Noncommutativity at the foundations of discrete physics — Part 3

0:00 If you form the volume vector m1, c1, mn, cn, that's that sub-matrix multiplied by the permutation, and that rewrites the matrix into an. And it shows you also that an in this sense is bigger than the matrices. And then you could ask yourself, well, what are the matrix representations of an that they actually look like? Since they're linear transmissions, you can't look at them as matrices. That's this picture. It's a nice picture, and I think I don't need to say any more about this slide. I want to just go on and quickly give you an example. Well, here's an example, but it's not such an interesting one. It's just a small example. If I had to make this 1, 1, 0, 1, with Jordan vectors, then it's 1 plus p sigma, where p is 1, 0, times 6. And p has the property that it leaves e1 in there. And then you see all the properties of that matrix, playing with the algebra, that sort of thing. So here's a more interesting one. Let's take a look. So SU2 is the group of matrices with complex entries of the point ZW minus W by C bar. So if you turn in the ZZ bar plus W by C bar, and that's SU2. And if I put in, now I'm going to start by working on this as 2 by 2 matrices with complex number entries, and then I'll shift to 4 by 4. It's a good example. If you write it out with A plus IB and C plus IB and C and W, then you see that this is A times the identity matrix plus B times the matrix I minus I diagonal, which is I times epsilon, plus C times the matrix 1 minus 1, which is epsilon sigma, and D times the anti-diagonal matrix II, which is I times C. So you have these three entities, which I'll call i, j, and k. i epsilon, epsilon sigma, i sigma. And indeed they are i, j, and k. They're termed as i squared, k squared, k squared, i, j, k equals minus one. For example, I verify here that i, j, k is minus one, i epsilon, epsilon sigma, i sigma. The i commutes and comes out as i squared, epsilon squared is one, and sigma squared is one.

2:30 So there's the quaternions. And on the other hand, if you wanted to get rid of the complex numbers and see the quaternions there, you could put in for the i and the minus i what they are as 2 by 2 matrices and go to 4 by 4 matrices, and then we're writing over the symmetric root on the 4 letters. So i, for example, is i minus i, 1 minus 1 minus 1, 1. And now the vector, of course, this is indeed just an overlay on permutation methods. So this is 1, minus 1, minus 1, 1 times this permutation that you see which switches the first two coordinates to the second two. So, that's all. And what about J and K? See, it's a nice exercise and you might enjoy doing this sort of thing in some favor and you can tell. Great. So here's I. I turns out to be, as you see, this set of signs multiplied by that communication, J is this sign, this sign is that communication case. That's another good question. And that's the quaternion view this way or this way or that way. To see it a little more graphically, this picture is the same thing. I'm using... A dark circle to mean minus one and a light circle to mean one, so that if you were to multiply a dark circle times a dark circle, you get a light circle, and a light circle times a dark circle, you get a dark circle, and then here's the permutation on the equal, so this is vector times permutation, and the first permutation was interchange one and two and three and four, the next permutation is this, and I won't go back and show you this, this is what it says, this is first to third, second to fourth. And the last permutations here, and so those are the peternians finally. And we, you see, so we went from the matrix algebra via this translation into thinking about sign permutations. And then from sign permutations to their diagrams, and then the whole structure of the peternians is given by multiplying these permutations. For example, I times j is the limitation of the other one. Then you look and see, well, let's say we get just 1, this one would be minus 1, and minus 1 is the limitation itself. It's exactly the same, it's the limitation. So you can go, you can use this method to go underneath the structure of something that's algebraically familiar, but with a different kind of choice, and so on.

5:00 That's it. Thank you very much. I have to talk to you more about this philosophy. I mean, we got rid of the complex numbers and we kept the quaternions. If you don't mind the quaternions, maybe there's something about that because we have, well, quaternions are interpretable in three-dimensional, four-dimensional space and they're much more geometric. The complex numbers are sort of in this transition phase between, not really geometry, sort of planar geometry. I don't know why. That's all. Just for the record, I think that was almost exactly Hamilton's own attitude towards the relation between the quaternions. That's the position that, no, the quaternions were more intrinsically geometrical. Didn't like the quaternions. Yes, sure, sure. But he also had rather an aversion to the complex numbers, I understand. I times AB is minus B times A. Anyway, now I'm in another digression because I started thinking about what Bob was saying and I realized that some of the doctrine that I was curious to explain was related to discrimination systems if you think about it. Suppose that you decided that A times A would be equal to A. Now, X plus X in our Boolean situation is XOR. The star is XOR. A times A is equal to zero, right? Or unmarked. Here, I'm saying that an entity combined with itself is just itself, which is also suggestible. So if you started with that and kept the other rule, the other rule is the one that we were using when we were talking about So now I want to combine these two as axioms and ask myself whether I can get any of these

7:30 Well, sure, there's one that we all know about now, which I call triplicity. I have three entities. Each one distinguishes itself by being itself, A times A is A. But A applied to B is C, and C applied to A is B, and it's commuted very similar. So that's an example. Are there other examples? From the point of view of thinking about discrimination, there's a couple of examples. But one thing you'll notice immediately is that intricity is not associative. Right? A times A times B is A times B. C is A times A times B is B. So you won't have to place non-associatives if you wanted to do them this way. Even if it was commutative. I'm not assuming necessarily that it's commutative. But then we move right from the left to the right, and you're doing A times C. What? In the second line. A times A is A. Right, A times A times C, and you're doing A, almost like from your second tone. Do you think you have a contradiction there? No, no, no. I was saying that you can look at it and see if there are any contradictions. I didn't mean doing this, did I? No, no, no. I'm sorry. Of course, in this system, I did some connectivity. I wasn't keeping track. My rule here is that this is the simplest possible example of a triadic relation, right? A and B combine to C, and B and A combine to C. It is the same. So here are the axes, based on days, and based on people, and I'll call this an item of discrimination. And the question is, do we have examples of this? The answer is yes. I'll have to note them down. Not theory. I'll show you why. I'll try writing an operation with 0.8 times b is some multiple of a plus some multiple of b. And I'm thinking of a and b as things that I can add. Like, for example, if the integer is not 12, I'm multiplying by some numbers. Okay? Let's see if we have these.

10:00 If you make this film, you just find out what you need. A times A is equal to A, so that says that A times A is R A plus S A. R plus S times A. So R plus S must be equal to 1. On the other hand, A is equal to A times B times B. So that says R squared A plus R plus 1 times S times B is A. So R squared plus or minus 1, R plus S should be 0. If r was equal to 1, then s would be equal to 0 in the first equation, and that's not very interesting. On the other hand, if r is equal to minus 1, then s would be equal to 2, and that makes r equal to 2 b times 7. I was just trying. I have a possibility that this might be minus you. That's not a minus. You can't tell me. You have to put some minus in it. I know. I know. I wasn't reading. I reread my slide. Okay, so you arrive at the rule of 2b minus a. It certainly works. a times a is 2a minus a. You can check that out. For example, in z3, a times b, if you use this rule, right, 2 times 2 is 4 minus 1 is 3 minus 0. So if you combine any two of them by the square root of the third, this is what you get. So there are lots of these. You just take the integers from mathematical math and do this.

12:30 Now there's another fact that's amusing. If you use the rule a times b is 2b minus a, then the operation distributes over itself. a times b times c is a times c times b times c. I have a proof here by calculating and you can check the two things on the right are equal to 4c minus 2c. So it distributes over itself. And now I can reveal where this comes from. A label not diagrammed with elements in this algebra is discrimination. The operation of discrimination is represented diagrammatically by A comes up to B, and when it goes underneath B, it turns into A discriminating B. So you see, A times A equals A means you can undo a little loop, and A times B times B equals A means that when you come under B and then you come under B one more time, you can pull it apart, and so that's another topological loop. And finally... Substitutivity corresponds to taking the middle line and sliding it underneath the line and across the middle line and up and back. When you go from the top, you get an A times B and an A times B times C. When you go from the bottom, C comes down and you get an A times C and then B is operating on that C. And so this is telling you that when you write alphabets at this time, you are writing alphabets that are What he's saying when you move the knot diagrams around by topological moves. So these can give you variants of knots and things. And for example, uh, perplicity is a good way to label the topological knot. And if you think about it, it shows that the topological knot is not a move. So topological multiplicity basically. If you go to other knots, like the figure of A, the figure of A knot, then you find you need other modules, so for example here I started with 0, 1, 2 times 1 minus 2 is 2, 2 times 2 is 4 minus 2 is 4, 2 times 4 is 8, minus 2 is 6. I came around to the beginning, but I find that 1 should be equal to 6, so I have to be in the integers 9, 5, and 5. There are other systems which are not necessary, which are not just the integers module and the sum n. Which come from knots and links. So any knot or link gives rise to one of these algebras, and so there are lots and lots of them, and you might think of the algebras of these knots and links as sources of discriminations. Where does this go in relation to quantum metaphysics? I'm not quite sure, but I wanted to bring it out in this form, because when you start with that idea of discrimination, where A times A, or N times N, itself is different.

15:00 This system of labeling the non-diagrams by an algebra where there's an algebraic operation of the value, and you go through the line of that, and sometimes they're A and B, is due to Gatlin rate of John Connolly around 1960, but never written up by Gatlin rates of Sussex, John Connolly's a prince of math, rediscovered by David Joyce in his thesis in 1979 at the University of Prague. Wraith and Conwraith call the system the WRAC, W-R-A-C-K, Wraith and Conwraith, W-R-A-C-K. David Joyce called his system the quantum. I've been puzzling for a long time about why I spoke of Blanco and somehow I think I understand because he's a student of Peter Fry and those people who studied quantiles, which are called linear logic, here's the category, I'm willing to cap, not to call out David Jones to find out, that he was transposing quantiles to quantics. So that's not that this is linear logic. Anyway, he called them quantals, Q-A-N-T-L-E-S. Then, quantals, quantals, quantals, not quantals, quantals. Well, I did some work on these in the 80s, but around 1990 or so, Roger Fenn and Colin O'Rourke started thinking about this again, and they decided that they wanted to think about these algebras where you can move the A to the B, where you can have these sub-movies together, and they called those RACs, R-A-C-K, as opposed to W-R-A-C-K. Removing the W, W stands for ride, which is the twisting of a knot, so we remove the ride from the rack and got R-A-C-K, and so these are often called racks. The category, by the way, has its office down in Hallsville, Merge, and Sussex, right? I have a paper that I think is full of them. Yeah, absolutely. I can't remember them.

17:30 Yeah, that's the terminology story. So anyway, there are these two ways of thinking about discrimination. Whether or not you like the algebra systems, I was just showing you, since there are these two ways, that as an entity provided itself with advantage, or an entity provided itself with equivalence. And you might say one is for me and I didn't know that both of these things correspond to those kinds of things. This one could be Boolean XOR. This one corresponds to what I call mod sets. I'll show you in a second. The other one corresponds to the specificity of the quantum system. It also is like standard sets where you have a set of elements and you have a mod set. So there's something fundamental about thinking about both of these kinds of things. A dot set is a similar variation of the same thing. The lines will be labeled and they don't change their labels. It's just the relationship between one and the other. So if something is under something else, I'll say that it belongs to you. And then you can write down sets. For example, here's the empty set. Nothing's underneath it. And here's the set consisting of the empty set. And here's the set consisting of the empty set. But you can also read things, like here is A is the set whose member is B. B is the set whose member is A. Or omega is the set whose member is itself. So you can express some non-well-founded sets in this language. If you want to make it topological, you need to have the bosonic rule. A against A. I mean, look at it. If A and A repeat twice, they disappear. Thank you for your attention. And this is an amusing model for some non-standard sectors that we've been giving rise to in the form of sub-reference, so for example here, this red, the black line is the set A, an innocuous member of the black line is the blue line, little a, and a less innocuous member of the black line is the red line, which is the set whose member is A. So A is the set consisting of A and the set whose member is A. And this one doesn't go away topologically, whereas we have the topological groups. Self-membership can be put in and taken out as well. So let's just say, this is proving I'm a senator and not a senator. Okay. Now I'll go to what was originally talked about in the beginning of this talk.

20:00 This is, well, I've talked about this before and I'll start from the beginning again. This comes from here and my thinking about the Biden-Weiss derivation of electromagnetism and we wanted to discretize it and we wanted to think about it in terms of time series. So here's a time series written, I mean, written as x-prime and x-prime. And x could be a number or it could be an operator, whatever. I'm going to have a time-shifting operator, J, and you interpret this in the following way, that when you see J, it means a particular plot, and when you see a W, it means you do it in an a-list, like you do it in X. And then I will define W, J, and J double prime. That's the first one I'm going to do today. It's an obvious one, particularly for the point to do with this interpretation of sequences and observations. Because if you pick the plot and then insert W, you get... Thank you for your time, and I look forward to hearing from you in the future. And then, it's natural to think about the commutator of W and J. Why is there a commutator? It's W J minus J W, which is J W prime minus J W, which is J times W prime minus W. So, the commutator is a nice thing to define as an operator on W. Commutate this commutator to somebody else because that satisfies the Leibniz rule. And on the other hand, you see that this is the discrete derivative of the prime and the prime multiplied by that operator. The discrete derivative itself wasn't satisfied by an institute also, but we were very happy to make this construction because it lets us do calculus and still have a wide institute at the same time. And it's simple and fundamental in the sense of telling people about the structure of the time series. So starting with time series and that three-order calculus derivative, you're motivated to assert a very little use of the fundamental level of temporal process and to therefore investigate the consequences of non-computation. And so, for example, you could assume some coordinate time series, a three-time series.

22:30 I assume that they can use the fun numbers, but write the xi and xj dots on the dij on the plot and investigate the consequences of that. In particular, I'll try to do this quickly. If you look at the case of air science, j dot delta dot j, and this leads to our derivation of the final pricing method of reducing the formalism of electricity in languages, namely, if you let x double dot be e plus x dot plus h, and then we find that e and h satisfy Maxwell's equations and we collect Maxwell's equations for h. They're not commutative, but you get the formalism of electromagnetism coming out of almost... The formalism of quantum mechanics. I want to take a second look at this and then we'll stop. The second look has to do with the fact that you might have said, well, it's not really quantum mechanics. We've got the momenta identified with the xj dot. And they don't believe in the momenta. That's the commissary of the momenta is what gives you the electromagnetic field. So it's not really coming from quantum mechanics. So what is the formalism of quantum mechanics in this abstract sense? Well, why don't we take the formalism of quantum mechanics. And then, if you change the momentum by x, you get some potential. And this is what people actually normally do in physics at the level of both brackets and compares, as was reminded to us by Ed Oshin when I was telling him about this. And it's also in Dyson's papers, as a matter of fact, that we know from the ad that he could have looked at it this way, but I think he wouldn't have looked at it this way, even in our discrete context, because he wouldn't have looked at it this way. If you did that, then you would add this extra guy, and then the PIs and the PJs don't commute with that. But these people, the new momentum, and the old positions, they still don't quite change, of course, because the potential is assumed to be just a function of x. It commutes with itself. The q is x's q.

25:00 So if you do that, then you're right. You want to look at qi dot and qj dot and you identify, and this is where it isn't the usual thing exactly. This is a climate model. You would identify the derivative of q with this momentum, with this uniform momentum. And then if you look at qi dot, communicate with qj, you'll see what happens. The P's commute, the A's commute, and so you just get the cross terms. The cross terms are, in fact, the derivatives of A with respect to Q. So, you get the QI dot QJ dot is equal to the UAI to QJ minus the A to the QI. And so... This is the short form without going through all the axiomatics of decades of certain things, like I might assume here that E is just a function of X and H is just a function of X. And then I'm going to try to figure out what is H by using the computators on the epsilon problem. So here's the period, here's this right now. So the acceleration is B plus x dot plus H because there's epsilon in there. Epsilon i j k 2 i dot h j. And if you pick a commutator with QL, then this part gets knocked out because of the use of that, and here you get the commutator of QL with QI dot, and you can use it in each case. Now this is delta. This is a delta. And so we end up with H epsilon bit, and QL, QK dot, with a dot on the other side, and then because we're doing calculus, that's the same as QK dot QI dot. And therefore, you get from QK dot to L dot is S-L-I-L-J-L-K-J-H-J. On the other hand, it's that partial derivative with respect to A, and then if you do a little vector algorithm, then you see that that's the same thing you need to program. And this shows you what's really going on in finance derivation, which is hidden by trying to do it the other way and saying, I don't know about the A, and I'll just work it out correctly. So that, and I'm happy about that because it solves the mystery of what it means. And there's still a lot more to discuss about this and its generalizations. Yeah? Just a little note here. In the present, the thing, I mean, the other one, the paper, you've got the form, haven't you?

27:30 Thank you. In one of the tendencies, it shows the forms for H, which you are, I hope you are now, And one of the general forums contains functions of the co-organizers of physics, such as vector and Taylor, and Jules, and the vectors, and all of this comes out of something I've done in practice, something I've made exclusive of that paper that's just about particle physics theory. So if there was a little bit of this, we could play a lot of this on the page. I actually played a lot of this on the page. You can work out a lot of these things. Because you don't have to start with the copy-paste, you can start with the copy-paste. So that comes from an academic perspective. It doesn't seem like an academic perspective. So the point that doesn't follow that element is that it just follows the common theory. It throws away the superstructure, which I just put back in here to show you where it came from. I'm saying this because I wanted to point out that this is just gauge theory, and this is a nice way to understand how gauge theory works. Standard gauge theory in relation to quantum mechanics, written in copy papers, is just a mess. And it shows you what climate is doing. It's just trying to add some time to that setup to throw away where it came from. When you've got the topology, you get the equations of function and you get all the combinations evaluated for you. You know, and you don't actually need commutators to do this, you need something to check, so it's probably a handy tool, and that's why I spoke back and forth across, across non-graphics and commutators, everything I said, the commutators will work with non-graphics, as long as they satisfy the Leibniz rule, and that's an interesting point, which I'm going to go into later. The cross non-graphics satisfying the Leibniz rule is the same as assuming the templates are satisfied, so, so the first is, the first is trying to do any graph. And we have these derivatives. Now these are not, because this is not a delta IJ, this is a sort of un-normalized derivative, but they're derivatives of a sort.

30:00 And then you find that at once, in tiny computation, using the Jacobi identity to rewrite the common denominator, you find that if you take the derivative with respect to the constant variable, and so I've got twice the acceleration, you land on the levy to the form of the levy to be the connection. Let's do three lines. Now, I kept puzzling over this and puzzling over this, but it's amazing that this footprint of geometry derives out of nothing like that, and there's no time to tell the story going back through the plus one brackets and the geometry of the Hamilton's equation to explain why this had to happen. But it does have to happen, and it does happen. And if you didn't know about all that superstructure, it would be amazing. And so it's very strange that the only way I am able to conceptualize why this happened is, it's strange for me that the only way I am able to conceptualize why this happened is to go from all that superstructure across mathematics and physics. I would like to see a more direct understanding of why I'm getting geometry out of this. So I'm leaving this with a question. I'll just remind you what you're saying. I'll have to talk to you afterwards about what I understand and what I don't, because we're out of time. But I'll just remind you a little bit of that. Because you see, if xk double dot is equal to some term like this, involving two velocities, and maybe some other things, which I'm going to ignore, then when I differentiate it twice with respect to xi and xi dots, Then I get the gamma IJK. That's what I computed abstractly, not the form of the Web-Street connection. So that's showing you that formally this particle is moving in a geodesic, in this abstract space. Oh, it said to this that they don't have to find the answer. Yes, I know, I said it. I don't want to, I'm sorry, I'm not going to cut you off because I said it already and I'm out of time. I don't think I can become the B.E.C. I'm trying to block you. Because I already said what you just said. Okay? And we're pretty out of time. But the point is that this has been interpreted by mathematicians as being what leads to parallel trends, of course, and then there's geometry, and what's preserved under parallel translation is the, at angles, the unit product given by the vector. So there's an abstract differential geometry that comes here. And as Tony says, you can explain it in terms of the Hamiltonian and the Lagrangian.

32:30 And so on, in motion and across the practice. But I would like a different kind of explanation in terms of just the discrete derivatives and the geometry. I want to review the differential geometry. So, in other words, I want to do some kind of non-community geometry. Something like what people call non-community geometry. But it's not quite clear how you want to do that. You can conceptualize it in terms of ordinary physics, but it ought to be understood mathematically at some point. I'm sorry to jump in. I would just want to comment, which is I've used the Tchaikovsky idea. Probiotomy says this, if you think of A times B as a commutator, then that's another connoisseur of the population, and there's the probiotomy. I just wanted to end with my favorite way of writing the probiotomy, which is you've got two lines going from the Y to the output, and that's the problem. And then this is the probiotomy, you've got two lines coming down the fist, and you switch their order, and you take the difference. We're going to move by here. That's a rewrite of the C number. So that's the footprint of the L. Is that a new one? Yeah, here's the new one. So everything in some books is hard. At once, in the non-community, measurable contents. But to understand what came out, more or less is needed. Particularly, needed is better understanding of the relationship between the plus and minus. And the role of the discrete and finite systems. Thank you very much. I propose we split the agenda by exactly 30 minutes, so we start now, 30 minutes later in the next talk, and then 30 minutes later we should probably wind up.

35:00 Thank you for watching. Thank you for your attention. Thank you for watching.