Noncommutative structures, physics, analysis, calculus

0:00 ...out of the 64. Okay, I'll just leave that. And let's Vanessa take over now. So that's just some rough ideas of synagogues in physics that may re-emerge as a I think I can't, I don't know which direction it points it, I think it's my encompassing. Vanessa Hill takes over this presentation now. Just press the red button at the top and point it to hold it in the right direction. Okay, just start by introducing the tonic cells for anybody who doesn't know what they are. tetrahendron, 4 sides, cube has 6 sides, octogenon has 8, dodecahendron, 12, and the icosahendron 20. I'll be talking mainly about these for the first half of the talk, and these for the second half. These actually include the golden section. I'll explain about that literally. We have photonic solids, they nest two-dimensionally and show this, just to brush through quickly, the pink, the tetrahedron is shown by the pink one, or the yellow one, the cube, basically we have the pink here, and some of the yellow ones as well. the actual face of this one face there, it's two face, two face. Star texahedron, we've got a large pink one, we've got a really yellow one inside. Octahedron, basically going pink and yellow.

2:30 I'm just out wearing it, but it's there. The dodecahedron is this inside, you know. And the alcohol you do is this female. and then will continue to nest Adam from Aiton. Points of interest, the tetrahedron is a reciprocal of itself, and the optahedron is a reciprocal of the cube, and vice versa. The icosahedron is a reciprocal of the dodeca hedon, and vice versa, unless Adam from Aiton, and just a little curiosity, a 4D image of the tetrahedron protected onto a 2D plane, It's a star pentagon. What do you mean with 4DMAs? 4DMAs. 4DMAs. Yeah. So, biology, the main bit, I suppose, is the expression of DNA into protein. So DNA is basically composed of four bases. You just usually use ATGT, adenine, thiamine, ryanine, cytosine. and it's a double-stranded helix of plus-sense and anti-sense strands. The RNA is copied from the plus-sense strand, this is a single strand, and T is replaced with U. This is known as transcription. Proteins are encoded within the messenger RNA by what we call triclet codons, which can be translated into amino acids. This is called translation. Basically there are 64 possible triplets that can be made from these 4 bases, which is 4 to the power of 3. And the one standing question for me has always been, why of the 64 triplet codon, Nature only uses 20? This is my interpretation of what I'm going to give someone's grammar DNA. We have a cube and inside a tetrahedron and we can place the base on each vertices. Okay, so now we can consider that this tetrahedron contains all an inflammation level.

5:00 The tetrahedron can now be considered to contain on an inflammation level all... all the 64 possible triplets. So, I mean, you can use it in any order, you can go C-A-G, T-A-C, T-T-C, etc. So you can get all the 64 within that tetrahedra. Double-stranded DNA, my interpretation, would be a star tetrahedra within this cube. So now we have the pink the anti-sense strand, the green in the anti-sense strand, and we can also place the bases so that we get the correct base pairing, so in DNA the G pairs with the C, the A with the T, and as you can see here we can get this correct pairing. Right. There are two ways we can label these tetrahedra. We can only use the vertices or the lines, which is best. So to say maybe this is messenger RNA, but there's many different vergencies as to this one, whereas this one is fixed. So what else occurs in biology that uses this tetrahedral format, and that affects the amino acids themselves. All of them apart from protein, each amino acid exists in two forms that rotate light, in an eleval and dextral rotatory way. Nature chooses to use the needle, so we simplify the amino acid structure. We've got the carboxyl group, amino group, basically the R group, of which we've got one of three possibilities. Non-polar, polar positive and polar negative. Okay, so just to get back again, we have 64 possible triplets that can be made from these raw bases, and all proteins are constructed from 20. Why is it that nature translates them into 20? The start codon is also an amino acid methylamine. We also have three

7:30 stop codons that don't come for any amino acid. So here we have all the 20 amino acids, and the variations of the current community to translate into each. We also have three stop condoms here, and the thymine is the start condom. So, what structure could represent these 64 triplets and 20 iron acids? Come back to the start tetrahedron. So we've already shown that a single tetrahedra can represent a single-stranded DNA, and two are interlopping, a double-stranded. But let's have a look at the actual tetrahedra themselves. What happens when we start to build up to higher levels? It wasn't until I actually sat down with some tetrahedra dice that I realised you can't put together tetrahedra with tetrahedra, you have to start improving octahedra. So in the first order we just got the tetrahedra, in the second order we have four tetrahedra and one octahedra. And in the third order we have ten tetrahedra and four octahedra. So, I'm going to consider each octahedron. Each face can be considered as a triplet codon. Each octahedron has eight faces and therefore each octahedron makes up eight triplet codons. And of course there are many possible combinations of triplets that can exist. Now, Right, so now, on the first order we've just got one tetrahedra and no octahedras, and the second we've got four tetrahedra, one octahedra, which gives you eight triplets. In the third order, we've now come up to ten tetrahedra, four octahedras, which is giving us 32 triplets. way there to our 20 in 64.

10:00 I couldn't remember the bloody man's name. We have now a star tetrahedron. We actually just multiplied it up by 2. We've now got 20 tetrahedron which could represent our 20 amino acids. There's eight octahedra which could give us a seat to hold faces, and there's triplets. Just to show you what's happening inside this beast. This is how the tetrahedra act. And then, I'll show you what happened when we combined the two tetrahedra. So now we've got a double-stranded DNA enclosing all the information of 8 octahedra, 8 times 8 is 64 triplets and 20 tetrahedra which is 20 amino acids. It's all coming from the packed octahedra, just giving us a 64 triplet codance. So, I'd really like some help with this one. What I've started to do is try to place these triplets to give us our amino acids. I've taken the corners and basically just use 3A's that's giving us lysine, 3U's that's giving us phenylalanine, I think that's 3C's proline, 3G's glycine. I need assistance to be able to place all the others to give us all the possible 64 triplets and quickly 20 mm. Like Stark and three stock codons. Stark codons are in AUG and we have the stock codons UAG, UGA, UAA. Now interestingly, we can define all those onto one tetrahedra. So we have the AUG, AUG, and again we can do another UAG the other way, UTA, UGA, and UAA. And I do sometimes think that this is the very inside tetrahedra, which is totally surrounded by the octahedrons.

12:30 Maybe, maybe not. Just to give you a few more visions, this is what's happening with the tetrahedra. And this is in three dimensions. May I ask a quick question? Do you think there's a relationship between the 20 faces of the top platonic solid and the fact that you only get 20 proteins? Yeah, I did have a slide before that was asking those questions. Sorry, I'm being a bit slow. Yeah. And again. Right, and you get this cute iteration. Fascinates me for some music. Right, now I want to introduce the golden section 5 and point 6 1.8. Up to now we'll be looking at the cube in this perspective where you have one corner lining up with the middle to the back corner, which gives us this alpha hexagon. Now I'm going to change the perspective on this one, to this, basically we move this point up and we now introduce, if we consider this line here as 1, this is now 0.618. So we've got, if we take this diagonal cross here, there's one, if you look there's a cross maybe another face over here, and that cross point would give us point six or eight. It's the damel on the front face crossing with the damel on the back face. That's right. It's quite interesting because the way you do it is just to take it from this lining up with the back point, but take it up so that this now lines up with where the red lines cross.

15:00 So you can put them back to the direction? Yes, it's a perspective. But this introduces golden section. So basically, here's your one, here's the five. And this seems to be like a breaking of symmetry, so we get polarity, not only here, but here. something interesting and you can see now we've got basically two styles overlap and we're getting more golden section here if this is one that's 0.618 there and so forth with the lens of the star but there we see the overlapping pentagons. I want to digress a minute we've got another This concept, I suppose, of breaking symmetry, you can stick five tetrahedrons together that make up one contagonal disk. So we're reducing the eight and reducing it to five. So now we're going to add the contagonal disks. And what we actually are looking at now is in the cross of Egypt. Oh, sorry. Oh, sorry. Right. Also, the line is defined. It's reciprocal, the di-decahedon inside. Okay, so at present there are two theories from nuclear attack pairing on DNA. It's the one that everybody knows, Watson and Crue, in 1953. But now we have a new one, which hardly anybody knows about, by Mark Curtis. So from 1998, it's been published. It's actually an artist who tried to reconstruct a DNA model using Watson and Crick's base pairing. He found that he couldn't actually do it. So then he went right back to the very beginnings, took the X-ray crystal ancestry data, and formulated his own theory. So, here's Watson and Crick's pairing. They have only really considered rhyming and rhyming in the ketone forms, where in reality they exist as a mixture fluctuated between two forms, the ketone and the enol form.

17:30 Now the beauty of Mark Curtis's work is that his theory actually includes both and shows the bonding for both types. And the other interesting thing is what he's done is actually turned this molecule around and we now have this sort of a contagonal pairing. And when I first saw this I just instinctively felt that something important. I thought it was This is just another diagram of his. He's actually got pentagonal blocks that he stacks. And this is a smile. If you look down from the top, this is the view. So now we've got the... These represent the pentagonal parents. So, and then we did this, which was the object of this study. And it ended up on the millennium stamp. So, now, I've just overlaid my cosahedron. And here we show, this is the actual dodecahedron part. It's very sardine. Okay, getting back to that, I want to now concentrate on this, so now I want to basically overlay the icosahedra, and you've got to imagine this as a spiral, okay? So I'll just show you how they overlay, and eventually you get this. The reason I put scars in is just to emphasise these base pairings. Now a second way that this could be done is to go back to that original diagram where you've only got the overlapping pentagon. I thought we'd try overlapping pentagonal discs made a tetrahedra.

20:00 This is what we get. So imagine this actually coming up out of the screen as a spiral. And you see it just naturally packed as a spiral. You know, if you take paper, a strip of paper, and keep tying knots, that's the form you get. Yeah. And then it just curls right up. So, if you just see inside, here's the pentagonal pairings, okay, it's tricky to see there. Okay, so now what I did was try to build this one myself using this pentagonal disc I did. I made two, basically it's five per strand, and then I realized that these, you could just lock them together, they're actually 20 lans, to give this. And you can build that out of one sheet of paper, the whole thing, just by cutting a How are these labeled? Well, I just went to the commission because there's no specific label. And how are they? They're just numbers. I mean, they're actually dice. So it's just... It's actually religious that it's in the science science. Actually, if you throw one of these dice, it lands with the points upwards. So how do you know what your score is? Yes, I have. So you went around perceptive people's lights? Yeah, I thought... I thought I was mad. Now, this fit isn't exact. Do you have some kind of exact fit? Sorry? Well, I mean, it's not true that five don't get accepted people. Five don't get accepted people. They don't. They do. Oh yes, they do.

22:30 That might be good enough anywhere involved. It still allows you quite a few attempts. So what I quite like is this group that you have here, and I do wonder if we pack those with the more tetrahedra. Maybe this is the messenger RNA, possibly. Then what we get is actually the completion of the icosahedra. So now we've got two and a half icosahedra. So what's this composed of? You consider it's been composed of stacked pentagon discs, This stacked hyposahedra that are incomplete, and the stacked internal dodecahedra, which is where Mark Curtis' base panel is shown. And we've got 10 nuclear types. So it's basically a five-fold symmetry packaging that is producing this spiral. So Kirstie's black contagonal blocks are now redefined as components within larger contagonal discs and tetrahedra. These larger discs are in a spiral structure by the way they're packed together, and the double strandiness is also produced as a consequence of this packaging. So, in conclusion, all 64 triplets can be applied by a single first-order tetrahedron. Single-stranded DNA can also be defined by a tetrahedron. And double-stranded DNA can be defined by a star tetrahedron. 64 triplets and 20 amino acids today can be defined within the third-order star tetrahedron of tetrahedrons composed of 8.0 tetrahedrons and 20 tetrahedrons. And finally, the Curtis model of nutrient-type pairing, can they extend further using the contact with discs and tetrahedrons to govern the pattern to produce the Australian helix of 10 nutrient-type patterns. I'll just finish your...

25:00 the master of the governing section. If you'd like to... Can I just ask something that's difficult to formulate? My trouble over this is something like when I go to a concert and enjoy the music tremendously, but feel that somehow I've not got out of it what the composer intended. So here, I'm delighted in all the geometry as a man who used to make carpool models of polyhedra and so on. It's all love. I think the real thing that's getting in the way of my understanding is that the platonic solids are defined in a very rigidly three-dimensional space by having every place the same as every other place and every vertex the same as every other vertex. And so a terribly metrical situation. Then in your talk, you went through a whole lot of things which really seemed not to be using that metrical characteristic at all, but simply a kind of combinatorial structure which was carried by the geometry. I thought, yes, that's fine. Then the golden section came in, so there's something really ever so magical, suddenly coming into this kind of convictorial discussion, and that was the point at which I felt, I'm enjoying this, but I ought not to read it, because there's something I don't understand. Can I interject the question inside your question? Why do you say the golden section is necessarily metrical? When it is the... Oh, it's a number, yes. But it comes from a numerical recursion. For example, the Fibonacci series. But it isn't necessarily associated with spatial methods. In her construction it is. But I always think of the golden section as coming from

27:30 from one of the early possibilities of recursion, but long prior to the emergence of geometry. So that, in my view of what she's saying, whenever I see the golden section inside of geometry, I think, yes, that's because the geometry comes from something deeper or more abstract, but that's a prejudice. But it sounds like it's similar to a prejudice of yours, Then you want something non-metrical to happen. Well, yes. That's the end of my reaction inside you. That's right. There is an extension to this you might consider and that is surely the same sort of looser but the same sort of geometrical considerations would come in in the case of the development of embryos. Sorry I can't remember the technical term. It's where shape develops. Morphid, yes, exactly. You see the cells are rather like the molecules, they're identical to each other and they're trying to stack in various ways. Now, there are all sorts of ways in which they might stack, and probably they're being encouraged in various different species in different ways, but it is remarkable that we finish up with certain geometrical symmetries and so many digits and so on and so forth, and this could seem to be a grand extension of this sort of idea. In fact, you might be in connection with what Thomas just said, wonder whether you derive a great deal out of the Trotonic Polyhedra, but there are also the Archimedean one, or some people call them the Uniform Polyhedra, where every vertex is the same, but there's more than one kind of face.

30:00 So you have another 13 of these things, which could be the answer for Tony's question. Yeah, yeah, sure. So what insight into biology is this suggesting to you? That looking at it as seeing that all the patterns of the and we can fit into geometry this way. Does it suggest something about biology? It's just the numbers. It's the numbers. Biology, I think the fascination has come from the fact that so much information can be derived from the cube. The other way around. The fact that we apply nature, but possibly physics and chemistry. I built a lot of this up from the beginning these lines in the face. You can get all the geometric solids from those. And it seems to apply to all these other shapes. That's the fascination of it. But the way it stacks into spirals is significant. What drives biology. That's really what drives it. What drives it. What's it trying to do? Perhaps we should finish because it's five plus seven. Shall we thank Vanessa and Peter again? And tomorrow morning we start at 10 o'clock. It's now 10am on Saturday, 6th of August. Luke Halfman is going to talk on non-commutative physics. See if you can destroy the equipment here. That's on. Yes, you can. Care for the strap, that one. That's on. Otherwise, you destroy all of them. It's been recorded there.

32:30 Yeah. It's adjusting in position. It's nowhere near the right position. What do we need to? Do we need to push it near and over? It's a lot near after not there. Yes, it's just... Yeah, but we can also push... Push that down. And I think we're now, except for focusing, I don't think you're allowed to stand up. The focusing is that. That's the focus. Yes, I seem to have a dedicated possibility of destroying equipment. Can I just adjust the street side? Move it back a little bit. Or no. Move it off the back. Like that? More? Can this be made just a little bit wider? This snake can go off a little bit? No, I don't think you can. Well, you can move this up. You can move this up. Like that. I think that's one page. Okay. Good. So there are a couple of new things in here, and some of this is things that you've heard me talk about before, and some of it is things that you've heard me talk about before from a little bit different angles. So I thought the best thing to do was to just start from the beginning again, all right? And the beginning, and I guess I should also recall that, you know, this goes back to work that Pierre and I started a long time ago, about legend management. So this first transparency is a very concise summary of how you can fix the Leibniz rule in discrete calculus. So this is a very elementary kind of fundamental thing, that if you define a derivative in the usual way, discreetly, so I'm not taking any limit. And for purposes of abbreviation, I've defined f tilde of x equal to f of x plus h, all right? Then when you check what happens to the derivative of a product in discrete capitalists,

35:00 you find out that it isn't satisfying the Leibniz rule. What happens is this, or you can permute this formula in various ways, but I like to put it in this form. F tilde has to appear here. And, of course, if you were doing ordinary calculus, you might, in fact, use this as your, and you were teaching it, for example, you might use this as your derivation of the Leibniz rule, and then you're taking the limit as H goes to zero, and so F tilde turns into F, and you have the Leibniz rule. When I say Leibniz rule, I mean the derivative of a product to the derivative of the first times the second plus the first times the derivative of the second. And when you're doing calculus in various situations, it's nice to have the Leibniz rule. People are aware of this, of course, and sometimes in modern parlance, people say that this is the braided Leibniz rule because something passed by something else and caused it to change, like a braid going, like strands of a braid going around each other. And, in fact, I'm going to ungrade it here. And the way I'm going to ungrade it is shown on the rest of the transparency. I'm going to take this ostensibly commutative situation here, and I'm going to embed it in a non-commutative situation by adding an operator which doesn't commute with things, and does something by indignant its non-commutativity. And we're the following, that if you put f, a function of x, next to j, then that's going to be the same as if you put j on the other side, but shifted the function to f of x plus h. So another way of putting it would be that j inverse fj as a function of x is f of x plus J acts as a shift. Is H absolutely fixed? I'm sorry, H is fixed here. Yeah. So it's just one discrete step that you've chosen beforehand. You could imagine something more complicated, and that might be useful. And then,

37:30 I'm not sure I want to remark on this, I should have erased it, but Basel likes to think of a diagram that's similar to this, and that came from a conversation with Basel. Basil likes to think of going from the implicate order to the x-plicate order and back to the implicate order. And that the j could be thought of as a kind of an operator like that. So that's the reason for that diagram, but I'm not going to try to talk in that framework. So you can look at that box and think implicate or x-plicate if you want to. Anyway, I'm going to shift the derivative by just multiplying on the left by J, and then you'll notice that that does indeed unbrave the Leibniz rule and give us back an ordinary Leibniz rule. There it is right here, because Nobla applied to FG is JVF plus JF tilde. The JF tilde is FJ, and now the J is applied to the D, so you can absorb JV together and get Nobla, and here's Nobla over there, and there's the Leibniz rule. So by some little miracle, the Leibniz rule gets restored by putting in this non-communicated operator. Okay? Everything that we're talking about somehow is turning around this theme. And the next point is to ask, well, what did you actually and you get the answer here, take a look at what happened to the derivative. You have JV, and that's J times F tilde minus F over H. But JF tilde is FJ, so what you did was you made a commutator and divided it by H. So that this derivative that we just shifted an ordinary discrete derivative into a non-commutative framework and found that the derivative that we had defined commutated with some fixed element. And, of course, commutated with some fixed element indeed satisfies the Leibniz rule and can be thought of as a derivative and has the properties formally of the derivative. So, it all... No, I'm sorry. You preferred that I stand on this side.

40:00 And then I'll try to come back here. Alright, so So the pattern of this is very clear, and it suggests that it would be a good idea to take all calculus in whatever situation you're looking at and shift it over into non-commutative calculus done by taking commutators. And there are people who do things like this also, anyway, maybe not exactly what we're about to do, but for example, Alain Comte, in his so-called non-commutative geometry, has lots of derivatives, operators that are being done in this form. That's done in a more, that's done in a different context than this one. But in any case, that shows us, that gives us a hint about what we could try to do in various situations. Now, one other thing is the background to this is this interesting derivation that Dyson wrote about from a conversation that he had with Feynman, and you heard me talk about this before, and I guess at the end of this talk we'll go back through this again in a different form. But what Feynman and Dyson observed was that if you had some operators which were supposed to be like position, one another, and you assume that the position and the velocity, that is there is a time derivative defined in this situation, so you have this commutator possible, and you assume that it's not zero, say delta ij, then you can define another field, h, which means a triplet of three things. And I've written it this way. Writing it this way is probably the fruit of various conversations with people because you normally wouldn't think of writing a noncommutative vector cross product. But when we were talking about this long ago, Keith, among other people, was thinking, why don't we write it all in terms of vector cross products? And of course, we know immediately, once the idea comes up, that we can calculate and that's not going to necessarily be zero because things are not commuted. You can write this out in terms of some commutators and epsilons. Anyway, that turns out to be the magnetic field, and you could define, in the Feynman-Dyson situation,

42:30 you would define x double dot to be equal to e plus x dot cross h, so that the force law is true, and that defined e, and then Feynman-Dyson proved that Maxwell's equations are satisfying. So Maxwell's equations somehow follow from this curious sort of mixture. So long before you get to Maxwell's equations, the time derivative gives you the possibility of electromagnetism. The time derivative gives you the possibility of saying anything about something that would be like Maxwell, right? Is that what you're saying? Yeah. Yeah. Yeah. If I didn't have a time derivative, I would have... More primitive, before you get to that. I mean, it gives us the possibility of going to max. Absolutely. Right. So, you might wonder, well, what's the role of the commutators here? But, of course, the derivatives in their derivation are the kind of derivatives that I was just talking about here, namely derivatives defined by commutators. D by the xj is going to be 1 over kappa times the commutator with xj dot. That makes sense because if I could, I was eliminating constants, I should have eliminated Kappa as well. Xi and Xj are delta ij-ing one another, right? So that says that if I interpreted d by the Xj as commutated with Xj dot, then the derivative of Xi with respect to Xj is delta ij, as it should be. So then the formalism of derivatives will work out right. but in the Feynman-Dyson derivation the derivative with respect to time wasn't represented by a commutator it was just given to you and what we thought was oh we could probably understand this better if we made it all discrete and then in trying to make it discrete we found our way into this and realized that we could uniformly make it commutators and think of it as discrete so that's the background for this which I'll come back to You know, there are some general remarks that are interesting to make. Here's one. Suppose you were working in nonstandard analysis or surreal number analysis, like John Conway's method of constructing nonstandard real numbers. Then you could define a d, which looks just like a discrete derivative, where epsilon is infinitesimal, right? And what you usually do to do calculus is you say,

45:00 real part of that. That is, you throw away any infinitesimal part that lives here, right? And then it will satisfy the Leibniz rule. That's the usual solution to getting the Leibniz rule if you're doing infinitesimal analysis. You throw away the infinitesimal part. That's the equivalent of taking the limit. On the other hand, if you just defined it this way, then you could run through the formalism just like we did. You would have b of fg is f tilde, and f tilde isn't f, and the Leibniz rule is not satisfied, but it only differs from f infinitesimally. And then you could correct it by going to a non-commutative domain, and you would be taking a commutator with something that was related to an operator that caused an infinitesimal shift. So you could do the same thing. And so there's some kind of ground of thought here between discrete calculus, which is really very much like infinitesimal real calculus and non-community calculus. So I think that's worth thinking about. I'm just mentioning it. So let's look at the simplest example of this kind of discrete analysis. So here's one variable, and I'm assuming that the time step is indicated by x prime, so I just have a time series, x prime, x double prime, and so on. And I'm going to assume that the x's are perhaps just ordinary scalars commuting with one another, so there isn't any extra non-commutativity in this situation, which is just all the non-commutities. Commutativity is coming from J. And the X dot is our adjusted derivative, so it's the commutator of X with J over ch. And XJ is JX prime. That's the next time. And then in this situation, you can think about what happens when you multiply X by X dot or X dot by X. If you multiply X by X dot, you see, you use the shifted derivative, but then if you bring the j all the way over to the left, you get x prime. And on the other hand, if you do x dot times x, you get x prime minus x times x over time. And x is not x prime. So this is actually a picture in the algebra of something that you could have thought about without any algebra.

47:30 Namely, if you are in a discrete world and you want to measure the positions and the velocities of some particles that are moving around in your discrete world, then you measure position, you could assume that it just doesn't take you any time, all right, just for simplification. But when you measure the velocity, you have to use two times, present time and next time. So if you measure position and then you measure velocity, the position you measure will be x and the velocity will be that formula over there. On the other hand, if you measure velocity and then you measure position, the position will be x prime because it's the position at the next time. And so position and velocity don't commute with one another. And so from the point of view of the discrete, position and velocity don't commute with one another. And you can ponder that in relation to the fact that position and velocity don't commute with quantum mechanics. I'm not saying that's quantum mechanics, but it's worth pondering that that position velocity commutator can be interpreted as just something coming from the discrete. It's well on the way. Well on the way, in some sense. So this algebra captures that exactly, right? So we can think then that we can interpret this algebra as telling us about that position velocity commutator in the discrete. And what did we find when we did this little calculation? we found that the commutator of x with x dot is equal to j over tau times x prime minus x squared right, take the difference between these two and you get another x prime minus x and you factor it out and you get that so x prime minus x is the step you took so this says that the commutator of x with x prime is j multiplied by delta squared over tau, where delta is the size of the step that you took. And so if you wanted the commentator to be constant, which is a natural equation that people like to think about, I think there's a formula on that equation. I'm setting it equal to j times a constant so that I can get back to the real values. Then I find that the constant is delta squared over tau. Now, that kind of might wake you up when you see it, because delta squared over tau is what's called the diffusion constant, if you were thinking about a random walk. And there's a reason why the diffusion constant comes up in the usual situation, which I'll remind you about, but you see why it came up here, it just is what has to be constant.

50:00 And that means that if you have, I said Brownian, but of course it isn't necessarily Brownian. It's just any walk which has a constant step. And it could be always stepping to the rut, or it could be stepping back and forth according to some perfectly nice function, not necessarily random. Or it could be plus or minus, and plus or minus is random. Any constant step walk that satisfies delta squared over tau is constant will fit into the family of walks that satisfy this equation. So this equation has as its solutions a huge family of things that are all fitting together in that way. Have you subtly brought us into an area where there's a multiplicity of particles or not? Now, I feel that suddenly we've made a step into thinking with Browning motion. Thinking of something additional to what you described before. Well, you can think of that X as the coordinate of A, of something. I'm thinking of the X as the coordinate of something, you know. That's something I just added to make a physical interpretation of the math, right? That it could be the coordinate of a something which has different placements. And then Brownian means that the plus or minus isn't unknown to you. It steps back and forth. I think this question depends on my second, but really I'm thinking of one positive part. I'm only thinking of one part. Thank you. Oh, sorry. Yeah, I could think of the plus. But the interesting thing is that, of course, Heisenberg's equation, you can't solve it in matrices, right? So here you're solving the Heisenberg's equation, but you're not solving it as linear operators anymore. So it's definitely a discrete thing that you're looking for. It's essentially discrete. You cannot take it down to the limit. As you say, you get into mathematical difficulties. Right. But on the other hand, I could think of letting tau go to zero and delta go to zero in such a way constant, and all the while we'll be able to solve this. Yeah. Luke, what about the possibility of thinking of a system in which the deltas are, say, Gaussian distributed or something, and this may well...

52:30 Where the deltas are what? Gaussian distributed, so you've actually got, you're taking averages then through some sort of process like that, which is halfway between being continuous and discrete then, isn't it, sort of thing? As long as those integrals behave well, wouldn't you have a result in that sort of thing? about different values of delta. Yeah, but then looking at the process in terms of its averages. That would be a deeper random walk. But it would still be a random walk because the probability of zero delta would be very, very small. Yes, quite. And that might make the integrals all right then, in that case. Yeah. But they will be anyway. There are certainly variations that you can do on this. And they will lead you to more complicated commutator equations, or maybe ones that are awkward to write. You can try different things. But just for the record, let's remind ourselves where the diffusion constant comes from usually. Usually, we're thinking of something like this. The particle could step to the left or step to the right with equal probability, let's say. That's what I'm going to do. And P of x at time t is the probability for the particle to appear at that point, x to t. So you have this equation. This is the fundamental equation in that situation, right? It just says that you have one-half probability of going to the left and one-half probability of going to the right in terms of the previous probabilities arriving there. And then if I take the difference between P of X of t plus tau and P of X of t and divide by tau and put the same thing on the other side, I see that I get the expression for the second derivative going on the other side. And I get a delta square root of 2 tau. So this suggests the differential equation dp dt is delta squared over 2 tau, second derivative of phi with respect to x squared. And then you say, well, therefore, I'd better let delta squared over tau be constant so I could try to solve the equation. And then you have to take the limit of paths like that that satisfy this, and that's the confusion equation. So that's the usual reason. Now, I don't think it's quite obvious. what the relationship is between having arrived at the delta square over 2 tau by this argument and having arrived at the delta square over 2 tau by the argument on the previous slide.

55:00 So there's something more to think about here about what's going on. Luke, can I just point out one wonderful thing about this, if you imagine two time steps, I suppose when you put in a rule that says, okay, two time steps, you switch the sign, so you have, in a sense, the coloring of the path itself, which has this rule of two time steps, you switch the sign. You're adding that in as an extra rule that two time steps will switch the sign? So this is like I squared equals minus one. Absolutely. So instead of the real constant, you get the fusion constant. The fusion constant then looks like a dimension. Yeah, go ahead. So there's the transition from the fusion to the shorting equation. Yeah, that's right, of course. If you put an I in here, and think of this as h bar squared over 2m, then you're looking at shorting the situation. And you can, it's not hard to just write a process like this and put an I up in there, but it doesn't, it isn't true what it means. But you're suggesting that if we put some restriction on the process that made the signs change in a certain way, that would be maybe a better way to think about what the I meant. Yeah. And then you'd have to ask, well, why does it do that? Right. But at least you eliminated the eye into sign changing, which is what the eye does every two times. Right, right, right. And a little later, you may well see one way you can put that in. Yeah. The reason why you should do it is because it's particle vacuum. It's only on the vacuum at the same time. That's why you have to do it. A very small statistical remark. If you imagine a symmetrical random walk where the step is stochastic, it is random with, say, a zero mean, then the variance of how far you get after time t, or sorry, the standard deviation of how far you get after time t, is proportional to the square root of t, which is really what you've got there is the ghost of that particular thing.

57:30 And that's certainly a normal kind of interpretation of what this means. The question is, if you're looking at it from this other point of view, we found out that structurally it's what you needed in order to make the Heisenberg's equation be a container for those solutions. So, it's a little bit different. Here's another example. Suppose that I define J with now a support clock to be equal to 1 plus H over I H bar delta T. Where H is somebody else that's non-commutative and delta T and these guys are commutative and are commuting stillers and that's just one. Then when you take the commutator with h, with j, it's the same as taking the commutator with h over i h bar. And so, the time evolution equation is a Heisenberg-Former-Schrodinger's equation. So, that's another example of if you put in the i, then that's the way things will work. And you'll notice that because we're working discreetly here, I didn't actually have to use the exponential series. I'm only using this much. That's all I need in order to arrive at H, which is amusing. Normally, you could write the same thing, but then you would have some extra terms that you would decide to throw away. But here they don't get thrown away, they're just there. So that's that. Now, we want to talk about a larger context for this. So I'm going to take a bunch of variables, x1 through xn, and assume they all commute with one another. So I'm trying to write down calculus. The thing to imagine here

1:00:00 is that somebody noticed ahistorically that you could do calculus by using commutators, they decided to do calculus, and what would they find, right? So you would have a bunch of variables, and this person decides that they want all their variables to commute with one another because they're trying to just represent ordinary calculus. And dI will be d by dXI, so I need a PI which will delta IJ my XI's in order to represent derivative with respect to XI, so I'll postulate And then I'll have bi of xj, it's delta ij. I also want, because I'm trying to do ordinary calculus, that the derivatives i with respect to xi and xj will commute with one another. So what will be sufficient for that is that the pi's commute with one another. It's not necessary, but sufficient. Well, all that's necessary is that the commutator of pi and pj should commute with everybody. That's all you can really do. So we'll assume this, and of course it looks familiar. Just a remark about what I was saying about orders of derivatives. If you differentiate with respect to J and then with respect to I, you see what happens, right? Differentiating first with respect to J, and then with respect to I. And then there is the Jacobi identity for commutators, which is useful all the time in this kind of thing. Namely, if you have A and B and C, you can write cyclically the repositioning A and B and C and add them up, and it will be equal to zero. So that's what I've done here. I've cycled I back to the beginning and F and PJ over, and then I cycled PJ back to the beginning and I and F over, and minus minus because the sum of this and this and this is zero. And you see that this one is dj, dI, and f, so that's taking the derivatives in the opposite order. And here is the commutator of phi and pj, so you get the formula that the commutator of the derivatives of phi to f is the commutator of phi and dj, commutated with that. and so if you think that well we could take as our most abstract notion of curvature that derivatives don't necessarily commute so that's and measure that and that so in that sense this is courage and so the curvature should be

1:02:30 in the center of the algebra if you want if you want these derivatives to commute and we're going to assume this so I'm repeating myself here there's there are calculus equations and I also want a time derivative so again with notational malice of forethought I choose H to be the representative of time derivative and then you see we can do a little calculation you can find the derivative of H itself with respect to time which is the commentator of H with itself which is zero you can find the So that says that h is not a function of time. And if you like the word function, of course, things are not actually functions. You can take the derivative of h with respect to xi, which is its commutator of pi, which is minus pi dot, because the commutator is in the opposite order. And you can take the hdpi, which is the commutator of xi, which is xi dot. And those are Hamilton's equations. Right? The thing is, I still think that's kind of astonishing, because you have to remember what we were doing, right? I mean, it's not astonishing, because we all know Hamilton's equations, but it's kind of astonishing that you could say, well, I'd like to do calculus, I'd like to do discrete calculus, maybe. I see that I could represent all my derivatives as commutators, I'll just write out what that would look like, and then I find that space and time are constrained by Hamilton's equations in this form. So the constraint of Hamilton's equations follows from the constraint of writing calculus non-commutatively, and it isn't necessarily something that came from physics. But of course it does come from physics. So how do you handle that? I still think there's something to think about there before we go on to anything more complicated. So, for the sake of our thinking about it, let's remind ourselves what Hamilton's equations usually come from. You might have standard Newtonian mechanics, where the Hamiltonian is, the energy is the square of the momentum divided by 2m plus some potential.

1:05:00 And now q is the x's and p, of course, is the momentum. So then P is MQ dot, and P dot is MQ double dot, and Hewton's law says that that's minus dvdq. So this is a little summary of standard Newtonian mechanics. And then you differentiate and you find that dhdq is minus P dot, and dhdp is minus P dot, and Q dot. There's Hamilton's equations. So Hamilton's equations are a consequence of, usually, a consequence of Newtonian mechanics. So is Newtonian mechanics a consequence of just mathematics? Curious, right? And then the other thing that Hamilton observed was that if he takes F dot, namely F dot in the form, the fdq q dot plus the fdp p dot the usual formula for f dot and substitutes in for q dot and p dot the derivatives of h then you get a an interesting gadget for doing differential confidence namely you get the fdq the hdp minus the fdp the hdq interchange in the order of q and p and that behaves rather like a commutator and it's called the poisson bracket so you can write the time as the Poisson bracket of f of h. If we eat five minutes into coffee, you've got half an hour. So, among other things, the Poisson bracket, as I've defined it just in calculus, you have two special variables, and you can do that. That satisfies the Jacobi identity, so this is really very close to being commutators, and of course the transition of taking taking croissant brackets and replacing them by commutators which started with Dirac for quantum mechanics is very well known. So this is all about this interface between that and somehow getting ordinary coordinates is very closely related to shifting from commutators back to croissant brackets by averaging or whatever means you do to get back to ordinary coordinates. So So it's mysterious, and I still think there's more to say here.

1:07:30 Now, what about dynamics in a noncommutative world like this? What's it going to look like? Well, I have to think about xi dot, right? Now, xi dot, of course, is equal to the commutator of xi with h, and we'll look at that. But on the other hand, it's very useful, and this is an idea that physicists have used for a long time, to look at the relationship between that and the momentum operators. I mean, in quantum mechanics, people do that. They say, well, let's suppose that you could write Xi dot as a shift away from just Pi, right? And if you think about that, it's just a definition of the AI from our point of view. From a historical point of view, it meant adding a gauge field or putting in the beginnings of the electromagnetic potential into the situation, things like that. But from this point of view, it's just a matter of referring ordinates back to the PIs. So, for example, if you were to take the derivative with respect to, I'm sorry, if you were to take the commutator with respect to Xi dot, that would be the commutator with respect to Pi, which is the derivative with respect to Xi. Minus another commutator, right? So you would be writing a new derivative, which is a shift from the standard derivative by something. So you're writing a covariant derivative if you take the commutator with X on top. So what comes out of that just formally? Again, the imaginary mathematician who is supposed to be writing down calculus is imagined to have invented this on priority, So I define a covariant derivative, delta I, nominal I of F, to be the commutator of F with AI. Remember, you could jump from here to here. You could say, I want dynamics. Dynamics mean, how do things change in time, and there should be elements in this noncommutative world's algebra, which can include temporal things as well as spatial things, that describe this. So on the one hand, this is described by a certain commutator, and on the some world element in the algebra. So anyway, taking this commutator, I get Di of F minus the commutator of F of Ai. And the question is, what's the curvature of this operator,

1:10:00 which will no longer necessarily commute with itself? Delta I and Delta J will commute. So I can look at R of knobli and Alpha J, which is just the commutator of what represents it, which is AI. So this is AI commutator AJ. And when you write that out, of course, you see the PIs create derivatives, and you get DIAJ minus DJAI plus the commutator of AIAJ. And again, hopefully, you get a slight shock on seeing that, because that's the well-known formula for the curvature gauge field. And if AIs commute with one another, it's a well-known formula for what happens to the electromagnetic potential. of the electromagnetic p. So these things come out of the mathematics. Comments? Yeah, but what was your initial intention? Suppose that you knew nothing about physics, would you? Yeah, of course. If I knew nothing about physics, would I actually walk this line? That's a good question. On the other hand, some of the line you can, and of course it's not necessary to try to tell a story where it would be, I don't think it works, right? I mean, the line, when I was telling a story and trying to show how, I am telling a story and trying to show how much of what you think about as physics is actually in the mathematics. But whether you could actually invent any mathematics without having some experience in the world in terms of physics or anything else is a good question. It's just a way of discussing where these ideas fall. Can I make a suggestion in that context? I think, from my point of view, coming from the maths that is used in engineering, which is all of the commutative sort, I then found, and this will come out tomorrow night's talk, that you can't just go on down that road, Otherwise, there's a preferred direction in space as one would do with correlations in high-dimensional vectors and things in signal processing. And you say, this isn't right. There's got to be an SO-type rotatability. So you just have to introduce anticommutation.

1:12:30 And you say, well, what's the most regular way of doing that? And it just immediately gives you this thing back. So then you're coming from the sort of vanilla form of the math and saying, we just need one more precept here. And it happens to be the anti-commutation. It's coming around the other way, then, isn't it? Yeah, yeah. So you might, I guess really in a way what I'm saying and you're saying is there are multiple motivations for these structures. Observacratically. For example, I guess I'll write on this slide. I was trying out the following way of teaching calculus a while back. I was saying to them, well, we're going to have infinitesimals, but I don't want to teach you all of nonstandard analysis, so I will assume that my infinitesimals are so tiny that they square to zero, one possible approach to infinitesimals. Then you can't divide by them, but you can say f of x plus an infinitesimal is f of x plus the derivative times the infinitesimal, and everything stops there. to do the calculus there, although it's very nice. The only problem is, or how do you extend functions to these functions over these infinitesimals? Well, getting into technicalities, but then I realized, on doing that, that I then had a very nice way to explain to them why I wanted things to not commute, because it should also be the case that dx plus dy, being an infinitesimal, should have square zero. Right. And so therefore, being careful with the algebra, I get the dx dy plus dy dx is equal to zero. So the non-commutativity comes out of the requirement for throwing things away. And once you've convinced somebody to consider a thing, then And, of course, we've got areas of bodies and scopes there and everything else have differential forms. So that's another example of... The nilpotens, by the way. The nilpotens, right, yeah. So it's another example of finding a way over to something that's a little bit different. Peter. Yeah. Very interesting. I've seen this similar presentation by you before and I've been thinking about water mountains for several years.

1:15:00 me that when you're doing your discrete calculus you're doing it for space-like objects you're doing it for the vector quantities space momentum space-like things but you have x dot which is which is presumably a continuous calculus it's certainly not defined as a discrete calculus that x dot within it the so that's how you started i know i know i accept that and you see yesterday i was arguing for some sort of discreteness connected with space type things because community. Whereas time-like things, I don't believe in discrete at all in that sense. So this supports your space like that? It certainly supports that, totally supports that. In addition to that, you say you get your dynamics from mathematics, and I'm sure you must because you have effectively Hamilton's dynamics, conservation of energy and conservation of momentum, really can be related to the properties it is space and time. Through Nertus theorem, for example, you can actually relate, you can say that these conservation laws are a result of the properties of space and time, so you're effectively applying differentiation to space. Like conservation of momentum and symmetry with respect to it. Yeah, exactly, yeah. And so it's perfectly reasonable, I think, that you should be able to get, I think it's a fantastic demonstration, I think it's perfectly reasonable if one thinks back, that it should this type of mathematics which is in fact directly stating what the properties of space and time are within the mathematics but no it hasn't but to me to actually describe something as a discrete in the first place is the same as doing space that's what I understand it to be that's how I understand it whereas I don't understand that yet I think of discrete as could be time it would work. It would work in that way. Okay, we must get on. Let's continue on. you could ignore, well, you don't have to ignore it. The computator at the top is what you would get if you were doing the computation that I just was talking about, where xj is pj minus ai. Then you get the delta ij, and you get an extra term, right? So, you can think of getting an extra term

1:17:30 And then instead of getting electromagnetism in Feynman-Dyssen, for example, you're going to get something like a gauge theory because you don't have the delta IJ, right? And in general, you could think, well, what if I have a GIJ? Now, there's a reason why I call that a metric, and you'll see why in a moment. But it isn't obvious at the moment that you should call that a metric. One of the things that is a motivation for calling that a metric is if you do a lot of calculations with this, and you find out what happens at the acceleration level, which was the Feynman-Dyson idea, then you find that, more or less in general, with some assumptions, you get a scalar, and you get something that looks like a gauge field, and you get the Levi-Civita connection for the Gij formula. The Levi-Civita connection is the usual formula for geodesic motion, and it comes from a Gij, from a metric by this formula but what it is from the point of view of a geometer is the following, you want to have some notion of parallel translation to do geometry, so now I'm just talking about ordinary geometry for example, I wish to parallel translate this vector here on a sphere and so I do this, I'm moving it the north pole. And then I move it down over here to the west, depending on which way you're doing it. And then I come back to where I started, and you see that it's rotated by 90 degrees. That's a reflection of the curvature of the space along which the parallel translation was happening. And so if you write down a notion of parallel translation, you can do in terms of some matrix form like that. And then the fact that things are changing when you go around a loop or that the derivatives don't commute with one another is measured by that parallel translation. The formula here is usually derived by mathematicians. Question? No, I'm just doing it. Yeah, oh, you're doing it. I'm just doing it. This formula is derived by mathematicians by asking that the angles between two vectors are being preserved while the parallel translation happens, because a metric locally tells you how to compute some angles between vectors.

1:20:00 So that's the usual way in which this interesting formula gets derived. Here it comes out automatically by just thinking about the quasi-physics of this situation. And I want to show you how it comes out again, by just taking a piece of things. So I'm assuming that I have the xi, xj dot is gij, and that these commute, and that the gijs are commuting with things so that life is not too complicated. And then in the whole derivation that I used, I used this principle. I think in the interest of time I won't, I'll just talk about it rather than worry about the formula. The question is, when do you say things are equal? If A commutes with all the XIs and B commutes with all the XIs, then I could assume that A and B, F and G here, commute with one another. I could assume it. are, in fact, actually functions of those things, then you can justify that. And that makes a very strong principle for doing calculations with this sort of thing. I need that in order to do some of this, but not what's below. What's below is just what it is, blocks it. We're assuming that this is a gij, and now we're going to do two derivatives of the dual kind, that is, with respect to the xi's, not the xj dots. I'm taking commutators with respect right? Two of them. This guy here can be re-expressed in terms of gij because you see there's two derivatives here. And if I took the derivative of this, I get a dot here or I get a double dot there and I get the derivative of this. So I get two terms on the left. And what I've done is I've taken the two of the three terms in the derivative of this and made a replacement, right? So that's what this is, just a replacement. And then expanded it, and I have a triple commutator here, which, oh, I didn't do everything out. I have a triple commutator, and I rewrite by the Jacobi identity. And then you look at the terms that you got. This derivative

1:22:30 on the GJK can be shifted to the XI because GJK commutes with the XI, so it shifts with the sine because you just differentiate and shift it. And then you reinterpret these turns that you got as derivatives, you see. This one is DIGJK once it's been shifted. This one is a DJ of a GIK, where DJ is now in this form the derivative with respect to XJ dot. it's a co-variant. And there's the form of the leditivita connection. So that's the simplest way to see it happen in front of your eyes. So the leditivita connection comes out of mathematics of this non-commutativity. It also can be interpreted as geometry. So in a certain sense, it's, I'll finish on sentence, in a certain sense it's really an expression of the jacoby identity in this context but it's also geometry so now we're in in this kind of knob where things could go in different ways right yeah sorry i just i'm not sure if i'm making a trivial question but you know there's a matter of a factor of a half in the usual expression oh yeah and i wonder yeah yeah you're right there should be a factor of one oh it's just a mistake I don't know. I mean, how did the heart come in? Or where should it be? I got twice the leverage to do the connection. Oh, it probably comes from the fact that I should have averaged a couple of things left and right when I was doing the communicative analysis. Don't worry about that. Okay, so now I want to show you something that's a little newer. Here's a possible H. Maybe this is my same problem about 4 and 2, but I put a 4 in here to make things familiar. You don't have to worry about the numerical factor. I'm taking This is Tony's H, and this has to do with some work that he's doing with Clive. Maybe this will turn into some further collaboration, I hope.

1:25:00 But look at this H, all right? Now, I put a G-I-J in here, and I have P-I-P-J, our old friends P-I and P-J, right? And here I'm following the kind of convention that Tony uses to make sure that everything is working by averaging all the different ways that you could write something non-commutatively. non-commutatively. I'm assuming that the Gij's commute with the X's, and I'm assuming that they're symmetric. And I calculate XK dot, and I won't worry you with the calculation, but this is an easy calculation. You see, you just put it in, and then you use the fact that the Leibniz rule is true to separate some things out and you'll find that this is what you've got for xk dot, alright? That it's gkj, pj, and the other way around. Alright, that's fine. But then the next point puts it into our context directly. I want to look at xi, xj dot. And you do a little calculation find out that it's G.I.J. So this fits. So this tells us that from a certain point of view this is an example of what I was saying, right? If you take the Hamiltonian to be given by that formula in terms of the G.I.J's then you'll get this. And now what's interesting and new is that Tony analyzed that Hamiltonian putting on conditions that should follow on those equations if you assumed that you had replaced croissant brackets by commutators and follow out the results on the commutators that happened from the croissant brackets, which is the kind of thing he's been doing for a while in other contexts. And he finds that with that Hamiltonian, you're going to need some restrictions on the derivatives of the Gijs. Now, we know the G.I.J.'s correspond by the previous thing and this remark to a Levy-Tubita connection, so that's a fast way to see that you really should think of the G.I.J.'s in the Hamiltonian as a mentor. But then he implied that that differential condition on the G.I.J.'s is equivalent to a curvature condition like this,

1:27:30 where that curvature is the curvature tensor for the, curvature tensors for the W2D connection that we were just talking about. So this is the Ritchie tensor. And then this turns out, remarkably, to be a consequent, a fourth order version, I shouldn't say version, it's a fourth order curvature equation, something like Einstein's equations, which looks like this. And if you started with Einstein's equations and plug them into this, that it's still true, but this is some, therefore, generalization of Einstein gravity, if you like, in the vacuum that is being asked to be true by the non-commutative world. And so maybe there's really some new general relativity physics that comes out of this way of thinking. This is very new, so it's hard to tell. I would make one comment about the mechanics of calculating with the ones that are commutators with the P's or the X's, and that is you regard them as partial derivatives, and we all know how to do partial derivatives, as long as you keep the order of possibly non-commuting operators as you do your calculation. get the results which Louis has worked through step-by-step you get themselves in one one move as long as you believe this particular magic you see it's the way I do things because I'm lazy but and it's the way that I produced this differential thing involved connecting the elements of the the GRJs and that is supposed to be because of the way I've derived it it's supposed to be true only at the pole of geodesic coordinates so it isn't the tensor equation but it's true at one point and Clive very brilliantly and expertly said it must be that tensor equation which is true and then as you can see if RAB equals 0

1:30:00 which is Einstein for empty space it actually satisfies that so that is a of Einstein's law for empty space. Thanks. One other didactic comment, down at the bottom of the slide, just for the sake of connecting up the ideas, if you have two covariant derivatives and you take your commutator, then the coefficient that you get if you rewrite on a vector what happens to it is the curvature. That's the usual formula. It's slightly different from my other formulas. But that's what's meant here. And then when you contract this index with the middle index, which looks like an odd-intuitive thing to do, that's called the Fitchy-Tex. Well, we have over four or five minutes. Four or five minutes, okay. I want to make one further comment about the Feynman-Dyson. I won't go through it. Actually, there it is. there's the question, and then we could maybe have some questions. So, the Feynman-Dyson generalization that I, I think I talked to you about this last year actually, but I was interested in the life of thinking about it. So now I have three variables, and I'm going to define EI of F to be equal to F commutator Xi dot, and F dot to be FH for some H, and that's it. I'm not assuming any commutator equations. It's generalized. And yet you're going to get back something very similar if you use this formalism to what Feynman and Dyson got. And the key thing that makes it work is this, that f dot is by, that d by dt is defined by this equation. Everything else is going to follow from that. What do you things are not functions of time this kind of goes back to your remark about well it's time continuous or discrete or what in this situation there is no function of time I all I have is I have the time difference that corresponds to that commentator that's that's one form in which time comes in and I would like

1:32:30 to have the continuous derivative with respect to time or something like and so imported by using the usual formula. And then everything works. It's kind of remarkable that everything works. And I guess it's not possible to go into any detail on that. So I wanted to show you something. Let's see. Yeah, okay. Just to remind you what happens. what happens. It turns out that if you if you take that, which is a very slim looking starting point, there it is, there's a starting point, then if you define h to be x dot cross x dot and you define e to be dt of x dot, so that's different from Feynman-Dyson where it wasn't defined in such a direct way, then the Lorentz fourth slot follows the divergence of h is zero, and you get these two Maxwell-type impressions. So that's what happens. And you might wonder, I'm sorry, I probably shouldn't do this. Yeah. I think I'll stop, because Because anything else I'll show you involves too many formulas and we're two minutes away from it, so it's better to take questions. We've got two minutes for questions.