't define this in the age to add it. I don't see much choice. So what I get in this very simple way is an operator representation of the arrow fields and the category. This is why I call it quantizing on a category. Originally I called it quantizing the category. But that's very, very different from what you would call a quantum category. You would call it a category, but it's not a quantum and not a category, it's not. It's somehow, well, it's like the quantum version of a category. It's not quantizing, it's quantizing a category. It's a version of a category. So that's a good start. I mean, that means we're on the right. Apparently, there are loads of times that a general candidate can be used explicitly for any example you're interested in. All right, what about the other phases, the configuration purpose? Well, here, I say it's not always what you do, but I mean, just as many systems and configurations use how these configuration purposes function. You're the same thing here. And representation is the obvious one. This is exactly what we've been all the way through the campus. There's a fire that retains such a function. It might begin out like this. And you get this algorithm. That's it. That is an algorithm, which is the exact algorithm that I wrote down before. You have a representation of the arrow field. The representation of the function is like a mental configuration. You've got this sort of twist. And if you remember the algebra I wrote down, which was exactly the same as the one I wrote down for the manifolds, only what you've got here is not the tau, but the l, just as the gx on the manifolds, and that twist is the representation, so that the l would go back to the category twist. Part of that is exactly the same, but any category has a gamma of a common permutation equation, because effectively that's what that algebra is, so there's a canonical algebra attached to any category.

37:30 And, not automatically, I claim, the representations that I have are the quantization system, because that's exactly what you did with mathematics. Technically, this is, I call it the cataclysm of quantization, one I was hugely aware of. So, it's this cartesian part of the system, like Chris, which is head-level, if you want to put it that way, this is that part. I mean, Dirac first wrote this down in the Manifold. It's not called Dirac quantization. I mean, there he was basically talking about quantization. Here, we don't have any possible brackets, nonetheless we set it up with an algebra which is very, very similar to a geometry. Now, of course, what about when we have an expectation? Well, algebra, that's the algebra, because it's not particularly an expectation. What about inner product? Because, after all, we're asking about inverse and so on. It's really one of the adjectives. What are we doing with the inner product here? Now, what are you going to do? Well, we know the ordinary way it functions. The inner product being psi and phi is the equal of psi star x, psi star x dx. So, we can find the same thing here. Suppose that any measure, for example, in the case of an object, we can find this. That's about the most general we can possibly have, actually. Now, this, of course, raises the whole question, what is the theory of measures of objects? Basically, that's the answer. Whether it's a general theory or not, it might be. It would be pretty hairy, but it is. But with individual cases, of course, we can study it. How common is it? If it can be tracked on a finite number of objects in the world, count them. Which many examples of physics and physics do. Then there's the obvious, which is just to sort of build your sum. Now, what I've already done, I've given you a... I've written down a simple... what I've actually done is I've guessed a representation of... The quantum of the video and the vector of the analogy is just the way it functions. And I showed you how you get a property algebra. And I said, right, well in general, representation, or quantization in general of the system, you might ask, are there any others that want that written down?

40:00 And I suggested, ah, just as well. In fact, what I've been doing down is not hating them. It doesn't do the job at all. Because supposing we just have the way it functions, literally. It does complex value function. Well the problem with this is it doesn't distinguish Arrows with the same domain and range. This can happen very easily in a category. I mean, think of two calls of the sets, there's an awful lot of these, all observing mathematical concepts at the same time, and one would want these to be distinguished in the quantum group, but this representation doesn't. I mean, technically this is called separation if you have, well, I mean separation, which you could, and then you don't get arrows with the same domain, then you'll say, I was going to say they're separated. Thank you for your attention. What really doesn't work is that all of the radio operations I've used at the start of this talk is over. I've let an arrow take me from here to here, and that's all I've used. But of course, this arrow and this arrow taken there, both of them connected there. They're not the same thing, but they're just having functions that I can't possibly tell. Because in both cases, I just didn't want side A versus side B. So this is quite inadequate as it stands. You can't separate arrows. In particular, you have to remember that it is perfectly possible, of course, in a category, to have arrows of the same dynamic range. So, order is only mapped and set onto itself. The number weight, of course, is set onto itself. You want to represent those as well, because that's the internal structure, so if you've got a causal set, it does have an internal structure, which is represented by the causal matrix onto itself, and that's kind of part of the causal set structure, and this represents the same part, it doesn't ignore it. So, the set of all arrows, the set of all maps, the set of all arrows of the object itself, you want to separate out, and this is what we're doing. So what do we do? Well, the most basic pie game of work is the manifold. You ask yourself, well maybe you should have vector space time. Well actually, what happens in fact for the manifold, given its size, is actually when you do it, the q is d to the h, what you actually have across such is the vector function.

42:30 It's not just vector space and function. So if you've got a manifold, let's say you have a manifold q, and we're going to reach 20. There are five of them which you can't really expect to be. And the actual space, you've been sighted by a whole section in this thing, so they're all just like that. It's a bit like a function, it's not quite a function, it's just a function. So this suggests, though, that in the category case, what we actually need is a vector bundle, in other words, each object is associated with vector space now, sitting over the object, and rather than having the way a function just takes its value to the context element, we take its value to that vector space. And that vector space will carry the eternal information, if you like, about what you say, call, and set, in such a way, we hope, as to separate it out of the algorithm, so that we can represent, fully, all the structure of the category, which represents the point of view as we're trying to do everything. There's no reason in a plot you can write down if you do this, which is just a generalisation of the game that we do all the time. So this is the key point. You want the vector space to carry a faithful representation. Well, there's an error that's constant in itself, so for each cause and set, you would like to have a representation of all possible causes of order-preserving maps, but now you realise there's a huge difference between the categories of manifolds, because in a manifold, what you do is each vector is placed at the same vector space B at each point, not with manifolds like homogeneous plates, so the categories are not the same. For example, the collection of all order-preserving maps for this thing on itself is obviously quite different from this thing. There's only one map. So what this means is that vector space has to vary from object to object wildly. It's not normally vector modeling. It's something that's shown as very wildly. I mean, it can handle it. The vectors are right. It can do it. But we need to get back to the line. So apart from that, what we do is, again, I'm not going to start with 4 and 3. Quantum theory is a game which is a game which is a game which is a game which is a game which is a game which is a game which is a game which is a game which is

45:00 Now, normally you can't say that that equals that because they've got different spaces. You've got to get back from this one to this one. So you've got to have these multipliers to map these general spinogram spaces back to each other. These aren't exact analogues of what happens again in ordinary quantum theory, except in ordinary quantum theory the vector spaces are the same. Whereas here they vary from point to point. You can't know. Let's take a look. Now, as a critical thing that applies to all people with group representation theory that applies here, is that in order for us to be representative of this arrow field of health and well-being, these things can't be arbitrary. We need to be able to satisfy these conditions. Listen, there's an exact, you can pick out one of those for group representation theory. Everything goes exactly the same, it's amazing, it's just exactly the same. Except, that you have this variation. Now, I won't go into this here for a certain amount of time, but in fact, what you can show is that it's really quite something to appreciate. Those of you who know about science, what in fact I've done here is to construct a pre-sheet of the film spaces, and that's the actual technical description of that conversation we've received, which is not often effective, obviously, in the capital, and you consider pre-sheets of the film spaces, those pre-sheets that are at the table are not allowed. You can over-sheet them, but you can't believe the joy of that statement if you don't. I see these at your best time. Terrorists don't know either, so it's all right. So we'll miss out the section on the chapter pre-sheets, don't we? So, what are adjoins? What do they look like? Coming back to your question about inverses, because like I said, the unitary question, you do dagger equals 1. That's when you've got a group. So, you dagger the g is a representation of g minus 1. Now, of course, that can't work for semi, but it's worth a lot of work. So, I mean, so, the statement that you, say, you dagger g is 1 means that essentially you dagger g... So you would use minus one. Of course, we don't have inverses, that wouldn't be true, that wouldn't describe this to be true. Now, I don't mean you can't use it perfectly well, but what is it? So just for fun, I've worked out the algorithm. It's in the simple case that it doesn't spell this way. Now, the algorithm, well, let's take the simple case where you have a part of it just this, then a good old dear apology, cracked down the NK to make it a little more respectable. You can write it out for us in the case. It's quite a useful trick, really.

47:30 We'll ask the Kecks at the end of the lecture. You've worked it out, haven't you? What does it look like in practice? That's what we'll be doing. Well, the way it works, if everybody had now... If I'm going to be part of a business, how am I going to decide if that is Keck-sized or not? Which I can certainly do. I have these things down. If you think of these, it might be eigenstates and modules. They're Kecks, or vectors, labeled by an object in the category. It just doesn't make sense. Yeah, it's a new experience. The way this works, that's the a dagger of x, in the very natural way, the visual-translated ket, is the a, which is interesting, the a of b, here's how it works, is the sum of all those a's which map into the, you see, if this, if the arrow x, here's the answer to the question, if this arrow, let's say, you've got this arrow x, which is your answer to the question, if this wasn't there, if it was unique, Then what we've got here with just one sum, and if that's one term, what you're summing over is all those arrows, all those objects which map into here, and generally there's more than one of them. And that's because this is not exactly one-world, so we've got that, this is basically where it comes from. So, let me give you an example which is worth getting a bit into abstract equations, which is this. Let's take again this very simple example, the category of five objects. And we've got these therefore, we have this big vector field which is what I mentioned. We've got these five natural states, A1, A2, A3, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A14, A15, A16, A17, A18, A19, A20, A21, A22, A23, A24, A25, A26, A27, A28, A29, A30, A31, A32, A33, A34, A35, A36, A37, A37, A38, A39, A40, A41, A42, A43, A44, A45, A46, A46, A47, A47, A48, A49, A49, A49, A50, A50, A50, A50, A50, A50, A50, A50, A50, A50, A50, A50, A50, A50, A50, A50, A50 That's a causal set. The other here, the sum of three causal sets. Like I was saying, the superposition of causal sets doesn't always arise very naturally, whether you like it or not, but it's there. But if you ask about this one, well, there's two arrows coming to C. There's B coming to C. Well, also, this is an identity error. Now, you can work out what A x dagger is and A x dagger is not. I mean, I don't know.

50:00 But if you do that, you can see exactly how much difference it makes. It's kind of natural. If you look at it, it's the obvious analog of the semi-group of the inter-operations, but you need to look at these side-by-side. If you see this in transparency, I'm not sure if you can see it, but you'll notice that A of x is like an annihilation operator. A of x annihilates certain states. A of x annihilates certain states. A of x annihilates certain states. A of x annihilates certain states. A of x annihilates certain states. A of x annihilates certain states. And that's the way to think about these things, is that they're created, but for me, these things, they've never created, it's most of one object to another, so it's creation in that sense, it takes one object to another, it's like creating, it's like annihilating one object and creating another, if you know what I'm saying, it creates an object, whereas the AMX is trying to take you backwards, but there's nothing coming in, it can't take you in, so it's like, so this is like annihilation, it's like all the problems, I mean this isn't just, this is pretty rough analogy, I'm not saying, but it's, that's the similarity to it. And that, again, is very typical of the reference semi-group, because they don't have any of those suggestions at all. Well, have I got five minutes, have I? So, I think, that's actually great. How do you use this? Well, let's come back to the after-examples I started off with, which were categories of sets. You see, there are many categories which aren't categories of sets at all. The odds are not sets, and the hours are not funky. It's one of the fascinating things about categories. Many obvious and interesting classes of categories are categories of sets. For example, the collection of calls of sets, right? That's a collection of sets. Now, how is all of this earthly matters? What topological spaces are sets? And many of the examples you will find, or all the examples you will most naturally think of, are in fact categories of sets. So you've got to ask, how does this scheme look like, like I've said? What that thing might look like is not what it's been equated by right now, that's not being traced, I mean, can you actually solve the problem, can you find, if you like, something appreciable, can you find over each Hilbert space, over each object, a Hilbert space which is sufficiently big, if you like, to give what you want, separately from all the others? No, that's not a tricky question. You see, I've actually made this clear, that each choice, if you like, appreciable, gives you a better representation of the analogy. So there are different representations. So you've got to choose one that's physically useful or important, and the obvious ones are the ones which represent all the others somewhere in the middle. So you want the smallest one of those in the way, and in fact you can do that with a basic balancer.

52:30 Here's the obvious question then, what is a good choice for capital A, if you have a space associated with A, and what is a good choice to multiply, you've got to go from A to B, you've got to go from A to B, and that makes the function, say, I want to observe it with math, if I say to you, you hold the space of this coordinate set, you hold the space of that coordinate set, how do I map back with this one? Now, as a golden rule, I was taught when I was undergraduate actually, which I've never forgotten, chapter one, mathematics. The reason research lacks is that always do the obvious thing first. I forgot that rule, so I started working on a seminar. That's more or less the only way to keep going, to do the obvious thing first. It doesn't work, so it probably doesn't make any sense or something like that. So I've always been amazed by how often this actually does work. In practice, I always feel I know the stupidest things these days. I always do the obvious thing first. Mathematics has its own somehow internal dynamics. It knows, somewhere up there, some platonic realm, something up there, some platonic realm, it knows the answer is, and it knows it's the simplest answer. Well, what is the simplest set? Most mathematics cabinets are fine, except that these are very simple causal set models that I wrote down. Now, what is the simplest vector space you can associate with each part of that set, which you can handle? What is the set of functions that you can see? That's always a vector space. And it is the primary vector space, which that's why it's more clear. This is the number of elements in there. That's the Hilbert space, so you can use it. Now, after all, nobody thought that this was going to work, except it's the obvious choice. Well, again, it's the obvious thing to do. If you do it this way, if you have a function that plays a thing, and you have another function that needs a state that belongs to this Hilbert space, how do you think that is a function? Fair. Well, like in the photodominator, if you put it back, the only one thing you can write down is you say that the effect of this function on that function here is just a different composition. It's just the usual pullback expression of the composition. Now, if you do that, you can easily check that mathematics is a different subject. You can prove that rigorously from this map. It separates out all the errors. This is a very good quantization scheme. It works. It really does work. It completely solves all the things you want it to do. Now, I'm not going to prove that, but I will give you a very simple example.

55:00 Like I said at the beginning, I'm not going to get a global prize for this, but this is it. This is Chris Huygens' quantum model of causal set. There are just two possible space types. One single point minus two points. Here's me in German discussing ontology. It's not very exciting, but there we are. It was still that complicated, you'd be surprised. So there are two objects, as Mr. Coulter said. What are the arrows? What are the order-preserving maps? Well, we go from A to B, there's only two of them, F1 and F2. Why don't you go from there to this map? Why don't you go from there to there? What about B to A? There's only one possible map anyway, and that's that order-preserving map, that's that one. What about order-preserving maps? Well, they're quite not so exciting, just the identity maps. The only interesting one, really, is this. Now, three different order-preserving maps, and this is it for itself. There's the identity map, that one. There's one which contracts those down to that line, and the other one which contracts them to that line, and they're going to observe all of them. One thing it's not allowed is that it flips it around, and that's a permutation of the set. So there's not an arrow in this category, but in this category we require all observing maps. If we were just doing onto the set theory, then we'd add to that as well. This is a very important point to realize this, is that the arrows really are all structured observing maps in the category. It's very, very important. Not any of them. So we just have these two. Now, I'll break up the previous scheme I told you about. Then, A has a single point, so the Hilbert space is just C. B has two points, so it's C2. It's all very trivial, of course, because it's finite things, but take the prosociation of that, and it's just C plus C2. So the quantum history space of this theory of causal sets is a three-dimensional Hilbert space. And here are the operators. We've accounted for the matrices, of course, because it's a three-dimensional quantum space. So that's it. Now what I've written down actually are the representations of the specialized algorithms, but you can build up any algorithm of products with the special effects you might expect. This is good enough. So that's what the algorithms look like, and you'll see each one of them is different. So they are separated out. And here's what the configuration rules are. If you want to, you can check the satisfactions of the algebra. So that's it. Now I must admit, I've not got that much time. I've got to step back and think. Okay, that's it. So the fact is, it isn't. And you could use this if you wanted to construct a real quantum theory, it's a system, and I must appreciate, write that down as a toolkit. I'm not saying how you write down a particular quantum theory of this and this, but how you would, or what these properties would do is you actually would write down a quantum theory history book.

57:30 You'd have to write down the Decanus function, which is built up out of these operands. You'd best put topological spaces, you don't have to write that down by the count of totals. And so, like I said, this is not really a theory at all, because I talk to people all the time. But the idea is that, you know, just these half ways, don't keep writing down so much stuff. So that basic example, you can write down others, and you can write down what you want to, but I don't want this one. So let me just conclude with what I think is that I've argued that, driven by one of our reasonable questions from our section, which is topology and quantum physics, There's a vast generalization where you think of a quantum theory of a system whose configuration space is the opposite of the category. You can think of that chronologically if you want, or you can think of it history-wise. It's the same mathematics, first of all. And certainly with that is this monoid, right, that you have this arrow field structure that you have these functions in. This is the angle of what the vowel group normally can become. It's the different interpretations you see. And arguably there are similar interpretations in models, like the way functions don't move. You can talk about pre-sheets, where you have extra structure, it's easy to make advantage of vector space, and certain examples I've looked at, they do work, and that's fine, even one example of that. And I'll also say that the finite set, you know, exactly, I've solved the problem of what a pre-sheet is. Now, there's much, much more on the list. I've just written three papers, or two and a half papers on it, actually, so it's a big subject. But this is the basic idea. One of the problems for researchers, and there are masses of them, but that's where we are for here, I mean, the first obvious one is to actually study measures, and that's a highly non-trivial question. Measures on the optical atmosphere. And if you look at measures on the vector space, except for its different dimensions, it's very complicated. You have to look at the dual vector space, the possible dual carriers of the measures, not the vector space itself. All that means is that if this is something I can't agree with, it's not going to be a dimension, it's not going to be big, it suggests there might be something like a dual, but I have no idea at all what the dual effect of consciousness would be. One reason I'm not thinking about it, actually, is because it's kind of a safe problem. Nobody else would. It's actually quite a big asset. Anyway, it's a very interesting question. This is the general way how you define simple equations because, like I said, there may be many, but most of them won't work.

1:00:00 Now I know that I've soldered the sets and added an important set to it, but I mean, I wasn't very interested in one subject, so I had to look at each one and figure out how to do it. So these are the finances that I want to explain. Now, the character of non-finite sets, this is interesting because it's all about doable, but you see I talked about finite calls and sets, but you could of course have infinite calls and sets, but it didn't happen. I mean, there's physical evidence, but you might decide to say it's unlimited. Now many of you have a problem because you can't just, I mean many of you have information from Hilbert space, there's a lot of space and it will get complicated. So this needs to be solved. And then finally, there's the use of talking for real. Like I said, with this you really can ask questions, it doesn't have to be much of a physical question. Of course, there's only talking. So what we really want to do now is to be able to do a physical question. That's possible. Topology, mathematics, mathematics, physics. I have a question for Chris. These causal sets are supposed to be entire space-time, right? But with this one, it is efficient, right? So when you're talking about the transition between entire space-time... And that's a good law of the problem. Yes, it's not a technical process. No, that's right. It's a part of what you're doing, of course. Yes, that's right. So that's why I say the word problem of fluctuations is it. You can use it. So the fluctuations start by something that's in time. If you say that, you can say that we take the curve of time. Because these iron fields, for instance, they point in a direction where the digital art would be, I mean, we suppose that each cos of 7, the order of the cos of 7, is one macroscopic algorithm, for example, to be temporal. So it's out of the timeline. Yes, that's right. So this intellectual transition is for the digital art. Absolutely, that's why I think we need to be careful. But part of the mathematics is the same, not the digital art. I'm actually very excited about this because everyone's always excited about their own work.

1:02:30 But I'm more excited by this than I've ever done in my career, which means everyone will ignore it. But it is, I think, as huge as possible there. The fact is, I've said this, but you can actually rewrite all the concepts here in this language. If you could think of the points of the manifold to be the opposite of the capital. You know, the arrows, all the group elements, are taken one point at a time. So in that sense is that, yeah, I mean, it's an interesting place to be. So they don't allow it, they've been responding to... Lots of different groups, different development, you see, because a group in general doesn't have a previous, I mean, such a different group. Yeah, that's right. You answered Janice by saying that I think it's easy. You're thinking then, clearly, of setting up a quantum mechanics of the whole universe. That's always struck me as very dubious, that the quantum mechanics of the complete universe is pretty much the quantum mechanics of the laboratory. Just as, from my market perspective, the market dynamics of the entire universe is quite different from the Newtonian mechanics which Sullivan needs. It would seem to me that you're retaining all the major things, above all transformation theory, operator, Hilbert spaces, of quantum mechanics in the laboratory, and I really am very sceptical as to whether that is appropriate for the whole universe. No, I'm sorry, I have to. I've had the most pain from that a few years ago. Say nothing of that, it's fine, we can hold on to different things at the same time. No, I agree with you. But of course, I'm obviously massively aware of this. History theory, because history theory is one of the things we can talk about, is a huge part of cosmology. And I quite agree with that. I have no other reasons for thinking that. If you ask a absolutely general category that doesn't formally know it, because categories are so general, that it's hard to see what extra-general structures you can have, at least I'm not aware of any. If you look at any particular category, when these things become much more complex, then you may see a specific one.

1:05:00 In one way or another, if you look at a collection of four sets, you may have an extra structure. But I'm not aware of any, I mean, single categories. Well, I have a follow-up to this and a few other questions. It struck me as soon as I was speaking that, though I don't know about the root for it, it's the word for the category in which every arrow is inserted, are amongst their enthusiasts, you know, quite a richly understood subject. So this would be a... Is there a way to think about doing it from a position for real by using mathematics as a result, which would have addressed the issues of adjuncts and not the issues of separation? I mean, everything you said about needing to have three sheets of multiple spaces to separate, would it still carry over to the issues of adjuncts? Well, I mean, there's nothing special about groups. I mean, semi-groups are just as good. The only difference between semi-groups and groups is the representation of semi-groups. You seem to have more exhortations. I mean, apart from that, it's very tangible. Yes, you could. I'd like to work out any models that possibly seem to fit in with what we're trying to apply in this case. So the thinking is quite specific. There's only one thing to say to each group. Yes. I mean, the groups I've written down... It's very nice that our I'm Ron Moroney from Bristol. I came here last year to give a talk on the subject of our graduate students. It had such a good impact that those of us who weren't able to attend were very glad to be back.

1:07:30 So I want to be speaking about measurement entropy and . What we've been talking about is the issue of the Szilard engine, and specifically what's called the Szilard paradox, that there's a problem in statistical mechanics that there are certain statutes that apparently by measuring a system, it could be said that one reduces its empathy. And this has been this variation on the mathematical theme of the problem, which... Traditionally it's been seen as a solution in terms of thermal fluctuations, but it's a target that in the Szilard engine, thermal fluctuation solutions apply. The currently accepted resolution of this is, in quantum physics, is that the answer is actually something to do with information processing by beam. This really actually is a sort of paradigm example of how information is really entry and entry is really information. It's the same. This has largely been put forward by physicists. It's been criticised a lot by philosophers, but to some extent the physical models that have been put forward, it seems to me, by a lot of the criticisers have been incompleted, certainly been criticised as being incomplete. So my kind of modest aim of actually answering this was just simply to see if you could complete the physical models of the anti-information approach. See if it's possible to actually generally analyze the whole Syllabe engine without being able to do information theory at all and answer that criticism, and I think the answer is yes you can. What ends up saying is that there's a correlation going on between the degrees of freedom in the system, and this is actually the additional thing that the Syllabe engine brings with it. This is above and beyond simply fluctuations in the system or states of existence. And usually the Maxwell's Demon, those correlations are destroyed by thermocouple integrations, so they're not apparent, whereas the cello-electron actually brings out this flow of correlations. So, I'm going to proceed by briefly reviewing the Maxwell's Demon and also how the cello-electron changes that.

1:10:00 Then go on to actually look at how a demon-less cello-electron operates and see how actually that can contribute to the diagram. And then look at more generally actually what's that saying and what it is. The role of correlations in statistical mechanics. So if you look at the matter of the demon, sort of the original experiment, you have this gas. The number of atoms moving around. I suppose there's a partition with a little gap in it. And generally, as long as the number is large, it's roughly going to be half the atoms on the other side. And then, obviously, you've got this position of the demon that sort of turns from there to a fault that's a trap door. And we close the door on the right and close it on the left. This is pressuring them that eventually all the atoms are going to end up on the right. And apparently this could be taken the same as the entropy is different. We can now then apply a piston to the system and use it expansionally to gas back to its original state and use it to create a wave which would vibrate the dynamics. So there are a number of things to sort of what actually is it that's particularly going on here. And it's a question of what mathematics is really trying to illustrate when we introduce this idea. And this has gone public. There's a lot of discussion about what actually is necessary for a mathematical theory. What is a mathematical theory? Well, one of the things we can look at is just simply random fluctuations. We just don't have to use shutters, but just move randomly vectors according to the column. And we should just suppose that, by random chance, it blocks all the attributes, rather than not collecting them. Well, again, we're going to actually see, again, still to the lower entropy state, I'd say, aren't we? Now, there aren't, there isn't any interpretation that's, that's used that this would happen. So there is clearly a possibility. Um, so one could suppose that there's actually, you know, there's some reason to do this, actually, and I can go back to what you were just trying to get at. It looks like there are situations where there are. But of course, this is actually very unlikely. It's like looking at quite a large number of types of indicators at the same time. If you're after a version of maths that's not being able to present it, then you're considering all the possible interpretations. In that case, the low entry to fluctuation is just one of the large number of possibilities, and the overall entry of the ensemble has not increased at all. So, from the point of view of the ensemble, this isn't a problem at all.

1:12:30 Can you share with me? No, it's fine, just put it further on. No problem. When it was noticed that these kind of atomic fluctuations were observable, could we actually find some little machine that was capable of capturing these fluctuations? Because again, possibly we could actually observe these fluctuations, maybe we could actually amplify them or not. And the particular way that it was actually supposed to absorb these was this idea of attaching a spring on a track to a wall and sink out of it. The spring holds the trapdoor sort of shut, because of its springiness. The fast atoms can move through the trapdoor, but the slow atoms blot, and this will create a temperature. But, as is well known, the spring itself is then going to have to have some kind of formalisation for it, and it all balances out to be sufficiently sensitive to actually the flow of the atoms. It actually has to be subject to a large amount of thermal fluctuation itself, and this actually wipes out its evidence, it has to be. So, this has generally been taken in the way in which Maxwell's demons were invented. There was a thought experiment put forward by Szilard. When this was put forward, Smolachowski first presented this as being the way of glock. But he left open this idea of whether or not intelligent observers would actually be able to obtain this. And in 1929, Szilard suggested a rather ingenious way of actually doing it. Reducing the number of degrees of freedom to such a way that actually you can capture individual fluctuations quite reliably. And you can treat this as being one atom of gas, and then you've got a shuttle which is slightly more limited than gas. It reduces the volume of energy by half. Maybe this is reduced essentially by k log 2, which is now half the amount of space being produced. The other thing is, well, we don't know which side the shuttle is going to go. And maybe therefore it has actually still got the whole volume of available to it, and probability distribution. But it's obviously the strange sort of minus nine line that is in the middle. That's a smudge. That's a particular one, yes.

1:15:00 It's not so apparent. It's probably better than that. Okay. And I think it's just as being learned. Especially if the entropy is being reduced, we should be able to actually extract an energy from that reduction of energy. And Szilard suggested that we could do it, because you could attach a weight by some mechanism to this shoulder and allow it to move as a piston. So look at the course of this atom moving around. If we exert force upon this, the force of this is going to move away from the net. And no one calculates the actual amount of energy we get from that. It actually does in turn come out in kt log2, which is exactly what we actually get from the expansion of the wires of the atom. But Szilard attempted to do this. Gerard of all regarded as competent scientists in the discipline of mathematics concluded there was nothing there, and that the problem was that we needed to know which side could attach the weight in order to be able to get this, because it would attach the weight on the wrong side. It helped if this atom had nothing to do with this weight at all. So it was this state of instant knowledge that we have to have. Somehow we have to acquire knowledge to perform measurements in order to get this knowledge. And this thing leads to this idea of information. And then somehow it's that by getting this information out of the system of mathematics, that this information then is the whole part of the observer, and that the information, the removal of that information is actually what produces the compensation, or is technically valid. Now there are some students who actually actually have this in place. The first one suggested was acquisition of information. This is not what's currently happening. The current argument on this is that... A general call by Bennett to build a local lander that states that if you want to erase a bit of memory, one bit per memory, then you have to dis-educate yourself to the information of the environment. This was the exact example of the topological convention. Now, the first problem... And really, it's that it's not quite circular, because Lambert, if you go back and look at his paper, he actually puts forward this argument that the erasure of the formative information is a case of paradox.

1:17:30 He puts this in the context of the theory of computation, of the dynamics of computation. But if you look at his paper, he assumes that the second rule of the dynamics is to derive this. And it's really an old point we've made. We have to see this as being a deal of making and measuring. We have to see this as an information processing. Can we perhaps actually see this in a completely new list? Can we actually have some sort of automata that has no real information processing in front of it at all? So, firstly we'll just look at a bit of land aberration on steel being intelligent. Not, no, totally unintelligent. No intelligence, no information processing. If you can do it without that at all, has information processing actually really happened? This would be the analog of the spring, wouldn't it be? This would be the analog of the spring. Do you need anything more complex than the spring? In order to put this system, which originally could be in state 0.1, into a deterministic state such as 0, it's necessary to compress this system in terms of the logical degrees of freedom. Of course, because of the fact that this is in an uncompressed proposed space, We have to then expand the degrees of freedom in the heat band, and therefore we have to dissipate Ks into 2mg per heat band. Well, this assumes that the evolving compressive heat, the closed mess, derives this from the assumption that it is more or less equivalent to the second law of expansion. If we suppose that Maxwell's demon can operate, then we can actually extract heat, network the heat, without any complicated expression. In that case, Landauer's principle technically would be valid. We could progress forward to the first question, but that duration wouldn't work, and its use of it comes to resolve the Maxwell's theorem problem, which is called Markham 2. So you're saying you could progress forward to the first question? In a kind of simplified situation, in a reasonable statement, it would work, but more generally, you'd have to have a way to do it.

1:20:00 Yeah, that's not strictly correct. However, the answer to that is that this was actually complete. This leads to a problem at the end. When Gatton was originally on the right-hand side, he used the left, and the right-hand side was the left. And then, put forward by the information theory, Yes, now we're somehow encoding that information. And the cycle has to be completed by restoring the crystal to the center. Well, what we need, therefore, is an operation that's going to put the crystal back in the center. Well, in terms of quantum mechanics, the industry operates out of the crystal to the left and the side of the crystal to the center. The problem is, we need the same operations to work if they've gone the other way. And this, there's no such military operation to do that because it's going to run in two or three different states. You need two separate operations. So, this would be the argument that you can't erase this information if you put it in this location of the test. There's also weights. Surely we can correlate the location of the test with the location of the weights. If it's on the left-hand side, then the right-hand weight is weighted. If it's on the right-hand side, then the left-hand weight is weighted. So we could form an operation like this, where this is a projection operator of the space of the weights. Now this correlates with everything. We could have an operation that would have been neutral, and we could put this one back in the centre. Well, you've still got the weights there, haven't you? You've got to reset the weights. This is getting a bit contrived, though, because raising the weights was the thing we were trying to do in the first place, but nevertheless, the problem is true, so what about the weights? Let's see if we can reset the weights. You can just drop them. Well, okay, you drop them, and that's dissipation of KT logs of energy.

1:22:30 It was stored in the height of the weight, it's now dissipated into some heat gap, or something like that. So, for our information theories to use their lagger error agents, they prove their point. This isn't really like something that's already gone wrong because the weights don't have to be in contact with the same heat-bath as the gadgets, which is a heat-bath for the gas, and if the weights dissipate in their energy, it's a heat-bath at a different temperature. And if the weight secret is higher than the gas, well we need to have got a cycle that's reliant on the transfer of energy, not too much, from a colder to a hotter heat-bath, leaving everything the same. So somewhere along the way, something has gone wrong. Again, it's far from clear where information theory is going to help us out. This is where, looking into it, it says, what actually has gone wrong? Well, it was actually that conditional operator that attempted to reset the test in the first place. That operator doesn't exist. It isn't here. The reason why is because, well, we've now got weights at some temperature. They're at some thermal space. And this means they've got some fluctuation in their position up here. If you look at this temperature T level, well it turns out the median height is given by this, the median is this. The height is actually raised by the expansion itself, which is this. So if the temperature is very similar, if the temperatures are roughly similar, then in fact the height we actually supposedly are raising is actually very comparable to the fluctuations in the height of the gas itself. We actually have a 50% chance. The two are potentially the same, but they won't be spontaneously found over this height in the first place. The way this prop causes the problem for any setting is that we want the correlations to be in this form. So that the location of this motor here, the raised height, is well correlated with the location of this one. Because of these fluctuations, there's always the possibility that the unraised weight is nevertheless spontaneously found in the highest stage already.

1:25:00 And that means that the projection is actually in its initial state. And in this case, the correlations can't be the same. And the probability of this happening, as it turns out, actually comes down to the square root of 4. It's a half of the value of 2gm. And all things like mass and reality are all just completely after-events here. What happens if the correlations can't be the same? Well, what happens is you can't move the distance back to the centre, it ends up staying at the sides. You have to start a new cycle in order to start a new science. And this gets what we should be expecting, is that the reverse situation occurs. Now, at this point we've already worked out the entire cycle in terms of, say, unitary operations, the grating cycle where we're lifting a weight. And we need the entire cycle to be defined in terms of unitary operations as a persistent problem. We're so constrained by this that in fact this pretty much defines our unitary operations when we continue starting. With the weights in some position up here. And what happens is pretty much the reverse operation. We'll now find one of these weights that's been captured in this high fluctuating state. Now the load begins to be lowering it and compresses the weight of the gas, transferring the energy backwards. Work your way through, you'll find at the end of this lowering cycle, which is transferring energy in the opposite direction to the original direction, we have again, we have the number of states. We have some possibility of the weights being caught on these fluctuating states. Now if we capture the weights of this one, well the weights are continuously tight, we can continue at the lower end of the circle. But, if both of them happen to be at the same height, it turns out that we actually have to move those back onto the rails at some point. So we've got a situation with these two different cycles transferring out one way or the other, jumping backwards and forwards between them. Can we have these probabilities at all? What we really need to know to show now, to make sure that this really works, is that... The mean flow of energy in the long term is always going to be the mass of the total. It's going to stay for a long time on one cycle or the other, depending on which one's the plus or the minus. And, of course, that's what happens. Very weird, isn't it? There's some degree of freedom you have in choosing your resetting operations, so the total is down to one parameter. When you go into the mean flow of energy, you can consider this to be, on each cycle it's on one determinant cycle, either raising or lowering.

1:27:30 Or you can take a different statistical distribution, which has got some probability of p-point. This same term comes out to tell you that the long-term mean flow of energy is equal to the mean flow of energy on an individual cycle. It comes out to be the same expression. And, if that's clear, the key point about this is that this point of the probability of a half is the value of p. Everything below this, the mean flow of energy is positive. And below this, the mean flow of energy is negative. This came about through some rather subtle balancing out of various things, which kind of should be rather suspicious, and we should think that there's something deeper going on in there, that we've looked at one particular thing, we've looked at the pseudometric in itself. Right, so we've got a demon-less version of it, there's no particular role for information processing going on here, there's no particular role to find out the ratio of the search. But, you know, maybe you could modify it further. I haven't really got to the heart of what's really going on. Also, how does this, where does this come from? This particular probability that some kind of distribution works. So we thought we'd generalise this a bit more to look at more general situations. One of the things we did was we turned the... All of this is linked from being simply a source of the word question mark to actually being an isothermal compression. I think that was a slight account of saying it is an isothermal state. It was actually an isothermal compression, technically. So, if you look at this from a general point of view, we'll actually just consider it a general state, which we'll represent by the kind of synodal H in the box, but this is just any system undergoing an isothermal compression, or isothermal expansion, with this symbol 3 of 3. We start off with this situation and we have the expansion. We extract this energy KT2. We've got a second system that names UT2. One thing we can do is we can just compress this completely to restore this single state. That requires KT2. This is just now actually operating as a heat pump or a heat engine. It's got to capture the same kind of cycle efficiency.

1:30:00 What's interesting, what makes this different to the typical Kano cycle is there's no particular heat fly direct between the two parts. This is a direct transfer of energy and it's mediated by the piston. So we're developing a correlation between the gas state and the piston state, so we're transferring that into the state of the piston and the energy. This is used to try and force energy from a colder to a hotter band. So this means we can't actually compress it completely and we're left with this situation. Because of the fact that T2 is going to be hotter than T1, it's trying to get out of the energy of the dimension, lower dimension. That means that we're always left with some region here where there's states having them. What we've got is now a partial correlation between the state of the piston and the state of the set of congruences. And this is exactly the same as what we had in our cellar dimension, where we had a partial correlation between the state of the weights and the state of the piston at the end of the dimension. We're more or less constrained by the inter-dynamics and the energy flow to what we actually do here, and it more or less turns out that if we have the gas there, then we've got this one there, and we've got the gas here, and we've got the energy in there, but we've got this property on the side of the reversal. So again, we've got this reversal between the lowering and the raising cycles. Why is it all going to work out so that it always ends up spending more time in the antitropical world than the antitropical world? Well, we're taking one system and we're capturing a state, some sub-ensemble that occurs in the problem itself. And what happens is that's always going to be in higher pre-energy than the pre-energy available on the sun. So we're gaining a certain amount of pre-energy by capturing a state in the sub-ensemble. And naturally, spontaneously, the problem is that it's a bit of a higher temperature than the first system, and it will try to do something that might be out of control.