Presheaficiation of Matter, Time & Space — Notion of t-Topos

0:00 You don't get any real deals, you see. I love that it has been a bad arrangement. When the stock market was good, people were very happy. Okay, I suppose we are ready to start the afternoon session. It's a pleasure to have with us today Professor Goro Kato from California Polytechnic Institute. And the title of his talk is Pre-Chifification of Matter, Time, and Space, the Notion of Tiktokos. Thank you. Thank you very much for the invitation. This is my first trip to London. But I wouldn't say, as George Bush said, this is my favorite country. Well, that's a relief. He said that for Amsterdam, right? He said that for Amsterdam, right? But Amsterdam, that's right. Amsterdam is his favorite country. A lot of people are the same. I would like to today talk about basically those topics. It's a very direct application of very standard topos of Grotendieck in a sense that domain category is a category where a Grotendieck topology is defined, and the target category a little bit which is the uh we consider a product capital and the connection with the physics i will make it clear as i go uh so anyway first i'll start with uh uh recall the definition of a site and that meant that after that i would like to talk about various states or even a pre-sheave in terms of pre-sheaves and topos language. After that, some explanation about double slit experiment.

2:30 I will use this factoring monophysm type explanation. The next one is rather shaky, since this is not my conference, I may be allowed to tell a lie a little bit. So dependency of two pre-sheaves, Kappa and Tau, which actually hypothesis I would like to make, those two pre-sheaves are actually sheaves, whose meaning I will explain a little later. The last one is this, I wrote subplank-scale objects. This may be much larger than, actually, plank objects, which I don't give a guarantee. But subplank-like scale objects may be better ways to say it. Anyway, that's a basic topic so I'm going to have to come back to. By the way, I never met Ioannis in my life before except email correspondence, and he asked me how I look. I have a great moustache, and I wear 172 . And I look like a samurai. But after I said that, I don't look like Yojimbo, you know, so I'm a little bit cleaner than Toshiro Anyway, so here's the first topic I would like to recall in your memory. Gold and Dick sight, simply sight, is the place where sight, sight is a place where we can play with shift theory. So minimum condition to do shift theory, the power, as for an object U in this category S, we define the families of monophyses. I denote it by fi. So that's the definition. That's the definition. And two things, covering over covering, and the composition gives the another covering w, which is the one, oh that's the second one. Okay, the first one I wrote one point one two is the four morphism for this one

5:00 Pull back again give them covering So this is actually Characterizing the usual motion of Covering in terms of political spaces except just rewriting into categorical notions so product of two objects is basically intersection in terms of separate topological space and the second one is the covering of a covering competition again gives a covering which is reasonable axiom for covering if you consider most traditional meaning of covering and the last one is we can so that we can say each object in this you have a covering just take if you're stuck you can take you that you prime as you itself identity isomorphism so we can talk about for each object in in a category s we have a cover if those things are given then we say this object This guy is called a covering family of you. Okay, that's a quick review of a site. Next, T-topos. T stands for temporal topos. I was thinking to rename it the wrong way. TG-T-topos. Two general temporal topos, maybe. because this S, category S, which now I can call a site, and actually now I may call this one even temporal site if it is used in this context. This S, site S plays a role of, in my opinion, more essential concept of time rather than usual linear time, way okay so this is the definition s hat is nothing but the usual definition of category appreciates defined on a site s where the target category is product category and how do we connect with physics is this this is the index set

7:30 And I choose just, I choose one, the one. So C1, I assume it is a microcosm discrete category in the sense that each object is a particle and no morphisms except identities. So C1 is a connection with the physical world. That's the definition of T-topos, which I call S-hat. And after that, S tilde, of course, we can talk about, which is the more traditional definition of topos. S tilde is a category of sheaves, which I will mention a little later. Well, in the middle of a talk like this, just a conversation came back to me that I have no fear of going to sleep since I'm walking and moving. but after lunch you may feel like taking a nap. If you don't snore, that would be very nice. Okay, so this is the definition of S-hat. Okay, the next definition is this. Now, this lower case M, this is a pre-sheave. Pre-sheave is said to be manifested when the pre-sheave is evaluated at and an object, which is the object of a site. Now, I would like to call this a T-site, or a temporal site. So this is manifested when MU is evaluated. And that case, I would like to, already ready to mention two types of states. Okay, before that, of course, this object, MU, is an object in this product category. And also, by the way, I'm a little bit sloppy in the notation today, in the sense that if I want to regard this guy in C1, the discrete category I should talk about, projection into C1 from this product category, sometimes, according to the context, it's obvious, but I have a little bit sloppy notation, so please be aware of that. Anyway, so MU, so this is manifesting, first, first, that's the definition of this.

10:00 Okay, so let me introduce already which I mentioned a few times, maybe not today. An object in this temporal site, S, is said to be generalized time period. Of course, this is almost emotional feeling is already embedded in it. Why do I call this a generalized time period? I hope at the end of this talk you may accept why or some excuse why I would like to call the object as a generalized time period. Okay, anyway, I may use the notation quite often in this talk. Okay, next, for a given pre-shift M, first I want to define a particle state. Particle state is this, when M is manifested at a specified generalized time period. So, for this one, where did I write? I don't know. I hope I wrote somewhere. Manifested? Manifested. It was previously. Oh, previously. Okay, so when U is specified, I call that state a particle state. Oh, yes, as it says. When M is said to be wave state, when an object of S is not specified. So we have basically two situations. So U varies in some sense, U varies. So I use this notation for indicating the wave state. or sometimes even not even unmanifested state when he's not even chosen at all is also another type of wave state I'm sorry yes the word this when it says in this context what does this refer to something on the previous slide is it oh yes in this context in a sense that this is just the usual site and when S is used as a domain category, with a target category, the product category as I defined, so that the first component, so to speak, C1 is a discrete category of particles. What's discrete category? I mean, does that mean a discrete category? No. Discrete, it's a category of particles which are considered to be discrete.

12:30 Is that a well-known category that we're meant to know about? or is it something you're defining? Oh, yes, it's perhaps new. Okay. Because, you know, we know what a discrete category is. I don't know what a discrete category of particles is. Oh, yes, so if I switch for the right way to say it. Okay, C1 is the category of particles, or the category of quanta, where the non-morphism exists except identities. Yeah, so it's a set. It's a discrete category. Yeah. Yeah. It seems to be the category of set. I stuttered in logic, sorry. Yeah, discrete categories are set. Clashes are sets, yeah. Is that okay? May I continue? So after defining wave states and the particle state, I would like to talk about. Oh, yes. Next is this. Say, a given particle, which I call the lower case M underlined. This is said to be pre-sheet-fiable, if you can find a pre-sheet M so that if the M underlying particle is actually a particle state of appreciation over this generalized time period. Okay, let's just another terminology. Let's see. Ah, let's see. Oh, okay. That's the next notice. Next, I would like to define in this way observation measurement. We say this, pre-shift M is observable by P over the generalized time period U. In this case, since I talk about the morphism, the observation is done in non-discrete category C-alpha, 7, not C-1. It's defined to be when there exists some morphism between them. So this is the definition of observation. So SU exists between the two. And also, a typical case I'm thinking in my head is that P is like a human being, is my personal feeling about this.

15:00 So M is the one that is observed. P is the observer. Okay, so the other case I would consider when preshape M is not observed, there are various possibilities. One, they may have same U, but the morphism doesn't exist. But this case, so this is not observed at all this guy is this is again in the way stick and this case so this is this is not fixed and no morphism between them so this also case that another case of a when that is not observed and even a manifested state of course not observe and for example this is a two different objects case we said another case of M is not observed. There's various cases of not observed. Okay, next I would like to talk about entanglement. All right, let's see. M, so M and M prime, lowercase, those two things, said to be entangled. When, as a pair of two pre-sheaves, is a pre-sheave again. This, I said over sub-site, that means a sub-category where the growth and new topology defines a sub-category. In that case, I call it sub-site. So this sub-site, I was giving a talk, and I made a minute of a talk. Professor Kaufman, Max Planck, how do you describe the two particles not entangled? And I got stuck. Oh, yeah, that's true. Then somebody said, just a sub-site. you're doing some kind person. So I should say two particles, two presheaves are entangled if they have, there exists a subset so that for each object in the subset the pair acts as one presheave. That's the definition of entanglement. Okay, next I'll go to talk about examples. So as you read here I am using E and E prime as two preserves associated with two

17:30 electrons. How long? Why electron? Oh, it doesn't have to be. No, I mean, what I mean is how much is this really quantum, namely meant to be quantum? I mean, so far there's nothing at all with mostly quantum about it. I mean, electrons are fermions, so we satisfy fermions statistics and so on. Is that built into this, or is it merely a sort of a façon to parlay? The importance of choosing electrons, for example? Well, even the importance of choosing quantum language, I suppose. Yes, if you like. To focus on the question, yes, electrons. Is there anything exclusively electrons? Just two parties? Yes, that's right. So M and M' maybe? Mainly with E. Two elephants. Why do you like elephants? Yeah, that's not two elephants in here. I see. So, two elephants, E and E. Suppose... Yeah, we heard that here first. So, let us assume they are entangled. OK, then let us observe one of those guys by P. So, by the earlier definitions, there is a morphism in a category where it's not discrete. That's SU again, I think. So, 2.62, I have two things to say, maybe more. One, since observed, we know that U has been specified. When U is specified, E is now in a particle state. And another thing is that particle state has been observed over the generalized time period U. And having that, this, since entangled, this E and E prime, they have the common generalist time period, which we call it, which is, which is we call U. So state of E over the generalist time period u is been determined simultaneously the state of e prime over the general generalized time period you also state is determined. So if you observe e over u by p over u, it

20:00 determines the state of E prime for value. Next I would like to introduce new pre-sheaves Kappa. Can I ask a general question? Yes. Because it's not the easiest thing to follow. I mean first of all can you show that only quantum physics fits this axiomatic structure? And secondly also can you do the same thing with the classical physics because classical systems can also be entangled simply by triad correspondence. I mean entangled is not actually a quantum theory. Have you matched this scheme against all quantum physics, for example, or against classical physics? Can one see that a particular example of this is all quantum physics, or is it something which is different from anything you've seen before? Actually, I don't know the answer. Does it fit? I don't know the answer. I have not examined it carefully. But you know, it's something even classical physics, right, you could think of this term very easily. So classical systems can be entangled because values match up and they separate and they're sitting entangled. And when you measure one and know the value of the other, in that sense, there's nothing that you're reading in quantum mechanics about. I see. It's just a very abstract mathematical structure. And it would be helpful if one could see, as well, I always ask a certain thing, a simple example, to see what actually is going on. It would be very nice and a very kind person in this audience can ask a question. What could you do from one of my students? Okay, so I would like to assume two things, hypotheses. Let's change the I to E. My wife is a teacher, so she will be. Two things I want to ask. tower entangled over again for over a subset and also another thing I would like to assume that cup and tau are actually sheaves not only presheaves and that means it's a sheave is a special special case of presheave which satisfies this well-known exact sequence actually restrictions from i and j O and K into this one same value but all those all those elements here this is as if I'm saying

22:30 that this is a couple of dollars into a kind of a sense there's a technique we can do without using the sheaf here and the compacter sheaf contravariant this is covariant here. Then you can have general pre-sheaf to in category A and this is the vertical one this is a category sets you can handle this way. So anyway this is to make life simpler. So this is this so this is locally it's glueable. So locally, it agrees, then there's a global section so that the UI coincides with traditional definition of a sheep. Anyway, I wrote the other way, and I quickly like the diagram, that's the definition of a sheep. This is, yeah, it's all right. Oh, I made some obvious comments, which I want to say. Okay, right. Oh, yes. This is, again, a little bit, how to say, a hypothesis, again. Hoping this will give us the right stuff. Okay. Tau v, tau u. So tau is a functor from a site to, now I'm considering a C1, microcosmic category that I talked about. Okay, so tau v is a usual physical time. Tau u is also a usual physical time corresponding to object u in a site. tau v precedes tau u, there exists a morphism from u to v, say g, u to v. Okay, that's another hypothesis, and I hope the direction of the arrow is acceptable to you a little later. Okay, let's see. Anyway, since I talked about the sheath, let's use S tilde as a full subcategory of S hat.

25:00 It's a u-jole thing. Okay, what did I write here? Oh yeah, H0, this is of course, I'm assuming more than that, since it's a kernel, so I'm talking about a million categories. It's a little bit cheating here. Just as a moment of our geometry, this is probably, in this complex, it won't make sense. But in the future, if such a thing's possible to define some category where the zero element-like object can be defined, then you can do more some cohomology or homological algebra just as you would do in Abelian category. And in the future, it might be important the kappa or tau evaluated at you and how kappa preserve the exactness that kind of stuff, which we can measure by cohomology cohomology well again hesitate to say group because it depends on what kind of category we are talking about anyway this is just a moment on my background so proverb is not true the proverb is not possible to define something like this ok this is ok first half is alright that's interesting saying that Next one is the one that I said, the assumption that tau v precedes tau u. In that case, I assume there is a morphism from d to u, which is this. I call it g. Okay, so this is, I need to be a little careful if I'm not wrong. I hope this is all right. So since M is a pre-sheave cell, there's a morphism from here to here, because G is from B to U, so this way. So also, let us assume that M is observed over this generalized time period by P. So there's a morphism. So we have morphism from here to here, here to here, doesn't come much. and I would like to well say this more here here this is I'd like to call it an

27:30 indirect observation of the state of m over u by PV okay in general it's this okay so okay so if you observe this state appreciate them over this period that's information is this that's all information can get by this observation okay since this is smaller it's funny right right here okay so how do we precedes this so this is this is in the future this is okay so I said this measurement of the past state does not provide the measurement of the full state of the this measurement of the past state oh yes this does not provide be a measurement of the future state, since it is a sub-object. And this is the correct direction then? Yes, it sounds forgetful, but in some sense, what you were saying there is if you want to gather information not directly on the past, but you, yes, exactly, but you go in here for observation pertains to going through it's consequent so use consequence yes yes so in some sense yes it's a how this observation how difficult it is to predict the state of the future that's what it means this indicate that thank you for your impression this it is indicate the it indicates the difficulties of predicting the future by observing the past, that's what it is. Okay. Oh, I have lots of time. Oh, I have not too much to say. Okay, next I would like to define this notion.

30:00 you change the letter to H. So let's consider this new notion, V to H, V to U, morphism H. I define this notation by the collection of all the W's where H can be Okay, I want to use this one or two pages later. Remark 2.10.1, I was reviewing last night. I don't remember what I meant. I'm getting senile or something. I think there's something I want to say, but somehow I can remember. so I maybe half one is too far away to fly to England sorry okay next I would like to consider this so consider I just as before I changed the H from G to H sorry so earlier expression is this how we increases how you think about that if you prefer the broken cube on the table on the table oh yes this is longer than my finger Yes, so this situation I would like to consider like this, let's, for example, may I use electron rather than elephant this time? Electron just launched that from here. So you, the location know the moment so if you can specify and nothing between and the land somewhere and so if you is a screen there so you know how to hit and then you know that time so you can specify the objective okay that's first case second case but oh yes second one is this same thing you let it go there's There's one slit, and this slit, this gap is just narrow enough, but not too narrow, so that only one electron can just go up here.

32:30 And you observe where it hit, so it's here. So this case, this is only one electron can go through just barely, in an ideal way. So you can specify that. So this case two is essentially, is case one. case so this is the job this this is of course this is the yard this is this earlier documents anyway so double case this case again give the same thing this case you know it leaves when you know it lands on a screen and then but you don't observed but this case even two of them so and there's no morphism from W to W prime oh maybe that's what it is so the W and W prime there is no morphism between them so there's no one is before after or same time there's no concept of one or after so this two possibility. So this, if you, you can observe here, you can observe there, if you don't observe that meanwhile, in a double-slid case, actually it's a wave state, it's a very particular one, there's only two choices. But still, it's more than one, since weakness gets by an object, it's still a wave state. So this case, if you send another one, send another one, send And another one, consequences, this behaves like whips. Just to clarify, before getting two-point down the technicalities of it, is this sort of intuitive idea that makes it connect with physics that the solution to the old riddle, what does the electron actually do, is that it goes through the two slits at two different times. It's just that different times doesn't mean earlier and later than each other. that it means sort of magically sideways for each other, is that sort of... Yeah, for my language, people just say there are morphisms between the W and W prime. So, one thing does not precede the other. So the electron goes through the slits one at a time, it's just that the times can be unordered and sideways from each other. So, when you try and translate that into linear time, it's as if it went through both at once.

35:00 I think this may be probably I hope that may be more precise not able to specify the object in a site it's the good maybe maybe one of the good pictures to say about the behavior when it's not Do you want to make this someday quantitative so that you get, you know, quantitative things like diffraction patterns, you know, predictions that this will be bright and this will be dark? Yes, yes. I mean, because at the moment there's no place where real numbers start appearing that can count as brightness levels or probability. Yeah, this one, this formulation does not give the probabilistic picture at all. Just indicates the more quality type information. Let's see, oh yes, this one is the one, I assume two things, Kappa and Tau, those two guys entangled, I assume that, and also I assume that they are not only pre-sheaves but also sheaves, I assume that, okay, under those assumptions, let me talk about this remark So let's do two objects in S-half, two questions. I would like to consider the first case not entangled. So we have MV and F-half, these two things here. And this is, this part I am not sure at all, but anyway, So this is the space and time around in a neighborhood of this, and this is space time and a neighborhood of this. Okay. I said the local neighborhood. Okay. Suppose next, M and M' are entangled. So now I can let you choose B equals B' and the

37:30 which picture is this picture is this okay would you imagine this is distance apart and okay so we have still kappa B tau B sent so I thought and the police again I learned but what is a closer to reality is this so it's entangled we have I thought there are two possibilities since kappa and tau space and time the same I thought maybe the nature determines space and time one globally than I thought so this is no problem it's okay it's still, still, same I can use kappa b and tau b, even though mb and m'b is very far away. Maybe that's my first case, I hope. Second case, if those guys are not so globally defined, then this should be only associated with this particular particle. This one is associated with this particular particle. Then, my conclusion, which I thought I made able to, or I'm allowed to derive this, is the sheaves, cup and tar, depend upon the, very locally, for a dependency of the local particle. That I consider as the second possibility. Again, I would like to learn what is the reality of this kind of situation. okay yeah this is the next one is so please allow me to be humorous and joking a little bit because I am using the words subplank yes I will explain why I want to use that one but it will explain minute okay first of all I would like to consider covering found this and the very first definition I talked about another covering family over covering pharmacist is so if I J J J means and so

40:00 forth so this is depend upon the oil so I I definition is this out I think something a direct element arrow goes arrow girls is the original covering and next one is the completion covering so the first one second one I want to this guy to that guy that version in that direction i took a direct and this one is the individual very subdivided so to speak this thing and i would like to call this one subplank generalized time period does not mean that If you evaluate this guy at the tau, really it is as small as somebody mentioned earlier. So this one, again, is a subplanned scale tau when you evaluate tau over this direct limit. the reason is this for this you this is in some sense smallest smallest possible generalized time and universal we go to definitions universal mapping property is a pretty most appropriate way to define this limit so this is in some sense smallest possible generalist time so you can evaluate on the top top over object that's as small as possible one for the given given covering of a object you so this is all these it's time as some sort of refining the progression of time is it some sort of in the way you have to find so far Do you imagine time, the progress of time, to be some sort of refinement of observations? Is it some sort, because when you speak of smallest time, is it something that you refine, you localize in some sense? it's okay okay is it some sort of with these coverings is it some sort of course

42:30 green okay okay let me see it seems completely counterintuitive that notion of time I see okay let me let me see let me connect with two things when there is a arrow from here to here then let's operate out to it so how this power is arrow goes our daughter okay in the early definition if this goes to this one this tree how of this precedes resolve you that okay this is it's okay this is again like this so you it's not the refinement the usual definition of covering this is in something smaller smaller yeah okay so it's so if you if you this one is getting smaller smaller smaller okay if it comes in so short it has the content fluctuation type situation and if you take a limit it's it's, it's a, when my radar evaluate this, evaluate this kind of thing over some kind of matter. Then, in tau, that is so short, the mass appears, disappears, and they end up with quantum fluctuation, many big banks are going on, in the very minute, for given local view. And if the entire universe is possible to cover by finite of those U's and B's and W's, and again, consider all the coverings, and the take of the left rim, and the cross product of all of those, that looks like, to me, the big bang type picture. So, it looks to me as if it is quite different. When you take the, I think I thought that in some sense, there's much discussed today, you know, sort of Planck length. Yeah. It's something that exactly the principles of quantum mechanics and relativity, in some sense, conspire to give you that irreducible language in nature, something that you cannot localize.

45:00 In the analogy of John, it would be something like a black box where things become fuzzy underneath. So you do not, in some sense, you do not have this ultimate refinement. You do not have, you cannot take the limits of ultimate refinement. That's why you cannot have infinite localization. Ah, exactly because there is that irreducible elementary length, so you cannot, with perfect accuracy, of course you said, you said there is fluctuation, but exactly that you use the term fluctuation is counter to the actual mathematics used there, in the sense that by these limits you infinitely localize. Well, that is not allowed. Okay. In a physical practice. It's an object. Direct element of an object. This does not mean this is very... The charm of this universal mapping property over category theories, an object we don't have to consider since it's not necessarily a set. There's points in there. So this can be something, in some sense, units could be very small. just satisfy the universal mapping. So that tau of that, for example, this can be very, very much defined. This could be much bigger than Planck-scale timeline, or even could be much smaller in that sense. But anyway, given you, this is the ultimate shortest generalized time period. And that's why I call the subplank. That's a skew, again, as I said. and I can talk about the kappa of those same thing okay so okay I would like to do for the pre-shift m not so this is just that it's not kappa it's not s Again, I'm treating like M, kappa, S as all pre-sheaves, so in some sense, though they may have an equal right, except I'm assuming two things about the kappa and ta, which is entangledness and oversimplified, and also their sheaves.

47:30 Those hypothesis may not be correct, I don't know, I've got feelings. Anyway, so I want to do the same thing about this M. So write this as a direct sum of the pre-sheeps, and also write each, again, direct sum of the pre-sheeps, all finite. And we can consider this bunch of sums, direct sums. And also, again, direct limit by defining that direction of error in the proper way. So this is, again, I would like to call, since it's a small given, and this is, again, by the universal market, this is the smallest possible, which could be small, doesn't have to be small at all. But for this, this is the fine, this is the ultimate sub-sheet. So, I would like to call that one as subplank object in this hat. And again, evaluate this and so forth. That kind of natural definition that I would like to use. Anyway, so last part of this remark is, so we have a bunch of things to evaluate. And I actually I didn't know there may be a very interesting physical correspondence to this kind of object. And also I talked about the inverse limit as well. Again, you're keeping the same direction of arrows as I talked about. And you can talk about the inverse limit of the cover. Mathematical intuition I have, but physical intuition I don't. I think I will stop here. Any other questions? You might like me. This question sounds really contentious, but I honestly don't see why there's anything to talk about quantum physics, except you've used the word wave-like, which is just a phrase in the English language.

50:00 Can you go back to a definition of wave-like? You put something on the first transparency, you have the definition of... Could you... I mean, I presume the implication is it's meant to be something to do with something, maybe part of the duality, but... Is this the one? Yes. Yeah, okay. Yeah, I see the... You know, why should that be anything with quantum mechanics? I mean, you can not specify a state fully classically as well. I mean, you can localise a system classically in a certain region, or you can not do so. But that doesn't mean that it's anything with quantum mechanics. Well, the word wave-like, if I really took that literally, would at least have something to do with, example, well, say, a non-computation relation, and if you really wanted this to be difficult, at the very least, you'd need something like a position or something. Otherwise, I don't see why, I mean, why don't you just say it's undetermined, you say it's a wave state, I mean, there's no... I see. See, even your two-specific experiments, you could just as well interpret pictures in in a classical way, especially as you don't make any predictions, you just have the probabilities of anything. I mean, there must be sorely much more precise specification you need to make as a structure that's really worth it if I respond to even remotely anything you think of as quantum mechanics. That's why I was asking if you've done it, will it be quantum too? Because I would guess that one could do this for classical physics just as well I don't mean because it's not sensible I suppose you assumed somehow intrinsically it's quantum-mechanical yeah I don't see any sign at all anything that's quantum-mechanical after you said that something's not specified that doesn't make it quantum-mechanical right that's right is that fair or anything yes actually I may be using the wrong names because maybe it was my wishful those terminologies to fit to those famous double-slip experiments, yeah that's quite possible that the terminologies are not proper at all. Classical is just a mechanic, and we certainly fit into this. We talk about knowing the system, not knowing the system.

52:30 That's why it would be very, very interesting to see if we could actually map this into just choose the ordinary real numbers of time, not the intrinsic, and just an ordinary part Any other questions? I wish you to thank again. Thank you. So we should have a five-minute break before federal... ...for you could be able to get a... ...for most in a particular type of system. It doesn't have the cause of sex, but that's why I'm used to love it. It doesn't have the sex. It's got nothing to do with one family animal. You can't ask a time. The question is, I know, is that the US has got anything to do with time. It's time. It's well adapted. It doesn't mean that because it's really rare that you can't. You can't understand this. It's just like a lot of those. Yeah. I'm not sure how you're trying to put them at my back time. It's just having more of a sense of the right now. That might be a difficult question, precisely because, consider the example you chose, you have to show, well, X to the Y, two manifolds, I make this construction for you now, then, of course, automatically, you have an understanding of the image. Is it preserved? I don't know. I don't know whether that's actually been gone into. Of course, it happens that the tangent bundle construction

55:00 but it's not an exponential, it's an exponential in S, but it's not an exponential in S. It's a big thing, though. No, I haven't, but I think it's Steve, it's this guy, and then Steve. He's a research student here in some ways, because he's got a span of scholarship. Well, not all of them, but he has. This guy has, I imagine, we're about to be introduced to him in a few seconds. He's been a research student. I could be wrong, but I don't want to speak to him. I could be wrong, I didn't go to every single talk last year. It looks like an impulse. Well, he is. Yeah, Chris is talking tomorrow, isn't he? Yeah, Chris spoke last. Oh, maybe it is the same guy. Maybe it is the same guy. Maybe it's the same guy. And I was a bit surprised, but you know... Yeah, yeah, yeah. No free lunch. Well, I don't know. Some of these things... He kept very good notes. should things be spin-up? That was Smarty Heimers talk, wasn't it, all about Vince's and the facts? That's your stuff, isn't it? Well, maybe that was when we were sitting down talking to him, wasn't it? No, I was there. I liked Andy's stuff this morning, but it was Vince's going to try and cover the ground not too fast for me, it was just too... I've got to do it a few times now, but I've got a lot more out of it today. I've been hearing it for the first time. It was rapid, but it's obviously extremely interesting. I can see why Bill hates it. Because, you know, it's that tension that we were talking about the other night, that between the people who think of the topological and geometrical meaning of categorical structures being the most basic thing about humanity, yes, exactly, and the algebraic structures in some sense could be explained ultimately because, you know, the natural geometrical meaning of instructions and people think of geometry as just an epiphenomenal something that can be in a structuralised way in favour of what it's typically deeper algebraic,

57:30 or more general algebraic notions. And that just seemed to me to be a pretty fundamental and conceptual divide in the approach to mathematics as a whole, as well as it's specifically. Uh, Leos' initial impetus is an algebraisation of connections. Sorry, who's his initial impetus? Oh, yeah, yeah, yeah. Oh, he's absolutely unalgebraic. Well, I'm pretty easy. That's the question I'm just going to ask myself. It wasn't obvious, but how you actually did it. There's a hard start to get it, as I said, there's some hard analysis to make it work. Well, that's maybe that's the perfect spot. No idea. Just for the ordinary person. We've gone a large effort. That's right. And this guy's got this thing about smashing all the doctor. I'm sort of interested because I'm working. coffee or something. I think he's got to go. Oh, no, I'm not worried. He's coming to work. No, that's all right. Just as Thomas said, I know you're looking for a picture. Some of the things that people speak about. I'll leave you. I'll leave you. I'll leave you. I'll leave you. I know they're just worried. I was heavily lying. It's a very crusty guy. You know, one of the problems is that you'll get... But I'll leave you, I'll leave you. ...a smooth cavity or pomegranate that part. I'll stop this again. The only problem is when I stop this again, I'm starting to put a star on again. to the fixed parts of the needs of those who joined together smoothly, you know, on the page of that, yes, and then we see affinity capsules. We've still seen the examples from... If you actually... I'm not going to look back.

1:00:00 I suppose... I'm not going to look back. So that's what's called... Well, he doesn't... Well, he doesn't have to ask that. He's not, he's not... I'm not going to look back. That's the beach, right? Smooth. I thought... I thought... Thank you. It was really strange. Yes, no, it's very good. It's very good. It's very unlikely. It's exactly what it is. My guy got on my bus and went upstairs. This guy was sitting down, St. Helen Derrick, and so-called his arms crazy or something. But the weird thing about this was that many of the years when he said Derrick discovered his ...who lived in the same city, and separated by people, and they haven't spoken since it was before, so... I got him. I got him. I got him. I can pick up. Norman. Norman. Norman. Norman. That's another reason for the thumply round. He'll look well. He'll come well, he slept with a beard. I'm very glad to see you. I would take you to have a look at some numbers in Christmas. May I have a three-way, please?

1:02:30 Not until five. It will happen. It's a pleasure to have Pedro Resende again this year. This year, the title of his talk is quantiles equal here's a question quantiles, do they equal half quantization? Is that h-powered? Thanks very much for squeezing me in into the program on the last minute yes, I want to stress that my title really is a question otherwise I will have to hide somewhere after the talk anyway so the point is Following last year's talk, when I was discussing fontales, I somewhat adopted the point of view that fontales are structures that are used to study CISAR algebras, and CISAR algebras are related to quantum mechanics, therefore fontales are sort of related to quantum mechanics, and this is a very inefficient way of conveying a specific idea. so the point is so these are these things are of lattice theoretic structures they are not commutative spaces in some sense When Chris Mulvey thought of the name in 1980-something, he had this idea of, well, also seeing them as a kind of non-communicated logic. I'm not going to say much about this, although what I'm going to say certainly has to do with it, and I think in his talk, much of this stuff is going to be brought up. Right, so my idea now would be to try to show you one way of looking at fontiles that sort of make explicit how they relate and how they differ from standard fontiles.

1:05:00 Well, what I will start with is by discussing something that none of us is familiar with. None of us has discussed in this meeting so far. Okay, so everyone knows that in classical physics you have these fixed trajectories that are determined by the least action principle. Then in quantum mechanics, you forget all about trajectories and you get this other picture where there's just nothing here in the middle. We don't know what is in the middle. So in a way, this is a bit disturbing. And in order to make physics more clear, we want to give more insight to people. Feynman came up at a certain point and produced a much clearer picture of what is going on. Now, the funny thing is, of course, this is not very clear. Mathematically, this is a bit what happens. One doesn't know what to do with this, except in those cases where, well, it isn't really that interesting. The thing is, there are things which one knows exactly how to calculate. These are probability amplitudes, so they're matrix elements of some operators on Hilbert space. And the point of the finite path integral is to provide a method of explicitly calculating these things, which is ill-founded. So why do we want to replace something we understand by something that, well, we sort of understand, we have some idea what it means, but it's otherwise mathematically ill-founded? So explicitly what's being done is you compute the action over all the possible trajectories and then you compute this integral over all the set of paths. And this should give you the probability amplitude of getting from here to here. Now I think, as far as I understand this stuff, the reason is that you have some ways of suggesting new paths, new ideas in when you study physics by considering these things, considering paths. For instance, if you want to think about an experiment, you could say something like,

1:07:30 well, let's suppose that I allow my particle, whatever, some state to flow a certain direction, and I have some black boxes that perform things on the particle as it evolves. These black boxes are localized in space like this one. One does this, one does that, and so on. think of this whole collection of things as an operator, which is actually a direct sum of operators, each corresponding to one of the little boxes. But you can also think of the actual paths that are taken, and when the path happens to go through this, you know how to calculate what goes on in here. So you would replace this probability amplitude by this stuff, and then you would compute this unitary evolution by, well, again, replacing it by this much simpler expression where you have all possibilities of finding your particle before you act on it with the operator A followed by when you have two different states beginning and for a matrix element of this operator A. Okay, anyway, so the point is that paths are a nice way of thinking about these things. Of course, you don't know how to put a measure to compute these integrals, but there are some consequences of this, which I'm going to get a little bit much longer. One could try to simplify a little bit by saying, well, let's really not think about all the paths, but just take homotopic classes of paths. So in this case, the double-slit experiment, now you have the path that does this, the path that does that of course a bunch of other paths you still have plenty of them but anyway you have a group point of paths up to homotopy so if you consider this as being a typological space and you have the fundamental group point and if you compute over each homotopy class of a path the integral then what you're doing is you're assigning to each arrow in a fundamental group part of space, a complex number, while composing various measurements. So it corresponds to multiplying operators, will correspond to multiply several

1:10:00 of these functions, which you do by taking convolution product. So the validus we'll take on some homomorphism will be the sum for all possible pairs of maps composing to give that homomorphism. Well I'm replacing a complicated thing by something else which is also complicated because in general this thing will not even be defined. So you need to have some extra structure on your underlying group wide. It should be something so that it can have a measure, so that it can make sense of this sum, which is very often an integral. So I'm replacing something hard by something hard again. But anyway, there's a lot of stuff on group points, locally compact group points, and their convolution algorithms. And somehow I think this is a part of mathematics that, if one looks carefully, should be related to this. But this is not where I have to stop. What I'm going to do is something really outrageous now. I'm going to suggest that instead of doing this, take such maps valued at those two helmets. Now, of course, now you'll tell me, okay, that's it. Now you really destroyed all the physics that you could get from here. Well, but I'm I'm going to look precise to this. I'm not going to see this as Zad 2, but rather as the 2 chain. So I'm going to consider multiplication here to be just meat and some to be joint. Now, if you apply the same formula to a convolution formula, what you get now is it's going to be the joint of all the maps one place and here we have the multiplication and this is always the fun this is a total paradise you don't need a measure or anything there are infinite joints in this set of maps in fact a map to that two chain is just a subset of the set of arrows so if you replace this condition by saying that F belongs to the subset whose

1:12:30 characteristic function is F, what you're doing is precisely saying that the product is computed point-wise to subsets. And this is just the set of all arrows composed in the group point, such that F is in F, G is G. OK, before you go on, can I ask again a question about this? I mean, it happens in this particular case that you have distraught homotopics. For example, there is actually one slit experiment. I was saying the group point, sorry, I was... What's the notation for a phenomenal group point, surely? You've got, oh, well, it's there, I1. Yeah, I1. The implications, presumably, is that there are paths there which are not homotopic to each other, is that right? I mean, you're in homotopic classes. There could be, yeah. No, what I'm saying is that there's also a one slit experiment, actually, where you have just a single slit. you still get quantum diffraction. So that's nothing talking about pi 1 of x. I was just trying to think, if this is a particular, I'm just wondering, is it just an accident, this particular case, you have some monotropic of the paths and the power integral? Or is it something fundamental? Well, I'm not sure what restricting, well, to fundamental groupoid, which effects this would have. So I'm choosing this experiment. I'm talking about homotopy just because this is going to relate in a certain sense to what I'll say next, but not in that sense. I think the point is that your physical system is not, this happens to be a topological field theory or something. It does depend on the path, not just the top of the path of the path. Sorry, I didn't hear you. The physical system, the amplitudes, do depend on the path, not only on the path of the path. Sure. But there's no reason why you suddenly only I'm doing it because it's useful for something you'll see later on, so yeah I mean I agree with you that it doesn't really simplify very much, and that's what I was saying, you're moving from one hard problem to another hard problem by doing it like this, what I was doing is splitting the integral into, well just compute it in each homotopy class and then sum everything else. Well, just a quick remark, it loops from what it's presented. I mean, maybe you're going to do some trick, but it looks as though this simplified world

1:15:00 cannot really have the concept of destructive interference. If there's two or more individual possibilities of something happening, and yet you want them to add up to, no, it can happen. Grant doesn't seem to be able to be possible. I don't agree more, but that's why the title of my talk is a question. I mean, of course, if you look at this at a first time, you'll say, well, this really doesn't do that. So you're not quantizing anything. But the fact is, we can, well, we can, from this kind of structure, you can recover other kinds of structure, which is very surprising. You can recover certain C-SAR algorithms, for instance. And there you have complex numbers around. So I'll just get to that. I think Ian's remark touches on the other half of quantization that you are not going to talk about today. Well, I mean, I'm going to talk a little bit about that, and my hope is that someone here can tell me the rest of it. That's just a third quantization. In fact, it is H-ball. Right. Well, you could use an arbitrary complete lattice, I mean, instead of 0, 1, I presume. Not that that might be useful to your purposes, but put a hiding algebra in there, anybody, well, any, you know, any complete lattice. So I'm by no means pretending, while trying to give you a complete theory of any sort, As I said, I was just trying to do this impressionist brushstroke, a part of it that is usually not conveyed in the papers. It's not something that you can't really write about, so I'm just trying to bring out some specific piece of intuition. Okay, then one usually writes it like this, because this subset of the arrows of the group oil, or the category for that matter, is known as a quantile. So there are one sees that a quantile is just monoid in the category of complete lattices with John preserving maps, which is what you get by replacing the field of complex numbers by this lattice-threat extraction. Okay, so the question is, what can quantiles do as far as a quantum system is concerned?

1:17:30 So the point is, starting from classical paths, unique paths, you get probability amplitudes in time and path integral. And this is somewhere halfway through. What you get here is not a uniquely determined path, But what do you consider such a subset, which you think of as an experiment, you're allowing for non-deterministic choice of paths. So it's not a well-defined path, but it's not quantized in the sense that you have complex numbers attached to each path. That's why I was writing that this is kind of path quantization. Of course, you can't argue that this is no quantization at all, but let's hang on for a while. So this would be my view of what else. right of course not all quantiles are like this and and surprisingly quantiles seem to be able to encode structure which at a first time you would not be willing to describe to them and in particular there's an example of well if you take any sister algebra there's a spontile which was by Chris Mulvey, which you call Max A, which is a set of all norm-closed linear subspaces of the C-star algebra. It also has a punctual structure whose multiplication is going to be the closure of the linear span of the pointwise multiplication of the NW. so you actually define a functor from the category of C-star algevers to the category of quantiles in this way and this functor is a completing variant so you do have that two C-star algevers are isomorphic if their quantiles are isomorphic Now, the proof of this does not rely on the kind of quantile that I'm thinking of.

1:20:00 It relies on more general quantiles of endomorphism of certain complete lattices. But, well, that's what we have. I mean, the structure, the Caesar algebra, is somewhere hidden in here. This is still something that is, well, understudy because there's a lot which is not known about this function and which is necessary for us to know how much this is really strong statement or not. But it's already surprising as it is. So in a sense, this kind of structure, which is all about valuing arrows on 0 and 1, somehow is able to recover this other structure, which is all about valuing matrix elements in C, the complex numbers. That's what I say. Well, quantiles are surprising. It seems that you're throwing away something, but perhaps you're not. Is C-star algebra sort of fixed by policy as seeing the complex numbers as a scale of field? Or can you have an R version, a termian version? I mean, how free is, I don't know what background knowledge, I'm sorry about this. Is C-star algebra compulsory to use the complex numbers is what I'm asking. Well, I usually send them on a complex number. Yes, of course, but what I mean is how much actually depends on that. You know, what I'm really asking is how far can we stretch this miracle? I mean, imagine, for example, there's a legitimate version of C-star algebra using something else. In the same order for the car, you can get a lot of that. That's a very good question, but I don't know the answer to that. Okay, but do you happen to know... You can ask the question, and it's worth looking at it, but I don't know the answer. Okay, but do you happen to know offhand whether C-star algebra as an axiomatic system force you, because of the way the axioms work, to use the complex numbers? So, sorry, what's that again? Well, do you happen to know offhand... Sorry, this is just a general knowledge question that I'm going to give the answer to, but you happen to, or does anyone in the audience happen to go on hand, do the sort of standard axons for a C-star algebra force you into a cardinal sense that the scale of... Well... You could, but there's no way to see it. No, things will move off in a very different direction. I mean, if you look at, in the commutative case, at real C-star algebras, real communities around the world, then the burden of proof is in a very different place to when you're looking at it in a complex case. Of things like it being a complete invariant.

1:22:30 Oh, I see. Oh, yeah, of course, of course. But what happens is there a lot to happen. Is this a kind of pointless version of the representation theory for C-star algebras, I mean, the classical theory? Well, how does it connect to the classical theory and represent even this compact house door spin-off, the old girlfriend representation? But this is not commutative. This is a non-commutative case, right. So is it, I mean, does it give you the commutative case? Does it give you the classical representation theorem? Well... No, because that's... Well, no, because the quantal is supposed to be pointless. I mean, you don't... No, no. It falls from there. I mean, it... Oh, I see. This assumes that. Uh-huh. Okay. I mean, this is part of the motivation, which is something I said last year. I didn't say this year. Chris has this series of papers with Bernard Bernachevsky about Gelfand duality in the And this one was devised to be the kind of non-commutative spectrum of Sitter algebra for non-commutative topos. It's one of those things that have a name that didn't exist. Yeah. Right. Okay, so during the rest of the talk, what I will do is give one example of a particular situation where you want to describe experiments in the sense of a quantel. Give axioms relating those experiments and see some interesting things coming out. Now, this example, which is again Penderel's telling as last year, but I'm going to say things which are very different from last year. This is not a physical example. I mean, Penrose tailings are mathematical things. I'm not going to say something that could be formalized in terms of some Hamiltonian of some sort. But the point is to show how, by means of this example, how this kind of playing with experiments could be done. And I'll see if there are any takers in really picking up some simple physical system and see if it can be really handled in this way. Right.

1:25:00 So Peno's tailings are tailings of the plane made out of those two shapes there, where is Aslan's golden number and the short sides have won one. What I'm going to say now is based on the paper with Chris Moby that has recently been put on the archive. And I stress that this is really being used as an example. This is a funny thing, just a couple of weeks before I came here I was doing some math search and I found some, well a series of articles, postings in sci-physics research, which are triggered by a comment by John Bias from last year. He said, well this is a nice workshop and I just heard these, well two or three talks, one of them was by Pedro Nezena and Andrews Tellings, and I think this is related to such and such. and there was a series of 20 postings discussing what people thought that we were trying to prove or not trying to prove, and, well, thinking, well, if they really come up with some theory on ergodic theory, then maybe we could... I was amazed. I mean, I said, well, if they told me, I would have told them, well, no, we're not doing ergodic theory or anything. You don't want to let them out. Why do you say Penrose Tellings? People sort of wander off in all sorts of directions. You said erotic theory. Pardon me? You said erotic theory. Okay, so just a quick overview of kind of the studies. When you put two of those things together, you have to respect certain things. Vertices with the same color have to coincide, edges with orientations have to coincide. So when you take a little triangle, there are in principle only two ways of putting them together. is by considering this oriented edge. So there is only one way of looking this edge to something else, which is either to another one of these triangles, or to an edge in one of those.

1:27:30 Why do you choose this way? And it's very easy to see that you run into complications, because then the way you complete this will leave you with a hole that cannot be filled by any tile. So the only possibility left is to join a large triangle to a small triangle, like this. And given the choice of lengths, this is tau plus 1, which is tau squared minus this tile. And so this is a tile which is congruent to the original small tile. And if you take the previous large tile, which now is small, we call it as for small, and we call this L for large. So you have one large tile, one small tile of a different type. Since all the small tiles are linked to large ones in this way, if you remove this edge in all of them, you get a new tiling with two kinds of tiles. Now this is a small tile, this is a large tile. And you can do the same thing again. a small tile can be linked to another tile in order to produce the tiling of the plane that's to link it to one of these and so on and so on and this after four steps you recover exactly just a little bigger those two tiles again so well you can go up to infinity this way and it follows that from any tiling of the plane you have this infinite this sequence of other tilings with bigger and bigger tiles. And if you fix a point in the tiling and you write one because this is in a small tile or zero it's in a large tile. When you remove this you write a number for the kind of tile you're in. Now this is a large tile so you write a zero. Of course following a one there has always to be zero because small tiles always disappear in the next step. But like this will generate a sequence of zeros and ones with the restriction that following a one you're always at zero. Another thing is that if you take any sequence like this you'll always be able to construct a tiling and choose a point in a tiling that generates a sequence like this. And finally if you take two points in a tiling and do this construction at a certain point the two points when some edge between the two points will disappear and it will be found in a big tile. So from then on they generate the same sequence. So what you get is a set K of such sequences

1:30:00 and an equivalence relation on them saying that two sequences are equivalent from a certain point on the code side and it turns out that this set this quotient set is precisely isomorphic as a of the plane, or, I mean, a set of equivalence classes of Penrose-Talics, I want to consider Penrose-Talics that result from another one, or rotations would be the same. Right. So how is one supposed to make sense of this, in terms of Montal? So the point is, we want the paper, the preprint we wrote down describes this theory of pendulous talons as, in terms of a quantile, as a noncommutative theory. I just want to say something else. This set of sequences is actually homomorphic in contour space. And when you take this equivalence relation, you destroy the whole topology. So there is not much topological to say about pendulous talons, which is another motivation for moving on to a structure which is non-commutative. So for quantiles, to make sense out of this, what we should do is consider certain experiments. And now I'm going to think of the tiling somewhere on the plane. Take a point in the tile which I'm going to consider as particle to be an electron, I'm afraid to say electron. Well, it's a very small elephant.