Theoretical Physics Research Unit Seminar — Roman Zapatrin (contd.)

0:00 It's certainly the way that Grassman thought about the... You see what I'm trying to get at, don't you? So you're building in this linear superposition without any talk about waves and particles and all that crap you've got in polymer caps the way we normally do. And conceptually you're trying to start from an understanding, a prior understanding of what the operations in the algebraic operations are about. Whereas in your approach I think you're just starting from the operations themselves, just given as the absolute bedrock. This is why, this is where the potential between you is, this is where the disparity between you is, you know. I sort of say, no, no, no, we've been trying that for so long, the operational point of view. You know, it comes from essentially the creed of logical positivism at the beginning of quantum mechanics. And it went through all the time, people say, oh no, only talk about what you can see. and then you see you get a lopsided view because in that no one ever sees the wave function, no one ever sees the vector in Hilbert's face, it's a purely artificial construct which in fact I had a little diagram up there didn't I about the magic carpet and on the magic carpet I had in the tapestry Hilbert's face and sitting on the magic carpet saying, fly me, no worries. Yeah? Sorry, that's just an aside. But the point is that if you look at the way Dirac comes into all of this story, he is more interested, this is not about Hilbert's face, although he pays lip service to Hilbert's face, what he is actually dealing with is essentially minimal, it is essentially left ideals and right ideals and he's structuring these ideals together to get his algebra. So that what we call the vectors are essentially the ideals, so they're not something which you attach from outside but they are the unfolding of the total process

2:30 and the total process is the algebra, everything is in the algebra. Now you get numbers because you use the dual relation between the left and right ideals. And you can show that if you multiply left and right ideals together you get the numbers that you were talking about here. This is my way of coming into it. That's why I've got algebras as the basis. You know, may I tell you why I'm speaking about this? because it was... I finished it ten years ago, because I felt quite a difference. At the time, it started in 1986, I tried to develop a kind of dynamic of space. And one of them died out, but it was, again, I think it's too good, they are so resembling, because, you may agree, if you tell me. Really? Yes. I was very much excited by the paper of Ayshon of 1886's quantization of the lattice of topologies and his idea to consider the lattice of topologies as a configuration space. But with this exactly, with this idea, I was not. The right advantage is because configuration space is something uniform by its nature. the latest of topology is something definite, you have something like an order. But already by that time, I was involved in the activities related to quantum logics, and I wanted to do the following thing. So, quantum logic, you consider the collection of the properties of the quantum system, and then you can observe that if you take joints and you take leads, you see that they are not Boolean operations. And I managed to provide a formalism of the following kind, that if you have any you may introduce this as a lattice of properties, called properties. So these properties, this entails this, this entails this. And I suppose you would like to have some black box who would have this prescribed

5:00 let's support and they introduce the following thing so you have a semi-group then you have something like a elementary action so So, something like operations anyway, and by a property I mean a collection of operators such which is testable, but what I mean by testability, I, it was a semi-group with zero So when I introduce the notion of mutual exclusion of two operations, the operation and then I use it. Right. No addition, only multiplication and zero, that means you discard your system. And I, if you have two elements which are equal to zero, it means I say that they mutually exclude each other. And then I called a property such a subset which is if you take a, then you take not a, zero signature, and then you take not not a, a which is equal to not not a, I called it a property. And, of course, these were left ideals, and these were right ideals. And I managed to develop, not only for quantum logics, which are of very particular kind, but you can't have any property matters. So, okay, any letters consider it as prospects and then simply give you a semigroup such that it's less idealism is bryzomorphic to the collection of this report, please. And then I have simply applied these techniques to the letters of topologies. Of course, the first idea was, well, first time I have found an application, I use this for some spaces of the Hilbert space if you have the Hilbert space, which is much simpler than they do. This is a bit different.

7:30 Unfortunately, some of them are so ugly. And then I finished it, I had nothing written on the table that was described, and that's all. My course, after that, five years, he tried to push me back to this activity, but I don't know. But for any occasion, if you might need such kind of techniques, then you're just going to simply send me a picture of the lips. Well, trying to use the ideals, left, minimal, no, left, and ideals in that context, I think, is not, I wouldn't have eliminated, no, it's very, very. Also, I'm surprised you say this, because this condition that you get out of double negation, would seem to restrict you to the billion case. Well, you get it in by the failure of the distributivity in this case. Yeah, yeah. It is Boolean when the semi-group is commutative, when the semi-group has so-called bare properties. It was proved yet by a foolish in 1861 that you have the automodular weightess. Yeah. And then I could build the general structure for... But even for the double negation, because I thought the whole point was that the double negation gave you back the... that as a property. Yeah. So you're singling it out there. Yes, you're singling it out. Double negation is a closure operation. So if you apply it once, you get it closed, then you may apply it many times. Sure, sure, sure. But then, of course, that doesn't capture the complementary structure. But the algebra is where the closure is... It captures. No matter what. If, again, if you have a semi-group which is a semi-group with involution, then you So we can complement it, and we can take all the complement, and such. But semi-group needs to have an evolution. Just for any occasion, I told you, if you mentioned it, I will say, but this was essentially based on semi-groups, because at that time I was really thinking that I do this, I do that, and I had no addition. I had no linear combinations, weights, and so on. But, of course, it's not. It's almost a moment. Fortunately, because even to characterize the set M,

10:00 it's already something, I should say. Because it is not a partition lattice, as it was in the case of Hildred's face. Because it is, for any Hildred's principle, that's something more complicated, even if you consider the logic of all operators. You fix them, say, in terms of product structure, you get a partition. You apply a global unitary transform, but you might have a different global unitary transform. It's really something... But it should have some structure, but I don't see it anymore. So, what you just said about applying the global ubitric transform, when you get back a different transuboric structure, is that connected, because unfortunately I know very little bit about operators, I know virtually nothing about operators in Hilbert's spaces, but is that connected with the so-called polar decomposition theorems in Hilbert's No, no, you can formulate the same thing just in terms of algebra. Suppose you have a non-trivial automorphism of algebra. And you have a given multipartite stretch that applies this automorphism and you will get something different. Okay. Which may be completely not compatible with the initial. Yeah. It is not something specific. Okay. I wanted to ask you, in turn, how did you, because you mentioned several times, incidents of Algebras, I'm very interested in how do they occur in some other context other than I described, because all the Algebras people, they... I was curious, we were actually just talking about this briefly before you arrived, we was mentioning about the incidence algebras actually was you i thought in your talk you mentioned the incidence algebras uh the incidence algebras are very much yannis's baby i mean he's always banging on about the uh rotor algebra the incidence algebra well it's his hobby horse i think didn't you start the instrument were you the one who we in fact didn't even know until i didn't know until tonight did you introduce yannis to You did? You did? Yeah, of course. Okay, well he's... And this is what I'll let us say, that Roman knows a lot more about incidence algebra.

12:30 Than any of us do, so I think... Okay, my apologies. I was... In that case, you certainly know far more than... My colleague from the Economical Department, and I was complaining that I didn't see any Yes, but don't you break such information? But look at the radius function, look at the... And they knew each of them, so they... You called them about the radius function? No, I told him about my problem. Then he told him... Don't you know the radius function? This was not in 1967, yes, yes, yes. Why are they called incidence algebers? just before you got here they don't really seem to have very much to do with incidents i mean incidents of what it's really a duality uh an algebra it's the dual of the um so what you know just before you arrive what are we talking about the what what was the let me explain a little bit about the skin It could be that the very term incidence algebra arise because they considered not any partially ordered sex, but some particular structures for which this relation of partial order was really an incidence of sex. It's a combinatorial paper there. I don't know, I don't know. It happened so, the computer spoke, probably just exactly the species of this paper, Janker and Levet, was a simple story. And I learned this notion from a book of Aigner over the reference of the initial paper. By a lot of time, then I think, of course, I should cite the initial paper, and I think, yeah, it's starting to cite. And the original papers have nothing to do with what you did? No, no. Completely, completely not. Amazingly. There's an old song there, a song about this. Not so developed. That's what you're listening. Algebras, they were not good duels. Sorry, they were not good duels. They were not good duels, right. They were not good in the following sense. Suppose you have to process one process and another process. Business process, you associate an incident of the brain, business you process, you associate an incident of the brain.

15:00 Then, if you call it a dual object, it must be not just numbers. But if you have a natural mapping between these two process, then you must have a natural mapping this way. Yes, yes. But if you have two process, the natural mapping between process are just monotony. You have no... No inverse. No functionality, it was... No functoriality, yes, yes. Yes, it was... but I was very... This is why we wondered whether it lived inside the category of some physical... Yeah, because I wanted to have a category, I wanted to have a function. I have to restrict the class of partially ordered, but why I needed this functoriality? Because I needed the classical correspondence to give limits, but to give limits I needed this composition. Yes. Well, we all like to have functorial, well, certainly speaking for myself, we always like to have a functorial framework. Yes, yes, yes, because... But how did you have to restrict the partial ordering to get that functorial? You said you had to restrict the class... I could not, not, not... Well, is that the answer that I keep, you know, has Keith answered that correctly? Did you have to restrict it to the case where you've got that simplicial... Yes, yes, yes, okay. It's one of the classes, of course, in such a past moment. Since, another class was, you have written a paper, as I was telling, that it's simply glitches, so called glitchy logics. Oh, no, no, no, no. So, it's bigger than simplicial. This is what I'm trying to get at. You don't know simplicial. No, no, no, no, no, that's why I'm... is bigger than the simplicial complex. No, no, the intermodics are not bigger, not more. No, no, the total class is bigger. It includes both simplicial complexes and the expulsions. No, no, no, because there is no sense. You can consider the extension of this class. This way or this way, you cannot unite them in the whole. You have to go away. No, well, I'd like to understand more about this. Perhaps this is not the time. Perhaps we could talk afterwards. but unfortunately I'm not familiar with the breachy logics, I am familiar with simplicity or complexes, I'd like to understand more about how you get the functoriality. I wanted to do something at least to... But did you keep it up? I mean all this working on the parallel to face time and connection with incidence algebra I mean, is it sort of exhaustive, because you just went both ways, and I mean, two papers, which we look at here, then...

17:30 Is it just that you're an operationalist at heart, and you didn't want to... No, just now, right at the moment, I feel... ...exhaustiveness, because it's really, I did what I wanted to do. You mean exhausted at the incidence algebra structure? No, I mean, just in this context of... He's moved on and become obsessed with this problem. Well, it's still not going to do computing. Well, it is actually. It is very interesting. This is the next issue of what we have written the big paper. Because in a sense, we summarise. Now, what we can do? We have a formal, just a formal operator. I know how to associate with states these non-numerical variables. And we, just within this formal way, we associate something multipartite with them. But now I would like to relate it somehow with something which could be viewed as a result. That's why I started. I only started. And obviously, particularly, with the separability. Yeah, but you see, the reason why I can't say anything here, because I wouldn't even get into the single part title. Yeah, yeah. Sure. Whereas, this is the problem I always put aside, because I know it's a bloody difficult problem, and I don't have any ideas on how to deal with it. I'm waiting, I'm waiting when the culture comes. No, I'm just apologizing. I can't say anything sensible. Well, Yannis, of course, would say that's right because in the end they fall from heaven, which is where they really live. But alas, I can't agree, you know, the level of philosophy of mathematics, I don't share, much as I respect Yannis, I don't respect, I don't share that particular view of method. No, no, I'm telling, I'm telling, I'm listening because it can happen that maybe in a week I will recall someone in the world, which was... But how do you connect parts of this? In what sense you say that you are one-partheid and he is a multi-partheid? Well, just simply that I've been struggling with this symplectic algebra to try and get it... I mean, if you talk to people like Gulleman and Sternberg, they know all about this symplectic algebra.

20:00 But to me, coming in from a physics point of view, there's a lot of structure there which is totally foreign to the type of structure I've been dealing with. And it was also that, because I came through this via clip and algebras, because I was working with Penrose when he was here at work many years ago, and I saw that he came through the spinner side of it, and I didn't like that. I wanted to see, once again, some of my prejudice for algebras, and I wanted to see the Clippet algebra structure very carefully. That came out very straightforwardly. But then the point I noticed was that underlying the Clippet algebra, you can make it up of two Grassman algebras. If you have a dual relation between the two Grassman algebras, you can actually construct the Clippet algebra on top of that. And then it seemed to me that if you had the two, And they were essentially the fermionic annihilation creation operators. And therefore the question naturally arose in my head, well, what about the bosons? I mean, why can't we use the Heisenberg algebra and then build up a Clifford algebra, which will then involve the symplectic structures? And I went around wondering why there wasn't a symplectic Clifford algebra, and in those days nobody could tell me. And then I suddenly discovered that, in fact, mathematicians have been working on these symplectic Clifford algebras, in the, I think he's a French mathematician, and actually wrote a book on the whole thing. But within that, there was a technical problem, and the technical problem was that when I'm dealing with the... Well, it's not going to look like an algebra, I've got my primitive idempotence, because I want idempotence in order to do what I'm trying to do. And there seems to be no idempotence in the symplectic algebra, because the Heisenberg algebra is nil-potent. And if it's nil-potent, there are no... For me, I'm always looking for idempotence. For me, idempotence are just particular subsistence, that's why... Yeah, but you see, my puzzle for a long time was, where the hell is the primitive idempotent in the, if you like, the symplectic clifford algebra? Because the symplectic algebra is the... It's the group of, it's the algebra of all amorphisms of the Heisenberg, you see the Heisenberg group is an invariant under these symplectic

22:30 transformations so coming on that side you see we had well I thought, let me start again I think I've got a better way of expressing this, I start with the and then Penrose introduced this idea of the twister. And the twister was supposed to be how you relate light cones at different points. In other words, there was a translational element coming into it. Okay? But it all came out of the conformal group. And what you do is you break the symmetry of the conformal group in order to get the Poincare group and so on. But everything was done in the orthogonal clip at algebra. And then I kept asking myself, Why hasn't he used the symplectic clippin algebra for translations because there you've got essentially a phase space and you're really looking at the, you're really looking at the, not the static structure, you know, but rather the phase space structure. And then I would have went down this avenue of saying, well, where the hell are the impotents in this symplectic structure? Now I've discovered them, you see. Do they exist? Yup, well, that's the whole point. No, no, zero exists and no zero. They exist in the finite case. See, the way I came to is looking at the finite case first, and now I really need mathematical help. Have I done the correct thing when I go to the infinite limit? But I get the same result that Weyl gets. In other words, these primitive hidden potents are actually delta functions. They're the equivalent of delta functions in Hilbert's space. Again, I am not a physicist because I understand them as positions. Yeah, and these are then my positions. In fact, I build my space out of these, I've got a very crude toy model in which these idempotents coming from discrete vial algebra are actually the points in my space. Now, my topology somehow is given in the algebra because I have a neighborhood relationship through the translation operator, which is a natural adjunct of the, of the, of the vile algebra. And... Sorry? Yeah, but your, the thing I liked about your work is that you'll, you don't have that, that algebra to carry you.

25:00 You see, uh, Clive Kilmer's actually... Well, because the algebraic structure already has these things embedded in it. You're trying to get them from the ground up, via the operations. I mean, your approach is actually more ambitious. Yes, much more ambitious. You're really trying to start from something absolutely primitive, defined in operational terms, and get the algebraic operation. It's just a clear, it's just a clear. Well, OK. I think you say, given an algebraic approach... Well, yeah, what I'm saying, I take, you know, when, when, if Chris Isham asked me which algebra, I would say I'm very interested in the, the relationship between the orthogonal clipper and the symplectic clipper, because that's where I think, you know, relativity and quantum mechanics actually come together, in a way which is not the, hoping that it's not going to be the artificial way it comes in. when you use Hilbert's faces, you've got renormalization problems and so on. In a way, the motivation is not that dissimilar from what Penrose is trying to do in his Twister program, except that in his case, he's got the rather than Basil's doing, trying to get all of his structure, see all of his structures already being there in the symphactic clipper between the Lycones out of that. He's helped himself to all of this machinery of the conformal, the upper and lower half planes of the complexified, conformal space. And then, of course, he sees the key to everything being in the holomorphic, anti-holomorphic functions on that space. And he's trying to get an algebra of directions, a complexified geometry of directions, and out of that, from which he then reconstructs the points, the points as well become a derived. But his aim was... But in fact, he's already built, it seems to me, too much and too abstract a geometric structure in. That's his starting point, whereas what Basil started trying to do by going back to the structures in the algebra themselves, it seems to me as conceptually much more clearly motivated. I think more than that, I think he's left half the structure out.

27:30 That's my biggest worry about what he's doing, is that he's left the essential part which will take you to quantum mechanics, i.e. the Heisenberg Archer. And this, of course, is why he's got a hell of a problem, why he can only get the linearised graviton out of his, and even then, by all sorts of rather arcane tricks, and why he's got this problem with the so-called... Ah, but you know, because I saw the point slightly different, that when you're given an algebra, you basically assume that there was an underlying space, and you just, when you're defining this algebra, in fact, you do, I think, as an algebra, which functions, say, with... Not, I think, not in the approach that Maslow's taking. No, no, you're not trying to build up the space from the algebra. is the approach that Roger had in the early days when he was here at Birkbeck and we used to talk together was the idea that you want to be you want you don't want to start from space and time yes you want to abstract space yes okay and that was what he was doing there and that's why the light cones the robertson congresses etc was getting a geometry which didn't depend upon a given manifold which is still doesn't in his approach but he does well except he depends down on a much more abstract, complexified, conformal. My worry is that he hasn't got what I would call quantum geometry built in. To do quantum geometry, you have to add in or relate the symplectic structure. That's my point. The physicists don't play with the symmetric structure. They do it in a way and they don't realise what they're doing. I think that's probably it. and there is a very rich structure that sort of reflects both the symplectic part and the orthogonal part and I now think from my latest and I don't know whether this is right because I haven't explored it enough you know either privately or in public talking to these guys to know whether I'm really talking sense or not but I've actually seen now how these two things are actually the same thing they're different aspects of the same thing but I mean that will next week's seminar next few weeks I'm trying to see but anyway sorry I'm just saying that what is missing the point you were making was why am I apologising because what I've always felt is missing

30:00 from this is that this is really looking at these arguments but ultimately we've got to get this idea of some sort of classicality coming in and this is a problem that I don't know what the answer is my natural inclination is to say we need some new physics but I sort of call that, what new physics you see, whereas you're saying look, forget about the physics, we've got a mathematical structure here, surely we ought to be able to discuss this within this structure in terms of the conditions on the stage, and this is what I think this is why I'm speaking to myself I liked your approach and the remark that you made when you speak at Imperial, about trying to get the points out as irreducible representations of the older ideals. I emphasize one thing, which I would like to learn how you would appreciate it, because, in fact, it was a point of very strong criticism of our Luciani's approach. I didn't encourage too much, but all the physicists, of course, they disliked it. The very essential point is that if we would like to get some non-trivial topologies, the algebra should be not a star algebra. It should not be inundated. Should not be inundated? Yes, yes, yes. Yeah, I'm sorry, I restricted the star algebra. Why was the criticism, what was the basis, why did they say they had to be non-involutive? Why do they have to be non-involutive? You can see simply when I represent them in a matrix form, they are all upper-triangle, otherwise you won't get... Oh, okay, right. So, just, directly it looks rather simple, but it was the main, main objection, every time when I spoke about it on physical seminars and in Italy, in Petersburg, again, yes, it's really nice, but, you know, don't use his star algebras, they are physical, that's all. I ask, but why, why, why this stuff? Because it should be symmetric with respect to time. Why should it be symmetric? What is it? We should be looking for something that's not symmetric in time. Yeah, Yanis told me the same, but... Yes, he's right! But I, I, nevertheless, I would like to reply something to people.

32:30 Look, there are two processes going on in quantum physics, and these two processes are entirely different structures. There is the time evolution, which is going on via the Schrodinger equation, and then every time you want to take a breath, you bring in this other process, which is the collapse process. And the collapse process is not discussed in the mathematical structure. It's something that you put in from outside. Now, what I want to do, and what a lot of people want to do, Bring that in to a total structure in which the collapse is an actual part of the dynamics. So what I would be looking for are semigroups and that kind of structure which has a one-wayness built in it. I don't want symmetry in time because we don't have symmetry in time. Ah, so for example, from your point of view, it does not look quite unphysical. I'm coming from a very, very different point of view from Basel's, but I think David Finkelstein would say the same thing. That's the reason for his interest in semi-groups. Again, he wants to break the symmetry because he also wants, as well, to build in the irreversibility right in the very basis of the structure. And this is what people like. For heaven's sake, surely that's more physical. I mean, why on earth anything less physical than time symmetry? The trouble with physicists, they don't want to be disturbed. That's what the magic carpet is all about. no worries yeah but there is look whenever you look because what they do is what they like is what they what they can do not what is necessary okay no the point is this whenever you look at the unitary evolution we know that if you start bringing thermodynamics into it the entropy is always non-changing yes but we do not observe it in our life exactly so what the hell are these physicists talking about? What we observe in our life is irreversible. Everywhere is irreversible in it.

35:00 So why do we, why do we add it from postulate 2? Actually von Neumann called it postulate 1, the collapse. well to save the phenomena of course there's a jump in there and the jump you see is related to this because you have to be you have to climb into the classical world to put the jump in it's very strange everybody says quantum physics is the basic physics from which the classical emerge but you can't talk about quantum physics unless you've got the classical but the classical doesn't exist without the quantum So there's a kind of a two-way process. Both are necessary. So the question, I feel something is missing. Two-way vicious circle. Now, the reason why I went on to C-star algebras, and I mean, okay, they're probably limited again, was because there, with this work of Umazawa, I don't know whether you come across it as a physicist's approach rather than a mathematical approach, but it's in the mathematics anyway, where you have, where you double, GNS construction. You know the Gelfand-Lehmann-Segel construction, the C-Star Algebra's, no? Probably, I... Okay, well, I'm not... Well, it's by standards in Marseille. You can tell me... Okay, well, the idea is that you double up the number of degrees of freedom. Ah, yes, yes, yes. Okay, and then you can find that you can actually get the thermodynamics taking place. You can actually go straight to the mixed states. you can go from mixed states to pure states and so on in that in doubling up that well no it's not it can be happened it's not that the idea was to do more than that there's a tracing out i mean some of the problems in this separability and decomposable is this this tracing act because you can you can make a you can start with a an entangled state tracing out and you get a separable state and vice versa. I mean, the problem is there doesn't seem to be any clear way of discussing this separability and entanglement, given the ensemble. I think you're probably up against it here with that problem. Well, I mean, that's an aside. Oh, yes, but what has not...

37:30 That has only just sort of been developed as an afterthought in the last five years, this business alone. And even now, this is going to happen. So what I'm saying is that they're saying you're building in irreversibility. Yeah, we are a contrabanda, you know. Gilbert, you know. We always say unacceptable things. But it's dated upon that. We are carbonari. Banditi. He shouldn't have his face to the wall. Don't stop me. It doesn't work. Well, even Galmar has given it out because... There are several answers to this, and I hear everything I say comes from a prejudiced man. These are the thoughts of a prejudiced man. Yes, yes. I tried to read the initial clip, but I did not understand. Well, I actually heard Robert Griffiths trying to describe what he was doing. Yeah, but the way he did it was to redefine what we normally mean by an event. He already had the collapse built into everything that he was doing without realising it.

40:00 And I once, and I, okay, so that's just one point. I had a long discussion with Robert Griffiths. And if that's the foundation of consistent histories, and he's supposed to be the man, you know, they always say pioneering work, you know, then it's a load of nonsense, physically. Now, I hesitate to say this in front of Chris, but then I think he is doing something different. And what? Chris, I should. Now, I think we, although when I studied Gaumann and Hartle, who I think is in a different category from Griffiths, so first of all, we've got to throw Griffiths away, that his philosophy, if you like, what he was doing, is wrong. When you come to Gelman and Hartle, the idea was that they were actually using these histories, and then they were bringing in a new principle. Or were they? Or were they bringing in a new principle? The problem is, in order to get this collapse of the wave function without a collapse of the wave function, there's a very crucial question. are you doing ordinary quantum mechanics or have you bought in some new principle yes is decoherence a new is decoherence a new principle or is it just quantum mechanics now i've been asked yes seem to be a new principle that is not because when i sat next to jonathan uh how well yes yes on a coach one day after i i've been hearing on this talk about this I asked him this question. I said, does Galway and Hartman, do they introduce a new principle or not? Instead of him. Yeah, and he said, that is a very interesting question. Okay, the point is not that you can help with a few... No, no, but you talk to Omnes and he says, no, I am not bringing in any new principles at all. I am just doing what Bohr asked us to do. So there seems to be some confusion. Now, I'd like to know what Chris Eichem's position on this is, but he does something different from Gell-Mann and Harkin. Of course, doing... Because I thought that it was a similar-transparent interpretation of what... No, because he goes straight to the logics. And then he goes to hating algebras and things like...

42:30 But now, of course, ties it all up with topos there. And in fact, to an underlying dynamical picture, that's why... modified logics or is he well initially I think he was attracted to top of theory just as a piece of machinery to express the the decoherence functionals but now I think he sees it as actually giving us an insight as to what is going on at a deeper level which one can actually have the dynamics built in and account for the irreversibility right from day one so as it were it is no longer the quantum principle day one it's something more according to Janis he sat down and learned the theory in order to do it rather than knowing the theory the topos theory he sat down and learned the topos theory in order to do this rather than knowing topos theory in the first place that's exactly what Chris Ashland told me yes there's no question about that that's what did happen in fact it said it was the first really conceptually piece of mathematics that he'd had to learn for about 20 years. That's why I got so excited about it. But, Basil, did you finish about, because you were saying the five points against consistent history? No, and I didn't believe it. Yes, you did. A couple, I'm sorry. Go ahead, give the others. Well, the other, I mean, I think there are a couple of others in our book, which I forget now. Because we did actually... You remember the chapter? I remember when Owen was here and you were pitying the Grifin that much, I don't know. Yeah, I missed that, but I know that Owen is also a no-fan of the consistency of the approach. No, Owen also has a lot to say. He probably studied it. This is a PhD student of mine who just finished his thesis, and he did, in his own, he's a very good, very, very capable man, and in his... Excuse me, you already gave company. I give all my students who deserve it compliments when they leave him when they are with him he never gives them but when they leave you never said anything right about Owen now first time I hear him you said a lot of very nice he said a lot of very nice things about Owen behind his back but when he's actually face to face with him

45:00 the exercise, that's what your supervisor is for. But I'm talking to him, and I'm putting a lot of effort into it, and if I'm putting a lot of effort into it, then he should know that what he's doing is very significant. I think he knew very well what you're planning a bit more. Any effort into yours, yes. That's because he's too much in awe of you, I'll say. You know that. If I could just reduce the evening to complete shambles, of course, asked us to do it's a bit like theologians saying we're doing what God asked us to do I'm well aware of that I want to get from one of these guys a very clear statement what we are doing which is different is and I haven't got it from and from Griffiths I just got more and more confused I wrote a 12 page email letter to him and he came back the next day which obviously didn't read it because it took me about a couple of months to actually sort the whole thing out telling me that I just didn't understand a word and never answering one of the questions I put to him. I mean, I just, you know, and then I just gave up. Because I think there was some connection between what they're doing there and what I'm doing, but because they've got a sort of a sharp, very, very limited view of the physics, the physics seems to restrict them. Rather than saying, look, forget the physics, structure is and see if there's some clues there. But it's one more question. No, it's a problem. Sorry. What I wanted to ask because, again, if I meet Chris tomorrow, I'll try to ask for him. What I, again, maybe somebody could tell me, what could be the evidence of the existence of superpositions, of histories? Because otherwise, again, to represent them as a kind of product, it would be a redundant structure. Yeah. Because superposition, it's just everything. If there is no superposition, it's just you're not dealing with linear space. You're dealing with something different, like direct sun or something different. You're asking for physical evidence? Okay. Well, any kind. Because, for example, if I have a homogeneous history, at least I can understand it. I did this, in a couple of hours I did that, in an hour I did that, and then I got this, and this, and this, and this.

47:30 I did homogeneous history. But what is a superposition? What is that? I don't know. Don't ask me. I mean, I mean... I tried to get from Griffiths what the basic idea was with these chains of operators. I said, are you doing a measurement? It actually is. Yes, yes. I understand it in the same way. What it actually is. But have you seen that there's been two or three very nice papers by... What's his name? Giancarlo Girardi. Oh, Girardi. Oh, Girardi, yeah. He's down there somewhere close to you. Yes, yes, that's what I listen to. Now he, if you want to know what's wrong with consistent histories, talk to him. Because he's written two beautiful papers and I agree with every word he's written in those papers. But then you need to read an issue. I didn't have your idea, did I? Yes, I had written about the... You've quite a particular paper, 94 paper. How accessible is it? No, it's not that simple. Chris is never that simple. He's a deep thinker and you've really got to stop and think. No, just before he got interested in top-iron. I think I know that, yeah. Is that the one on the coherence functionals? No, it's the next one. It's the next one, okay, I've got it. The next one was the criteria... But do you have some interest in physical questions? I see it. I mean, or do you... Are you into rather trying to serve some nice mathematics for physicists? So you're just selling stuff around? That's why I asked him here, you see. We have a little bit of money and we might like to try and buy these ideas. Because I don't want, I really do not want just to write formulas by themselves. You mean you feel it has to be used. Very interesting stuff, very interesting stuff. I am not a mathematician in this. So you are sort of like working with physicists? I like special cases, I don't like general constructions. mathematics as a service industry? Yes, yes, yes. I like to be myself a toolmaker, not just having a headman talking in my languages. I think much of the greatest mathematics is actually created by people who do

50:00 see it as a service industry, rather than No, I do not even get this in you. I just take existence. There are a lot of things which I've already done and run around... But you see, it's so many things standard, even when you find two things and put them together, It's very new. Nobody managed to read through all millions of papers, for God's sake. I see. It works. I mean, you know, in one paper you write that you are using enumerated combinatorics topology, but used the direct notation. Oh, yes, yes. So it's putting together three things. I see. So now, like, your greatest problem or whatever challenge is this quantum... To see the original structure of this. If I see it, then I can go to the room. Well, if you could manage to explain to him, he perhaps could provide some mathematics, since he is just a person who is serving his mathematical knowledge, Which is very cool. It's a general illusion among physicians that, now I will tell the mathematicians what are my terms? And you will just... Forget, forget, I don't know what you want to do. But you always complain. Is the mathematician interested? I'm complaining that you need mathematics, and then you find someone and you say, but we don't understand each other. Well, it does sometimes work out. Look at Monk, Minkowski and Einstein, they have been pretty, they have been very fruitful. No, that's, I mean, that's... Look at the interaction that Maurice de Goussons would have, just considering we were only... Absolutely. No, with me it's never happened so. I've always, I found, in Physicists, to collaborate with my whole purpose together, it was never that you accept one in some period being passionate, like a prostitute, like a gold, different science I should say, but all other examples of my collaboration, it always was from my side, you say, I'm a except karls watson karls watson from vienna but again simply he carried out some mathematical

52:30 formalism and we have written one paper just compared to the school I don't know one of my messages because I have written simultaneously with him the paper with his continuous substitute. Exactly, because you were using it so much often. Yes, yes, and I sent it to this group and I have already been about to submit it. And then I saw his paper and I sent it to him as editor of the International Zoology. Should I submit it or not? Because you are one of the editor and told me what to do. He didn't reply me for eight months and I submitted it. But maybe his system collapsed. You should always email him twice. You know, maybe his continent collapsed. It was Russia. What email? I had no email. I had to pay myself for mail. I couldn't send a letter. No, maybe, you know, people don't believe it anymore. People don't believe it. Unless you're in there, perhaps. No, but I mean, if you've got an editor, there's so much stuff coming. It's in there. I mean, there's so much... Is there a person who is... Yeah, it's because... Yeah, because I believe after we... after Yanni's talk here, and then we'll discuss with you, and then it will sort in like... No, but I would like to explain, because I think it's appeared before, before a while, and... Oh, yeah. Sorkin's paper was before. And in fact, these ideas, I simply was not aware, but he, of course, developed them before. Now, of course, I think. In one way, it was some sort of development of Sorkin ideas, because it was like what he had done. Yes, yes, yes. So, strange that he never commented. I was not aware, I think, about that, but after that he used it in a good way. With due respect, I find him a strange person. I didn't, you know, he was, I've only met him three or four occasions, but somehow, he didn't seem to be any, he was very nice, he was very nice, he was, he didn't open up to me, he opened up to you, that's interesting, that's interesting, because when I was with him and Yanis, I was just listening to him talking to Yanis, but I found it very, it's a chemical thing, sort of stand-up, yeah, I mean, because we discussed Ford, I don't know, maybe, maybe you got it on the right way, maybe, off in the way,

55:00 He's very, very shy. That's probably it. He's probably just... What was in fact shyness I misinterpreted as standoffishness is very often a very easy mistake to make. Especially if you're rather a kind of gregarian, sun-shy person. You may understand alcohol. But you have this champagne here, but you're showing us your photos. I don't know what you mean. Non-vintage of coffee. We need it. It's super much. You keep it for some event. We're going to keep a bottle of really good vintage champagne for the day that you finally get the answer to your... Oh, God. You finally get the whole structure out. Are you sure you wouldn't like a drink? I'm very suspicious because... What? ...the kind of social behaviour which I tend to be part of. What? I think that's... Because I'm afraid that people will think that I'm a normal person. What? What? What? I'm a normal person. People may think that they're afraid of this. Well, you don't want to be a normal person. Well, come and have a drink. You are not doing any of those things which normal people are doing. This is what you say. You give them a tea, coffee and you say, Oh, I don't. I don't drink. I don't do other people. I'm not so clean with it. He's nicotine. Yeah, but he wants something. He can have that as well. Well go to the bar and you can bring other people. I don't know what is nicotine. No, he said. No, he said. He doesn't know what is wine, what is beer, what is cigarette, nicotine, alcohol. And you are doing two projects in Italy now. I don't know whether you are aware of the webpage about yourself, or it's written that you are doing two projects in Italy, one is about something and one is about something else. No, no, no. You were not attentive. You were not attentive. I could show you, I could show you. No, you were not attentive. If you would read attentively, right, the moment I'm in the one. But the two were in Starlab in Brussels. It was corrupted and stored this in Stoke. It was corrupted. Yes, yes, yes.

57:30 Only I started, I could catch planes. I'd love to ask you some more about the functoriality do you want to go and have a drink now or do you want to just talk for just a couple of minutes more can I just ask you a little bit more about the functoriality and what it is you have to impose on this structure in order to to get the functoriality and particularly the question that you were dealing with about the distinction between the simplicity or complexes in the more general case you I was I didn't quite follow the answer there for the look so what is it generally see breaching process yeah is everything that you can You can't associate... It's really dual object. It's functional. Sure. However, they are dual. Dual, they are not... It's not a functorial dual. Why? Because if you consider the category of possets, what are the categories? Not only objects. Objects plus mappings. But mappings in this category are monotone mappings. And if you have monotone mappings, you do not have any factor which put you in any reasonable correspondence here. You don't have the odd vibrations, you don't have the odd... Yes, yes, and that's why you have to restrict in different ways. You have to restrict the category of perfect, but you have to restrict everything. And object and morphisms as well. Because I consider simplicial complexes and simplicial vectors, not repeat-heading vectors. Sure. Simplicial, and only with this restriction, it became a category. Right, well, that's the point I wanted to press you on. Because I had not understood your previous answer whether it was only by restricting it to Simplicial complexes. Did you get the functoriality in the category structure or whether there was you then went on to say that there was some more general Look, you have the whole... Yeah, well, you... Possets? Right. The category of Possets. I may select the category of Sympitial Complex. Here, with Sympitial Complex. They... It will become a fact. I can select another such category of this creature logic, they will become a fact. But I cannot unite them together, because they are different. Right. Well, you have united them. What would... What the question is? What's the biggest thing?

1:00:00 This is the biggest one, this is the maximum, but you cannot unite. Because the morphisms are different here, you cannot... Well, because you're using different mappings. Ah, okay, that's the answer. Simple. Okay, now we've got it. Now we've got it. Yes, and more if you need something particular, I have some experience already building this, especially if you really would like to have some special class of processing, you would like to make for the dual category, I could, I could have destroyed, or I do it quite quickly. I would be very interested, actually. Simply, if you have such kind of lead, just send me the description of this class of process which you would like. Okay, okay. Well, I've got your emails. Or I do it very quickly, and then I will enjoy it. Sorry, say that last sentence again. If you send me a description of this class of process for which you would like to have this material view, then either I reply in a short time and just tell you the answer, I have something. Sure. Well, I think there is a question like you don't see it. You don't? OK, but the point is you haven't worked yet. But in that case, I'm not quite certain why you're so certain that there is no general condition. Because we want to know how to construct the mapping. Right. So we've got a different mapping. We're using simplicity of mappings for the simplicity of the complex. It's what we're using is completely different. Different mapping for the Grecian logics and variety, any other? Yes, yes, yes. Okay, but there is no, so, and I ought to know the answer to this, there is no general condition that you have to put on the category of pro-sets in order to have a dual... Pro-sets and mathics. Oh, well, it's pro-sets. Oh, I said the category of pro-sets by implication, that's going to be into the category of pro-sets and mathics. Sorry, sorry, sorry, in order to define the dual category. No, you do not have anything general and non-trivial. Okay, no, fair enough. I brought your paper, maybe because when you came to Imperial College, I said your title is See, you have to look out one side. I just read it out for you. I have seen something completely different than you do, because he said that he will be talking about

1:02:30 topological entanglement and I just remember and that was printed off the LA archives you sure because it doesn't have the usual No, it was this, which I sent around. You sent around to me, yes. Oh, okay, so is it on the TV? Where did you get it on the TV? Sir, if I need a copy, I can get it from you. That's where I'm going. Can you send me one? Well, I have to just press the bloody button there. It's only cool. Maybe, have a look. Maybe it's already there. Well, could I trouble you to run a print? I try to get rid of it. Thank you about that as well, sorry. Not all the way, but of course... He's going to fast go on the way back to Dover. I need to do this fast over. No, maybe to heat, though, and then... No, no, I don't have to. Oh, I could attack... Oh, I don't like planes either. If you have an electronic phone, because anyway, I'm afraid to damage it, because the weather may be raining and... It's got a plastic cover, that's why it's got a plastic cover. Did you bring your boy from Italy? Mom, are you on a bike? No, it's in the ground floor of my hotel now. I see, okay. No, I mostly travelled in train. I have my own way of travelling. I will make a new website. But for some reason, something's already made me remember something I wanted to say to you. I've talked about a quantum log of the fact that it was on the LA archives. And you said it was, wasn't it? It was a completely different paper altogether. Was it? Yeah, I mean, you've not seen it ever. No, no, I haven't seen the one you... I made a copy, but I forgot to make it different. It's a reminder about it, so... Who's that? I don't remember the name at all. There isn't anybody we know. Well, if you do, let me know and then I can do a quick word search, a name search.

1:05:00 It's just a quantum holography. There's only... Oh, that's true. You can do titles. I would like to tell also, for a new question, that if you really would like to... So, I mean, there's some restricted algebraic thing in which you think that I can really help you to provide. You may just even come to Turinia for some couple of days and just simply to maybe speed up the things. Especially now, after beginning to make the business. Well, the bill of building is not going to be until next year. It's going to be in 2003. Where is it? It's going to be in Tuscan. It's going to be near to, near to... Parry? No, no, not, not Parry. I'm not dealing with your... You're not Parry. You're not Parry. You're not Parry. No, no, no. No, not Parry. No, no, not Mr. Pete at his, you know, $5,000 a day. No, I've managed to find, uh, Alberto Polizzi has a very good friend who I've known many or so-called, Umberto Mionchi, who wrote the after-long category theory in the Encyclopedia Italia, or Talia Cup, although, and he's got a place of old farmhouse in between Florence and, um, no, in between Florence and Zebedee, exactly. That's a private joke, I have to explain it to you. No, it's actually not in between Florence and Zebedee, it's going close to class, which sleeps about 12, and we're going to have a small conference down there next year, we have yet to decide the dates, in honour of Bill Walvia's 65th birthday. This is the more creative topos here, the guy's going to be coming in July to spend some time with Chris, who I've known as a friend for about the last 10, 12 years now. For any occasion, I would like to announce, again, if you like, in the first week of July, there will be a big conference, it's called Quartal Structures, yeah I know about that, and that's in Vienna, isn't it? Yes, yes, yes, I will definitely participate, very different people who come to this. Are you with this Professor Lambert? Lambert? Lambert? Yeah, Lambert. Yeah, he wrote a very nice paper. No, I know, I know, very well, of course, I went for a month or so. Are you, you're based permanently in Torino now, are you?

1:07:30 No, he's in three years. But you are in Torino. Permanent or just visiting? No, no, just visiting. So you're permanently in Torino? No, no. Good, okay. Well, no, it's just very interesting, this guy I just mentioned, Peruzzi, he's a philosopher, but with a very great interest in topos theory, quite a lot of papers on topos theory, is going to Torino next year to the chair in philosophy there. Ah, to the university? Yeah, he's been at Florence now, but he's starting... I mean, because you know everything, which is quite nice, in a sense. I mean, to put it in a newsletter, an electronic newsletter, I mean, I could... You can't use a computer. Yes, that's one, that's one, that's certainly part of the answer. But how could you have such a memory? That's right, isn't it? It's amazing. I mean, you... The memory is a bit trivial. Private joke. No, I don't think that's anything. I just keep things on the computer and... I think Lloyd has not learned the computer. No, that's for sure. Maybe this is why he keeps his memory. That may be as Plato made the same argument about writing. On the opposite side, because I, when in Russia, I was a head of the computer department, and I, more or less, well, with computers, but I trust them. Well, that's why you know enough not to trust them. The more you know about computers, the less you trust them, in my experience. was to trust, but that's what I thought. I suggest that we all go have a drink. That's okay. If you are a guest. You really don't. If you don't want it, just check it out. It came first page with an email of Lou, so if you don't find the paper on L.A., I could also thank you, but I don't know what is your e-mail. Is it this Italian e-mail of your body or Russian e-mail? We have several e-mails of your... Any works. Any works, okay, so... You could just watch me in the world, but... I wouldn't want to. I'm sorry we've not been able to help you, really. And, and, well, I mean, we can solve the worst problems. You know, in fact, how, for example, this reality arises? I gave a seminar in Napoli. And there were some questions, they said, they mentioned that, what did you consider, for example, this idea, or something like that?

1:10:00 I realized that I couldn't answer it in a proper way, and then I realized, oh, of course, I could not answer it, because it was not, and I had to do this. So maybe now I do not realize, but maybe you told me something very essential, right? Will be nice. Have you got something with your address in Italy? Because I will send you a paper that's right. Yes, please, please. Maybe, though, maybe... Why don't we have pure idea in the paper? Is it the one on the LA archives? You've got about 17 papers on the LA archives, haven't you? Oh, yes, this is true. And your email is on most of them. So the Italian email address? No, no, no, this is an old one. I didn't want an email address, I wanted an email address. Postal address. Yes, there is. Your email address is all over the internet. This is all on the archive. Postal address. He doesn't have postal address. Write it down. Bicycles! Bicycles! Bicycles! A bicycle. Well, why is it you ride it down in the drug pool? It's easy. How much other do you want? You ride it! No, I'm not. I'm not trying to. I'm not trying to. I'm not trying to. I've been back from Oxford, I suppose. Oh, yeah. Look, I'm sorry, actually, if David Wallace didn't send those bloody, uh, the lost statistics, because those things do. I did ask him last time, I saw him. He's very bad at that, what he's like there. Is that what he's doing? Yeah, well, his journey is actually in charge. But the problem is, Jeremy's been away so much this time. He's left it today to keep it up today. And when Jeremy's doing it, it was very efficient. David is more, well, a nice, very nice guy, very bright guy. He's not really the ideal person to be a star with anything. It's beyond that, but I think that... It's probably just that they just haven't updated it.

1:12:30 When Jeremy moved from primary shop, I've got an email. Well, actually, at that time, the list was still being done by Guido. Guido Bacigalupi, he was in it until about two years ago, and then when he left the States, he was showing it to get over. I have to say, since Jeremy's been doing it, he's maintained it very efficiently. Well, the last one I got this check was about two weeks into Oxford and so on. That was already about two years ago. It was longer than that. Longer than that, yeah. Probably more like four years ago. And that's the last one I got. Oh, in that case, you have been. Yeah, you have been left out. In that case, I'll just have another day with you. He should mail it to you here a bit. In fact, he should just mail it to the TPIU site. I'll have a go at him to do that, just to mail it to facilities guys. I'm on the Cambridge HPS thing. Yeah, all they ever do is to actually send an email, they'll always just send an email post to me, I just get that. Well, there's something, all sorts of junk, you know, including about the local politics. Let's have a drink up tonight and all this stuff. Well, the spirit is willing to... So I get about five or six a day, but... To be honest, the HPS of Cambridge now, apart from David Caulfield's local philosophy of mathematics reading, is really not a good start, actually, because he's always had the, he's all renaissance magic and, er... Well, there's Lipson, Lipson's a guy who often, you know, somebody Lipson... Yeah, Paul, oh, Pete Lipson. He's a pretty bright guy, but... Well, Pan told me it had gone down the street. The trouble is that all the really bright philosophers of physics have gone to Oxford, and the only people left are being historians for people like Stephen Duck and Stephen Schoenberg, who's in a very nice guy, if you want to learn all about what, you know, Hobbes, the Bible, and the airport, you know, obviously, because this was how, you know, my first time, when I watched it, I took X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X It's because they get so much money from the Wellcome Foundation, the history of medicine people. Really? I mean, there have been some very white people who, right now, there are no really good philosophies. It was when Michael Redhead left that chair. He took in with one of the black students who were alive at the...

1:15:00 At the CPNSS. Yeah, at the CPNSS. Or they went back to Germany because they had a lot of... And all these black Greek guys, too, and tiny offices, people who, again, are now all at the LSE. I don't think we've had any reason to go to Cambridge for a long time now. Well, I'm going tomorrow for David's reading group, but apart from that, which is every two weeks, I don't always go to that. It's only when they're discussing something particularly interesting. Last time they were discussing attire, that was interesting. to talk to them about Grotendieck, I'd help us. Not what, Grotendieck. What are you going to tell us about? Yeah, it's all right, I knew you were going to say that. I shouldn't have said that. I thought at the moment I looked at Basil, I thought, shut up, I shouldn't have said that. Will you talk next week, because you are saying now and again that you've talked about the question of the results and the people around the world? I do have something I'd like to talk about, about Gropendieck, Gropendieck-Law here, McClain, what's been going on in the talk about, homologies, and, well, you know, from space in... I mean, the only guy who tripped you up is... Yeah, yeah, that's okay. but it won't be a technical it'll be very very kind of qualitative and what's been the general kind of thrust of the ideas and changing shape of the subject and why I think it might be relevant to what we're trying to do that's the most important thing but it's that last part it's in all conscience it's simply because I haven't felt that I had anything worth saying about whether and how all of that is capable of tying in program for finding algebraic structures that signify the point of the process there, but it's three months' time. Well, let's talk about what I've got. I've got to go back down to the... What I am curious about is that they have to be in a very contact with the working

1:17:30 I'm thinking that they didn't work together, because Sorkin's most interesting piece of story was actually incident algebra and what, and connection was Sorkin. You see, I just don't know Sorkin. The only thing I've ever heard Sorkin talk about was the causal set theory stuff. To be honest, I thought it was just too much of a simple-minded toy model. Well, the only thing I've ever listened to Sorkin talk about, which is the causal set. Now, I know everybody says this is a very bright guy indeed, but he and Lee Smolin are the two of the brightest people in quantum gravity. So I know the guy's bright, but the only thing I've ever listened, heard him talk about was just this causal set, you know, getting back Lorenz invariance from the causal set model. And to be honest, I thought it was a bit... I just thought it was unspecified. today. I mean, old one, the 87th. Yeah. Well, it's... Are you really ready for either? So, space time is, of course, the 3rd of August, 1987. This is the first space world. This was intro. Ah, this is for review letters. Yeah. Ah. Are you, if you want to go and copy copy now? I would love to. Yes, yes, yes, yes, yes. Right away from the second. Yes. Instinctly. Instinctly. I'm very kind. I wasn't going to ask, but I've sent it to you, but it seems to be so bad. I knew that you have sent it to me. Can I just get my Natasha case out? Oh, sorry about this. It's all right. Excuse me. Yeah, just let me run down the corridor and do that. I'll give you a round, I'll give you a round, I'll give you a round, I'll see you next to your round. Is it open? I don't know, is it? Has it got it? Is there any way you can let me have the key? I don't know, I certainly don't, so I won't be able to do anything unless you do it. You shouldn't have to do anything unless you do it. The nice thing about Fidder over there is that the contribution's left to be very short as the pocket of four. Well, if you look on the east side, you're not bad, but you're not bad.

1:20:00 You're not bad. You're not bad. You're not bad. You're not bad. You're not bad. This is part or whole of the proposal. Thank you. Thank you.