Noncommutativity & Discrete Space (contd.)

0:00 I don't know whether she's going well in one of the people, so I'm going back to the wall. Okay, that's fine by me. Yes, I would be interested, especially in the whole thing. Uh, Pierre, um, oh, will we have to sit there? Uh, Lou. What? Well, Pierre is going to come. Oh, okay. Well, if you're in the middle, that gives us a... Can I sit in the middle? That's a good idea. Are you okay? Yes, that would be very helpful. My particular interest in this is the fascinating speculative ideas it raises about foundations of analysis you know you were saying about the alignments rule and uh many other things but anyway you you you just kick in as you i have if that's all right with you because i'd really like to be able to okay so here we go um this talk is designed to kind of begin where pierre and i began in the domain of what Tony and I and Claude are thinking about and what was pioneered Tony.

2:30 So this first slide is a kind of summary of where Pierre and I were quite a long time. We were thinking about a time series and it could be just numbers, commuting numbers, positions in the sun, the observed process, and, excuse me, so, you know, it could be just a series of observed positions of the particle in discrete time or some other discrete process. And I'll articulate some delta c here for my purposes. So then we have a velocity of the derivative of x prime minus x prime which of course I mean whenever you'll have an x at a given time you take x at the next time minus x is the previous time. D, that's the derivative. So this is very simple. But we're thinking of a discrete situation where there aren't any limits to the derivatives. We just have those differences. That's what's available to us. And for notation purposes, for sequences of observations, I do them from right to left. AB means observe in B and observe in A. So then you can think about what might happen if first I observe velocity and then I observe position. That's the more complicated one, because if I observe velocity at the top of the clock, I have to tick, and so the position that I've got is the next position. Right, okay, this is always at the screen. On the other hand, if I observe position and then I observe velocity, I am not forced to tick the clock when I make the position observations, and the value and then it's interesting to look at the value of a moniker if you actually compute the value of that moniker you see you get the square of the difference in position divided by the time right yeah you get delta x squared divided by delta t right so um so if you wanted

5:00 to think about the simplest commutator equation it would be that the commutator of x with x spot is constant the heisenberg kind of equation and that requires delta x squared over delta t equal to a constant. And if you're used to the diffusion equation, or situations involving that, you immediately recognize that setting that equal to a constant is what people call the diffusion constant. But this is not the usual story that produces the diffusion constant. I'll remind you about that story in a moment. That's what's intriguing here, and remains intriguing. I'll tell you the other story in a moment but what's the story here? The story here is that our process is that x prime is x plus or minus a constant assuming that delta t is constant that could be a random walk or it could be always going to the right or always going to the left but the usual thing that people often think about is a random walk with a fixed step length a brownian walk and it's the brownian walk that gives rise to the diffusion constant, but thinking in a different way, thinking probabilistically. What's the probability that I'm at x after time t plus tau, meaning delta 2, tau is delta 2. What's the probability? Well, I can get there by coming from the left or coming from the right. So I have 50-50 probability in the random walk case. And I look at this discrete equation, and I say, oh, if I take the difference between this and the probability for x at the previous time and divide by the time difference, then I get a discrete second derivative over here on the right, a spatial second derivative. And so I get dp dt is delta squared over 2 delta t d squared p dx squared. And if you wanted to make it into a viable differential equation, then you make this a constant. So it's called the diffusion constant. And a given Brownian process is controlled by the diffusion constant that you chose. And you'll notice that this makes the...

7:30 If you're letting delta t go to zero, then dx dt is infinite in order for this to... Diffusion constant to be finite. You're a Brownian particle, skittering all the stuff goes in. That's the usual story. Then there's this story. This story doesn't insist on a Brownian process. This is a structural feature of having the commutator equal to a constant. So this is somehow more general than that. They fit together. And I don't understand it. I still don't. This is just the most elementary part of the thing. quite intriguing i'm sure that if we think about this a little more we could do something okay that's sort of the miniature story yeah yeah that uh so the miniature story is is that if you start by thinking about discrete physics it has it's filled naturally with commutators it kind of looks like quantum mechanics or or the real real uh rotation of quantum mechanics right because this is shorting this equation except we need an i in there yeah yeah so we really are very close to a whole lot of physics just by thinking discreetly and that was our original uh project and it still is our project to think this way um and of course long people thought of Brownian motion as a possible guide to thinking about the sub-quantum medium and the origins of the Newton's potential did not. So, you know, this goes in a lot of directions that are very interesting in and of themselves, like there are these rather complicated schemes of getting quantum mechanics in relation to stochastic processes due to Nelson and so on. Sure, sure. And they have no locality built in. but they are really extrapolations of the fact that you have a diffusion equation yeah yeah whereas here you've got something which is always is getting it as a much more thing is that you have an eye in here and you could go back and put the eye in here and try separating out in the bowman way uh the radius and the and the phase and and look at it i actually keep wanting to do that. What does the discretization of the basic Bohmian theory look like? But I haven't found that much. That's stage 1, okay? Stage 2 is something that actually happened

10:00 to us, too, because we said, look at this thing. It's not good algebra, because whenever I make the substitution of x dot index prime minus x, I have to remember to shift the time on everything to the left right that's my algebraic rule i could live that way but it's not not good algebra because uh because uh because i i can't go backwards and forwards once i do this if i forgot to shift the time i'd be in trouble i can't do ordinary substitutions i have to do non-local algebraic operations i'd like to be able to do local algebraic operations how can we how can we do that? Well, you could replace the derivative by a shift operator. I'll show you on the next page. Because I just wanted to tell you the way we were doing this. You see, what we said was, let's have a shift operator which has the property that when you put it in and move it across to the left, it causes the time to shift. And then I'm I'm allowed to write out the derivative. I stick in the shift operator, and the shift operator will do what I have to do. It'll keep my memory from forgetting. It'll keep a memory of the fact that time has to be shifted because the clock ticked. So the ticking of the clock is J inverse X J. Already the ticking of the clock has become represented in an algebraic way. And now the same computation will go like this. get us the same result with a little symbol over here that i have to keep track of it's interesting now i'll tell you another story and as you say it's uh you know your your your symbolism is we're doing your calculations for you it's all good symbolism symbolism which is doing the calculations for me is wiser than we knew at the time and i'll show i might as well show you here and then I'll go back, because it was j times x prime minus x, right, divided by delta, but we won't worry about the delta, and that's jx prime minus jx, which is equal to xj minus jx, and now your non-commutative geometry in the sense that you know it has just risen

12:30 its head, because that's a commutator, and the derivative, this newly defined derivative to you is commutator with j. So it's commutator with j, right? So the derivative, this modification of the derivative is a commutator, right? So suddenly what we're actually saying is that our derivatives should be commutators, right? And let's make that into an abstract framework. So the abstract framework is simple enough. It's the same thing. I'm just going say it again um i have an arbitrary discrete derivative there it is yeah it doesn't satisfy the leibnitz rule uh-huh oh right well i like to tell the story the way we discovered it so i'll just say we discovered that when we did this trick which was motivated by making our thing work we discovered that the derivative satisfied the leibnitz rule actually we discovered the satisfied this new derivative, satisfied the Leibniz rule, we discovered it before realizing that it was a commutator. As soon as you realized it was a commutator, of course it satisfies the Leibniz rule, but that was a startling thing for us because we thought we were just adjusting this so it would make decent algebra, and suddenly the calculus fell into place. For the purposes of what we were trying to study, we wanted it to behave like ordinary calculus, and suddenly it did. So the moral is that if you have discrete differentiation, which doesn't satisfy the Leibniz rule, and that's how it doesn't. Okay, so now I'll prove it to you in another form. This is how the derivative in discrete calculus actually behaves, this little shift. And suppose that I put in an operator that does the shifting, like I told you, push it to the left, and it causes the function to shift, and I define a new derivative by taking j times the old derivative, then the new derivative satisfies the Leibniz rule, and the reason is immediately obvious here, right? You multiply this by j, but it moves back across and becomes jd, and so you have jd and jd in the ordinary Leibniz rule. So you would cover the Leibniz rule. Yeah, but in a non-commissioned setting. right so that means that i can start with discrete calculus and shift it into calculus living in a

15:00 non-commutative world and the leibnitz rule will be satisfied certainly good to have the leibnitz rule if you're trying to do various things whatever it is you're trying to do so um so very interesting so now this chronicle would have loved this wouldn't he chronicle would have loved I hope so. We could ask him. Well, a little late, but... Maybe Lee would have liked it. So we're saying a Lee algebraic context is good. Yeah. Well, that's no news to physicists, right? But this is a story you can tell. I like the fact that it's a very elementary story, because it leaves things wide open to discussion. But I also keep imagining that I could teach young students about the non-commutativity this way yeah the question is as soon as they learned about derivatives they can yeah yeah the only problem is the little the strictures of the educational system you know yeah well but but if I can get them outside like yeah like all good people I'll do their learning yeah yes because basically this is you can do from first-year calculus plus you know simple linear algebra you can teach non-commutators but anyway i'm just reminding myself on this slide that you know a b commutator j satisfies the leibnitz rule and if i define the operation of a star b to be commutator that's not associative jacobi identity though says that anybody is anybody acting on a commutator acts as a derivation right yeah that's the jacobi identity You usually think of Jacobi identity as taking one way of associating the product, A commutator, B commutator, C, and then cyclically permuting it. It all adds up to zero. But if you look at that, you see that says that this one, if you put it on the other side of the equation and flip this, it becomes that because of the change in sign. that minus this is that and this says that a star a times b times c a is the operator is a operating on b times c plus b operating on a time c so that's derivation and would i be right in guessing that you know that uh already in here you've got a category lurking in the

17:30 background with the appropriate um but in any case i'm just reminding myself yeah okay well go on go on go on um um we can skip this except i'll remind you because you've seen well all right i won't skip it this is just dear to my heart because this is the jacobi identity which is dear to my heart for other reasons you see if you diagram multiplication this way a b and a times b comes out it's a little processing network then lee algebra a times b this is the non-associative perhaps then a b times a which is what this is is minus a times b so the cost lines think of minus and this is the jacoby identity right and it's not difficult to see what this is you just feed these in yeah and um and then this diagram is is is the is the link between the algebras and non-invenients exactly it's the crossing the crossing moves isn't it uh it's it's well it's what's well it's it's the basic relation that gives rise to the sodium invariance but i don't i don't want to talk about that i just brought it up because i like the jacobian we want to think about calculus in a non-commutative framework okay so that means that all derivatives are commentators so let's have some notation we have variables but these are not necessarily commuting with anybody but we want them to commute with each other in any case there's some huge algebra here where everything is a function of these variables you might say and um and these variables commute with some things they don't commute with other things that's our context yeah yeah like they don't commute with the pi's because otherwise we wouldn't be able to make anything happen derivative with respect to qi is going to be commutator with pi the q's commute with one another because i'm trying to make ordinary calculus and the derivatives don't depend on order because i'm trying to make ordinary right now you do a little calculation and you find out that the order of the derivative is one way minus the order of the derivative is the other way is the same as the commutator with the commutator of P i with P j.

20:00 Okay. Yeah. Yeah, see that. So that's a good identity to think about. It's easy to verify because it says that if you had some derivatives which didn't commute with one another, then this is measuring how they don't commute with each other. Right. And the simplest way to make them commute would be to have the P's commute with one another. Yeah. Okay. so this is curvature in the most abstract sense right derivatives not commuting with one another is curvature so that's curvature in this case we're saying the curvature will be zero by fiat and then we want delta ij to be dqi dqj of course so that says that qi with pj commutator is delta ij these connotators are zero. It looks rather like quantum mechanics, but from this general point of view, it's just how to set up local coordinates that are flat, no curvature. Okay, so now we're ready to do some calculus, except we want this to be temporal, and so we need a time derivative as well. So there should be an H that represents the time derivative. Or maybe it should be I times this if you wanted that to look like quantum mechanics like tony uses i i have to sort out whether tony needed his i i suspect he didn't need his i that all the calculations work without the i but i have to check the calculations we'll get to that but then the next thing is that this is uh q let's just calculate qi dot is is the commutator of qi with h by definition which is d upper i of h that is there's d lower i which is the derivative with respect to q but if you take the commutator with q you get the derivative with respect to p so this is the definition dh dp that makes sense because dpi dp j should be delta ij so just do it and on the other hand PI dot is minus DHTQ that's Hamilton's this is Hamilton's equations so Hamilton's equations which you usually think of as expressing some physics

22:30 just express flat coordinates in this sense yes it's a very interesting way of doing it some of the talks I've been listening to in the last four days at this non-cognitive geometry workshop I think possibly connects quite strongly with some of this stuff that people are doing with triangulated categories where they're using kind of matrix decomposition to explain the source of the non-comutivity. Are you going to be returning to us in a few days or something? Probably not. I'm going to give away in Bristol. Well, the next time I talk to you, you'll have to tell me about it. Yeah. Well, I've got a whole load of notes on that one. But this is very interesting indeed. It ties up with beautifully the sound. getting clear enough so it would be worth talking to a group of people doing non-commutative geometry oh yeah that's why i want to go to that december meeting yeah yeah good idea well as i say freddy's coming back as well i think you'd be very interested in but but this is a this is a nice point if you tell it to i've had the experience of telling this to physicists and they said so what yeah that's nothing new because they know it already but they don't know it in telling the story this way no no they're they're just used to the fact was work but but telling the story this way you see it's it's showing that it's it's actually a mathematical structure just coming from the notion of doing calculus in a non-commutative way and then you have to say well i better remember what it had to do with physics i mean it did have something to do with physics it sure did here's class here's the classical hamiltonian and the energy is p squared over 2m plus the potential you do a little calculation with that and you you substitute into peter sorry we're trying to try to listen over here can you thanks very much but this is and then hamilton did more right he said hamilton said oh yeah and the time derivative of an arbitrary function is given by this usual calculus expansion in terms of its two variables but the q dot is dhdp and the p dot is minus dhdq so this then has this form which he called the Well, Poisson or Hamilton, somebody ended up calling it the Poisson bracket, but Hamilton then can write the time evolution equation for the system in terms of this. Right, I mean, the time derivative of f is given by, it's Poisson bracket with the Hamiltonian. And so this is Hamilton's reformulation of Newtonian mechanics.

25:00 And then, of course, as you know, Poisson brackets satisfy the Jacobi identity, so they're forming a Lie algebraic structure, commutator. And this was the situation of one way of formulating mechanics for a long time until Dirac came along and said, oh, this transition from here to here is telling me how to quantize. And that was the first time people started studying this transition. What we're saying is that this transition is worth studying, whether it's quantum mechanics or not, because there's something going on in between these two ways you're looking at things. Sure, sure. This classical way of thinking Maybe what he's telling us also is that the Sorry, it was that the Brahe-Ketz formalism was only This is some kind of transition between the continuous and the discrete. Absolutely, yes. Yes, well, maybe what this is telling us is that the Dirac version of quantum formalism in Brahe and Ketz was as it were halfway through the wood and that, you know, we had to go some deeper structural level to understand really where the non-commitativity coming from. There was a very, very nice talk last month of that CKC workshop in Oxford, you know, the one that Bob Cricker put together about symmetric monoidal categories, which, of course, involved quite a lot of citation of your work in topological computing, and showing how you could get the whole Dirac formalism out of it. It just very naturally falls out of their framework, you know. I've been saying for years that, you know, basic Well, they're saying the same thing. You'll see a nice summary of that in that paper on anti-ionic quantum computing that I put on the web, the long one. But that's fine, but we're not doing that here. No, but I see how all the motivations connect are. Sure, but these things are all connected up. So here's another one of these capsule things. What is dynamics in this abstract, non-commutative world? It is the temporal behavior of the coordinates, right? Right. And if that was given by some law, and a law in the algebra would be just some a sub i's, right, that are the functions of space and time, right, that tell you what the temporal behavior is. So what would they be like? How can we talk about that? Well, with 20-20 hindsight, I write a sub i as how far away it is from p sub i.

27:30 P sub i is a kind of momentum. This is a kind of momentum. So let's write it as a difference. Of course, that's the famous minimal coupling formula, but here it's just a definition, right? How far away is it from P sub i? And then we would have a new derivative, di tilde, which is the commutator of f with the q sub i dot. But what happens if you take the commutator of these? That's the curvature for this new derivative. Ah, right. Right? Similar to the one you had on the earlier page. Well, what do we get? Well, just look at it. The p's commute, and there are three terms instead of four because the p's commute. And this is the derivative of aj with respect to qi, and this is the j derivative of a i, and this is the commutator of a i and a j. And, of course, you know this formula. Well, that's the curvature of a gauge field. So you see that the curvature of this shifted situation is the usual gauge field curvature. In other words, electromagnetism or its generalizations will arise immediately from this situation because you have this. If, for example, this was zero and you did an epsilon ijk on this, you get the electromagnetic field. epsilon i j k time epsilon i j k of s gives you the appropriate for the usual formula electromagnetic field yeah um that's the fastest way to make the feinman-dyson derivation uh or something like the feinman-dyson derivation that's saying you get electromagnetism commutators but i'm not doing that here but that was where we started we were trying to figure out where that So, again, in the advanced calculus scenario, we can teach gauge theory, at least in this abstract form, in the first week, well, in the first semester. Without having to go through all that machinery of fiber bundles and the abstracts, you know, the vial stuff. This is near the end of the first semester if you were teaching a course, isn't it? It's interesting to see how this does connect up with the vial, you know, original derivation of the gauge. So Pierre knows all this, so I wanted to get to the part that he might not have thought about in our terms, but I want to go one step further here, okay, and then we'll get there.

30:00 I want to look at this. You'll see why in a moment. QI commutator QJ dot, which is like QIPJ, but shifted, right, because this is a shifted P. And what we get is delta IJ minus QIAJ, right? so in the simplest case we just get delta ij and that's the electromagnetic case but in the more general case we get something else over here let's call it gij and I want to justify that I should think of this as symmetric there are lots of stories about how to think of this as symmetric the one I want to tell you about is this let me jump to it Yeah. I want to choose a Hamiltonian. I'm going to do it in a different order than I did in the talk. You would think that the natural Hamiltonian to choose, if you think about a metric, thinking about a metric, the natural Hamiltonian is Gij, Pi, Pj over 2, because this is just expressing the momentum if you had a metric, right? Because the ds is given by, ds squared is given by gij, right? So this is actually just writing down what the momentum would be. Right. So then if we shifted it into the non-commutative world, it ought to be that. So this is the appropriate Hamiltonian. I'm throwing away the mass. I should assume that my metric commutes with the coordinates, because, in other words, it's not a function. In other words, that says a Gij is just a function of space. That's what it should be. It shouldn't depend on time. No, I said that's what you're talking about. So then we could do a calculation and see if it worked, and by gum, it's going to work. I take F dot to be FH, and I find QK dot. I do a little calculation. Calculation's easy. I'm just running my commutators. These guys pull out because these commute, and then I have this and the Leibniz rule does this and these are delta ij's and it all collapses and you get qk dot is the metric tied in with a pj. All right? So, in other words, this is a generalized pj and it got generalized by tying it in with the metric. That's what deformed the momentum.

32:30 And then when you take the commutator with qi and do a little calculation with gij. And it gives you You can prove that G.I.J. is G.I.J.I., but you might as well just assume it, and then you won't have to worry about my logic. So, you see, if you started with the Hamiltonian that actually corresponded to moving in a curved space, you end up with that. So that makes sense. Now we get into the thing that Tony started thinking about, And that is, we'd better be careful about some of the constraints. I'm out of order here. Where's number 13? No, you want 14 there? 13, yes. Ah, there it is. I decided to tell you this story in a better order than I did when I wrote the slides. This is the usual formula for the derivative, right? Yeah. You might think that I could just take it over to the non-commutative world by taking this to theta dot, capital theta as the operator capital Q's. So just take this to this, commutator with H, and I take this to this. Although, you know, I could have written it on the other side, but why shouldn't I be able to just do that well the calculation doesn't work right you can try um but it but it works almost uh and i'll show you the calculation okay here's our hamiltonian yeah um i calculate theta h and when i do that i get i get two different orders of things i get a di theta with a p on one side and i get a p and a di theta on the other side and i get a gij okay that's what i get now what will happen if i for the other side. I try for the other side and I get this, right? And because it's desymmetrized, because it's already got this order in it the way it is, and I work it out, I only get one of these and not the other order. You see, I get the P on the left and the D on the right and not the other way around. Right? So these don't quite match. They are missing, it's missing the other one on the other side. Yeah, I see that. And you see that I could get the other one on the other side

35:00 if I had also included in my derivative this written on the other side. Hang on, which... Oh, right, the... Yeah, say that again. I didn't quite understand that. Well, you see, in the commutative world, it doesn't matter whether I write this Qj dot on the left or on the right. Yeah. But in the non-commutative world, if I really want my theta dot to have this form, I'm going to have to do the following. the next page so um so here's the answer uh the answer is that i want you see i just wrote the usual formula and averaged it i said i don't care which side it's on i'm going to do them both and average it and then it will work but i also have to average the hamiltonian i have to write it on both sides if I do that then it satisfies this constraint, then the derivatives work right, but what are we doing what we're doing is following an ancient philosophy that is expressed in quantum mechanics books which is written here, that you have scalars and you take as your principle and I could have done this better page 11 we're looking at the principle is that x plus y will go as operators to x plus y and that x to the n will go as operators to x to the n you'll find this enunciated many quantum mechanics books that's how you go over to the operators but now notice what the consequences of that are It says that x plus y squared will go to, which is x squared plus 2xy plus y squared, will go over to x squared plus xy plus yx plus y squared. So that says, that tells you that it must be that xy should go to xy plus yx divided by 2. If you want things to correspond in a natural way between this rather corrupt world of ordinary continuum scalars and the non-commutative world, the pure, crystalline, beautiful, discrete, non-commutative world,

37:30 to have to average these uh in order to make them look a little more commutative in order to make things match up yeah okay and um and then as far as i know tony deacon is the only person who really took this seriously beyond a certain point um although most many quantum mechanics books will say this much so what do i mean by taking it seriously well of course you could take it seriously and some algebraic people will take it seriously what happens to a bunch of a bunch of variables well then you'll go to 1 over n factorial times the sum over all their permutations so this would happen right but then if we're going to take it seriously we have to go to higher we have to match the higher derivatives as well yeah right so rather complex but fundamental consideration what i just told you is how i became converted to tony's consideration because i would like the derivatives to match at least the first order and the only way i can get them to match and tell my story is if i did the symmetrization at first order so i'm going to do the symmetrization and then i want the derivatives to match and if i want the first derivative to match well hell then i should have the second derivatives match as well and maybe it matters less and less as I take higher and higher matches but at least I'd like to have second derivatives match well now we have something a little more complicated to say the least here's the formula for the second derivative that's easy just gives you a second derivative on d's and then two derivatives of q's you just iterate this is the standard formula what's it going to be in operators well you just average the same formulas I'm repeating the formula but writing it in both orders three factors, so I have to sum over six permutations. Yeah, page 15, yeah. I buy six. This has to be asked for. Now, where's this term? Of course, because you're... For any function. For anybody, theta and the non-community of the world. Yeah. So this is an extra constraint on how the derivatives are going to behave. It really is an extra constraint. If you take this guy and differentiate him, Well, you'll get some of these terms, but not all of them. I leave it to you to do this gore, but it's fun gore, but there's an infinite amount

40:00 of gore here because you could ask for the third derivatives and so on. But this much gore is quite manageable, and you'll find that just differentiating this does not give this. This is an extra condition. Now we're in Tony's domain. maybe nobody other than tony ever ever carried this out okay um but you can work out what this does and with our hamiltonian our simple g-i-j p-i-p-j hamiltonian um it puts a condition on the g's that's the thing it puts a condition on the g's and now i jump all the gore and write down the rather complex little equation about the g's that it corresponds to and this is actually giving So this says that my metric tensor isn't just an arbitrary metric tensor, it satisfies certain derivative equations. Right. And this leads to a relationship with general relativity. Tony shows this to Clive, and they think about it, and they realize that it actually corresponds to this, where RAB is the Ritchie tensor corresponding to GIJ and that this is an equation which if something satisfies Einstein's equations then it will satisfy this equation it's a more general equation a higher-rotary equation hence your question at the end of Friars Law and as you say you've actually derived the Levy-Civita connection here Well, that's a remark from a previous lecture that I made. Okay. You can see how the Levy-Javita connection is derived from the GIJs in this context. That's another matter, but an important one. But the point is that if you think that the GIJs are talking about the curvature of some space-time, then you get a generalization of Einstein's equations that should be satisfied by them if you ask for Tony's constraint and that constraint, the more you think about it is the next thing you want, right? I mean, as soon as you say I wanted to match the derivatives of the first order and you see that they really do match then you've become a believer and you really want to match the second derivatives at least

42:30 and matching those second derivatives lands you in the domain of general relativity or some generalization of it. And that's about where we are. Clive was describing, you know, some of Tony's speculations and calculations. And certainly one can see with this approach where these symmetrization conditions would naturally connect up with the requirement that you get the... One can see in this approach how the symmetrization conditions would connect up with the requirement that you should have some underlying variational principle out of which you get the... I don't know. I just don't know. Symmetrization is natural, as I said, if you want the operators to correspond in their algebraic properties, powers go to powers, sums go to sums, right? Then you have to symmetrize. but what we don't understand is what it means to keep going back and forth across this ditch between ordinary calculus and this what you might say is really discrete calculus or quantum mechanics depending on how you want to view it but it's very encouraging that it ends up kind of automatically landing you of general relativity. That is very suggestive. That's much more encouraging than seeing Hamilton's equation come up. Sure, sure. So I think that And that the non-commutativity is essential, obviously, that this really that we're going to look at this and look at Pierre if I can convince Pierre to become a believer. Pierre's been amazingly silent through all this. I'd be very interested to hear your reactions. Well, I think it's a little too complicated still. I mean, we're looking for some simple principle. You heard James' story. Well, I know it from experience. I mean, we started with such a simple idea way back here on page one, and we thought we could keep it simple and stay. And when it gets this complicated, then I get too old. Is it possible that the third order constraint doesn't exist, because in some sense the third derivative doesn't exist here, and so you would need only first and second order constraints?

45:00 I guess Tony's feeling was that the effect in terms of calculating things or observing things gets less and less with more and more constraints, so maybe you don't need so many. Maybe in quantum mechanics, the reason people in quantum mechanics don't even think about it is because somehow in quantum mechanics, which is more linear, it didn't matter at all. Right here, we're trying to bend it so that it matches classical mechanics, which is a funny thing to do anyway. But it raises all the old questions about what is the relationship between the classical and the quantum? and what's the relationship between the continuum and the discreet. But this is a fascinating story and it's still too complicated, right? Right. That's all right. I mean, I'm willing to be dissatisfied. I want to be dissatisfied. What I really like as a mathematician is to be so dissatisfied as to have an outright contradiction, right? If I see an outright contradiction and then I see that it can be resolved by a certain move, symmetrizing, and then I become a believer. Is it possible that there is some deep background to this discrete, I think that shift is basically a negation distinction from laws of forms and the this time intervals are a kind of paradoxical oscillation or this kind of things so what i'm saying is that if these things are not x or t but because discrete makes a step from x and t i think i think we're thinking along the same lines i don't think of the dot i do not think of the dot as time but rather process prior to And in fact, I think in the mathematics you have to do that, because you want to do Gij to be the metric in general relativity, which means that one of the indices is time.

47:30 But we still have G-dot in this formalism, and we're using it. So there's this background process, which is just a process. That is clear. And isn't the time that happens to these observers out here who are doing the measuring and everything. is could we be more specific and sort of try to specify this background process and about the fundamental variables like negation, which is analog of shift, and Yeah, like if I was doing it, if it was in law, the question is, what's your choice and what will work? Like if I'm thinking in terms of laws of form, I would say, well, there's fundamental cybernetic turn that's making that background oscillation, the re-endering mark, something like that, right? That's the war time or the war process. So it's not quite laws of form because you need, like in your paradoxes, you need to go back. This oscillation, so here is something... epistemologically I think this way this is Aviv would accuse me of meta epistemology I suppose you have a quiet void universe and somehow it becomes aware of itself but if it does and it's talking about then it wants to talk about everything there is about everything there is, or as soon as it observes, there's more. So time happened. That sounds very... You see, now there's always more. It's Russell's paradox, but I don't think of Russell's paradox as... I think of the epistemology of Russell's paradox is that when the observer or the universe tries to encompass all of itself, it can't, except in a condition of time. And so time and so at some primordial level that's why we have time because you can't encompass everything without it well this has been a traditional speculation you can't actually have something you can't have the universe

50:00 observing itself without time happening whether these kinds of thoughts are going to help this kind of physics I don't know but those are thoughts I like to play with they are interesting because they've certainly been some of the deepest thoughts in western physics i mean they're already in i said they're already in people like schlegel i don't know but it might not think of frederick's way of doing this which is to say something exists but this statement conveys no information so if you if you just say all right you know a quiescent universe you're disturbing it, but the disturbance is that something exists. But just because something exists doesn't give you any information. Then you have to work out the consequences of that, and that's how theory, Mark Rowe's got its theory. You're right, Frederick's starting point is very much that differences create differences, create differences. And the program universe is ticking because of that. So it's coming out of that same... When you made a statement that something exists, then it could be slightly modified to the question, does something exist? And then it goes to the denial of it, and then it goes beyond this... Which is very reminiscent of the idea of just drawing distinctions being fundamental in the course and form. I can even think of it as a topological significance also. Another way is that this non-something exists, so something beyond this something. So it means that this something beyond this something somehow naturally comes from this statement that something exists, with this kind of circulation. But if you take Frederick's point of view, you see, you can't just get something out of nothing, but if you can join this with two ideas, that is existence and information. And you can still have no information, so you have the universe with no information, and yet something exists,

52:30 and from that he gets his whole theory of indistinguishments. Now, I've heard that Chris Clark says that there is a mistake in that somewhere along the line. What is Frederick's next step? Well, I know how it goes with zeros and ones, probably back engineers and get stuff out of the zeros. Yeah, well, I did a paper on this way back, but it isn't at me after proceedings. But all that stuff that Frederick had in his theory with the three different and non-equivalent parity relations is so humbersome, syntactically difficult. I mean, I think there's going to be a much simpler way of expressing that. In fact, Frederick Van Eystein and I were talking about this in Ascloster. It's the simplest thing, right? Because there's a much. Okay. If you have two things, a zero and a one, right? And we're going to just be a Boolean addition, right? Then you have one plus one is one, and zero plus zero is zero. Oh, what? Pierre, correct me. What am I doing wrong? I want to see that zero and one are distinct. Now, one plus one is zero. 1 plus 1 is 0. We're doing it mod 2. That's it. So we're doing Z2. We're not doing Boolean. Yeah, it's Z2. That's the way. That was the right thing. Yeah, because one can think of this as connected with this idea of inside and outside. So if you combine 1 with itself, you find out that it's identical with itself because you get 0 in this little mechanical system. But if you combine zero and one together, the output is a one, and this tells you that you had something that's distinct, right? So something exists, but it conveys no information. But I can't agree with this, that nothing can come from zero or from nothing, because it's a kind of co-creation of two things, which could be the only reason for the creation of anything. from zero, a co-creation of two things. So it should be from nothing. It shouldn't be from nothing, but it can be from existence, yeah. But, of course, Frederick was an ontologist, and he didn't like a process view, whereas he liked to think of this

55:00 as a discussion. This does seem to be, yeah, but then you can easily read kind of processual metaphysics into Schlegel and Fichte and all those guys who thought this way, yeah, like if you add a 1-0 to a 1-0, you don't get anywhere, but if you add a 1-0 to a 0-1, then you get a new entity, and so you can generate new entities out of old entities, and then the move in the hierarchy is then you look at the patterns of mappings of those new entities, and rewrite as in terms of zeros and ones and generate more new entities and you watch the structure of that evolving. And then you get the time structure constant and the ratio of the Planck mass to the proton mass. And that's the weirdness of the whole Frederick. Yeah, you get significant physical numbers. And I think this issue of the AMPA proceedings has some genuinely good-looking, read them yet, but they look good to me. Summary papers on this, on the Combinatorial Hierarchy. Sometimes we have a year when there's nothing much on the hierarchy in the conference book, but it looks like this time there's something to read about it. That's where this whole seminar started a hundred years ago. Well, 30 years ago, right? It was about 30 years ago. Well, AMPA 26 was our silver anniversary, and this is AMPA 28. This is almost 30 years ago. It was in 1979 that I suggested we found it, and they joined me. Now, the laws of form consideration is very close to this, right? It says there's a distinction. And then at the level of the formalism, this distinction can interact with itself in two possible ways. It could interact with itself to cancel itself out, or it could interact with itself to reinforce itself. One gives you the part of the back, and the other gives you nothing. So it's a little particle which can interact with itself to either produce nothing or itself.

57:30 And if you decide to think of that in a quantum way, you get a lot. But the co-creation doesn't exist here. The closeness and co-creation, and that is the drawback of it. Pardon me? The kind of co-creation of two things doesn't exist here. co-creation what's co-created is the distinction and the two sides that's what's co-created this as I say is so reminiscent of traditional systems of early 19th century metaphysics as I say I don't see that one has to replicate the details of their way of thinking it just seems to me that this is I mean, firstly, I think the question of ultimate starting points is, I always have a very pragmatic attitude towards them. The real test is whether it allows you to do the kind of thing when tying up details of the formulas of IQ. I don't know that there is an ultimate starting point, but what one is looking for is a conceptual, a genuinely simple way to evolve a complexity, right? or to evolve the understanding of something. Like, take general relativity. This is going to be helpful to me more than I can say. Because you'll know how to talk. Where James is, because you see, my credit card has been canceled also. Yes, right. But thank you for listening. Oh, no, it's nice to hear it again. Thank you very much for going through it again. But for me, see, I've been struggling off and on to understand general relativity, and my struggle is that I'm lazy. I mean, you have to really convince yourself that these are the right tensors, one way or another, right? And there's lots of exercises to do. Einstein had a lot of trouble finding them, and even after they're found, you have to do a lot of work to see that it's the right thing to do. And why that rather than something else? And as he said himself, you know, everything on the right-hand side was a preliminary closed expression for some further stuff, which he never did get around to figuring out, as he said.

1:00:00 Because he could see that an awful lot had to be put in there by... And then of late, of course, in mathematics, we've been hit over the head with the fact that the Ritchie tensor is deeper than you might have thought, right? Yeah, what have you got in mind in saying that with... Richie Tensor in Three Dimensions is proving the Poincare Conjecture in the Perthurston Geometrisation Program. I don't know about that stuff. It's now at least published. I mean, yours truly has to actually learn that end of things. Fermat Theorem Proof I didn't have to learn, but professionally speaking, I guess I'm constrained to actually try to learn some of this stuff. while. There's this Russian, you know, you must have heard some of this. I've heard it. There's Russian Perelman who put these papers on the web, which excited a lot of people who like to think about three-dimensional manifolds and differential geometry, because they purported to fulfill the promise of this program of Hamilton, who was going to use the Ritchie tensor to create a flow on the three-manifold, which would geometrize it. And you have to prove that you can handle the singularities in the flow, because you expect when you geometrize the three-manifold that you'll geometrize part of it one way and another part of it another way, and so on. That's Thurston's idea. And if you could geometrize the whole thing, then you end up showing the Poincare conjecture among other things. Right, okay. And so there's a lot of analysis and geometry involved in this approach, and Perelman's papers were incomplete in the Russian style, And he never asked to get them published in a journal. They're just sitting there on the web, two or three papers. Just remind me, I know I should know, the Poincare conjecture is about classification of homeomorphisms. No, a three-dimensional manifold with trivial fundamental group is homeomorphic to the three-sphere. Right. Is there a general... Statements you could make about three manifolds, and it's been unproved up until now. Right. And is there a generalization of it to do with the classification of all kinds of homeomorphisms of manifolds in different dimensions, or is it just restricted to the... Yeah, and it's been proved in all dimensions except for... Except for four. Except for three, yeah. Four was very special, wasn't it? The one which they needed to use a lot of ideas from physics.

1:02:30 In fact, physics has fun ideas by Donaldson. It involved the confluence of Donaldson stuff and topological stuff from Friedman. Yeah. And it used a lot of very deep machinery from different parts of mass to the tools. And then three has been resistant, and I've always hoped it was false. I thought that would be interesting to have a manifold of trivial fundamental group that wasn't the three-sphere. But it seems like they've been proving it, using differential geometry, and the Ritchie tensor is the key character. Anyway, the story is an interesting one. This guy, Russian mathematician, writes these papers which are sketchy, puts them on the web and doesn't bother to ask to be published in a journal because he doesn't want to write anymore or whatever his reason is. he says he's not willing to take the prize if it turns out it's right either a million dollar prize a number of groups of people have been working on it and the whole thing has converged to a bunch of things now being published a 300 page paper in the Asian Journal of Mathematics by a couple of Chinese mathematicians a book by John Morgan notes by John Lott. You can see, you can find the whole thing on the world very easily. All just in time so that, and I wager, because I can't know officially, and I don't know unofficially either, that Mr. Perlman will be getting the Fields Medal this summer. Okay. For this work. And he did it in a way similar to the way Witten got the Fields Medal. That is, he didn't write down the complete proofs. Other people wrote down the complete proofs, verified that his breakthrough is correct, and the mathematical community will give him the Fields Medal. I'm quite sure. Good for him. Good for the community in this context. Good timing. And who gets the money? I don't know. they should just give us the money yeah they should just put it back in the pot and let it fructify but anyway that's good old Ritchie tensor

1:05:00 can you explain a bit more about the guiding idea I see but what is it in the structure of the Ritchie tensor that delivers this beautiful the Ritchie tensor has the correct divergence properties for it to match the physics I mean on the one hand you're writing down something which is just about the differential geometry But you have to write down something which looks like it's physical because it's, on the other side, you have the energy and momentum tensor, which has certain divergence properties and so on. And this tensor has the right properties. It balances the equations correctly. So that, as Wheeler likes to say, matter tells space how to curve. And matter tells... Space tells matter how to move. Yes, that's right. How does that quote again? Matter tells space how to move. space tells man how to move yeah whatever yes yeah however you want to think about it they reciprocate one another and and and this didn't land this business didn't land directly on this but it landed on something such that any solution of this implies a solution that so that's pretty good well sorry say that again the constraint on the metric turns out to be equivalent to this curvature equation this curvature equation is a generalization of Einstein's equation is a generalization and the generalization is when I say generalization that means that a solution of Einstein's equation will be a solution to this right yeah and that generalization is particularly connected to the way that the I'm sorry, how is this, and this obviously has to remain stable when you go to this kind of non-competitive treatment of the coordinates. Now you're getting to, what are we saying? We're saying that we're using... Yeah, there was a question mark at the end of that. Sorry. Do you go to the dinner or not? Yeah. Yeah. The dinner, that means we should meet people. Well, hang on. What's that? We want to meet people about 7.20 out there. Yeah. That's right. That's right. Yeah. Well, actually, I think Ted said we're probably going to just a little bit later, about half past. So it gives us another 20 minutes. So I'm going to go up to my room in a moment. Sure. Yeah. But you see, what are we doing? We're using the non-commutative, for this project, as Tony is thinking about it, we're

1:07:30 We're using the non-commutative world as a Rosetta Stone to define the correct equation, which we're then going to say, okay, now let's look at this equation as a classical equation. I don't know what I'm doing when I do that. That's like consulting the Delphic Oracle. Right? That's what I'm doing. I'm using the non-commutative world as a Delphic Oracle. I could have, but in this case, I'm getting some information that's beyond what I already knew. In the first case, when I used it as a Delphic oracle, it told me the Hamilton's equations, but I already knew them. Yeah, it told you the Leibniz rule. Well, it gave you about the Leibniz rule subject to... And then it told me to gauge theory equations, but I already knew them. but then asking for a genuine constraint about the way the ordinary commutative world talks to the non-commutative world actually asking for the constraints it starts to tell me more information and so it looks like the tip of the iceberg yeah I agree it certainly does and the fact that it connects up with this the great additional depth of structure in the rich utensil which is now beginning to show up which I imagine also has connections with some of the approaches to quantum gravity that people have. Well, right, and the fact that we're formulating gravity in a noncommutative context means that in a certain sense, we're talking about a quantization of gravity, but you have to be very careful. Yeah, sure, you have to be very careful, I realize. But I do believe that there is a particular, Lee Smollinger had an interesting talk in survey talk about, you know, the various obstacles that the loop quantum gravity program had encountered, and he spoke at the end about some new work in which they were covering some of these, you know, these new constraint conditions, and also the Ritchie Tensor came into it quite a bit. Oh, well, yeah, the Ritchie Tensor will come in because it's part of the formulation of general relativity in a standard thing, but I'm just saying that in my attempt to educate myself about general relativity, I keep trying to understand what's the meaning of the Ritchie tensor, so this is how, this is

1:10:00 going to help me a bit. But this whole program of Lee's is really deep, good, good program, you know. It began with understanding that you could Ashtokar's understanding that you could reformulate general relativity in terms of an underlying gauge theory and then you could try quantizing that and then it related to knot theory as you know. Yes indeed, that's what I thought you could write you could rewrite the states of that quantum gravity in terms of in terms of looking at the Wilson loops corresponding to knots and so in a certain sense the knots became took that seriously and started doing calculations with spin networks embedded in space to think about the quantum gravity, and then that dovetails with letting the networks start to change in time and you get spin foams and lots of stuff going on. Which is really conceptually exciting stuff. And also, of course, can dovetails with this insight that, as you say, the Wilson-lips, but also the know that the temporal Lieb algebra really gives you the correct way of thinking of the quantum form of the structure that's in Dirac, but it's actually, I think, getting at a conceptually more illuminating understanding of that and connecting it in with some of the deeper structural features that are now showing up in this modeling things in symmetric monoidal categories, or possibly in bi-categories, those are themselves quite difficult technical issues. I had a very interesting talk about bi-categories in connection, whether bi-categories or symmetric the right framework for understanding quantum mechanics at this workshop they had in Oxford last month, which Bob and Abramsky organized, which was where, of course, I met your colleague, student, the young Chinese guy. Jean, is it? Yung San. Yeah, yeah, Yung San. And there were some extremely good talks at that workshop. It was a mixed bag, but the best of the talks were very good indeed. Oh, you're talking about the Oxford one? No, there were some talks on bi-categories at White Point in the category theory meeting in Halifax as well, but I didn't go to that. But they were very, very... Sarah Mabrowski and Bob Cook are working very hard at interrelating quantum mechanics and category theory.

1:12:30 Yes, they are. I'm not saying they've got the right framework. I think that I have my issues with their strategy. I still think that, you know, symmetric monoidal categories may not, in fact, be the right framework, that we will have to go to sort of, you know, to bi-categories. Oh, you should still lose some of that. Yeah, as I said, at the moment, it's far, far too early to tell. You know, there are thousands of schools of thought contained. But the bi-category approach does particularly seem to have a whole particular promise for connecting up with some of this insight that's now coming from triangulated categories and from kind of ways of treating metrics, composition, you know, decomposition in triangulated categories that allows one to get non-comutativity out in a rather striking way. That may, which don't seem to operate in the same way in the symmetric quinoidal case. But this is all, no. This part of the story you'll have to tell. This part I'm just beginning to learn from, but of course it's interesting because I now understand, I think, a little bit more why Bill Lovere and Steve Channell have been so against the non-comutative geometry Oh, yeah, they've been dead set against it. It doesn't fit with their notion of the categorical... Well, it doesn't at all because, of course, they want to retain Cartesian closure as a condition on all their categories because, of course, they're committed to this really very practical... But they aren't physicists. They're not physicists, I agree. But I think I understand more deeply about where their motivation has come from and why they're so concerned to have a good general and what they would see geometrically reasonable notion of mapping between spaces in general and the bicategory approach if it works is going to allow them to attain that, it's going to allow them to keep marita equivalence as a general tool for controlling the way that categories interact that may be an aspect of the story, that may be an aspect of the big picture but certainly you can't get around this, not only to It's extremely interesting to see how it shows up, as you say, even in supplying a deeper understanding, perhaps a more direct understanding, of basic operations and calculus, of the Leibniz rule and of Hamilton's principle. It's beautiful, you get the Jacobian out. I'd really like to see somebody teach noncomitivity

1:15:00 to a first-year calculus course after they've done that. it's not really for first year calculus when I think of something nice and simple and I say gosh I could teach this in first year calculus the answer is I could teach it to my graduate students when you go back to redo first year calculus again when you get into grad school because there you have it's not that you couldn't teach it to first year calculus but the problem is that is that we teach them these courses that are basically recipe problem-solving courses for a long time. And in those courses, unless you have complete control over the course, and usually you don't, you've got five colleagues who are all teaching the same course, and it's expected that everybody... And you've got to cover the material, you've got to make sure they can solve the problem sets. And it should be pretty cut and dry. You aren't asking them to try out something that might not work. and you don't ask them to think too deeply either because they're going to get through the problem and that fun stuff comes late. But by the time you get into some of the upper-level courses or an honours course, see, I taught an honours course once, it's just you don't often teach an honours course, they want to know something that isn't in the book because that's what they're there for. Yeah, yeah, yeah. So that's right. You could show them yours. Yes, you could say, is this going to be an exam? No, well, why the hell are you teaching? I'm free. No, thank God, you've got to be on that point. See, it's the opposite in the opposite. or in a graduate course unless they're a little bit nuts they expect you to show them something new and different and interesting well this was very interesting Lou and thank you very much for your incredible patience you said you were going to spend 20 minutes in fact you probably spent more time on this than you did on the original talk I really am grateful to you but I got a huge amount out of this it's very helpful for me it's very very very nice stuff indeed gosh can I copy those notes oh yeah sure you can well let me see I don't know how you're getting it back well I can stick around tomorrow morning until we come what time do we come in tomorrow oh hang on if you want

1:17:30 Thank you.