Introduction / Alain Connes / Michael Atiyah / Q&A

0:00 And gradually, we will have asymmetric equations, with constant steps, until, finally, the state of a small temperature is the state that we are dealing with. And if it goes away, we will have no more. If it really goes away, we will have no more. We will have no more. Well, the state of a small temperature is the state that we are dealing with. Well, the state of a small temperature is the state that we are dealing with. So, we have a long list of terms, we have calculated them, we have calculated phase conditions, we have calculated phase conditions, etc. etc. and which is based on a notion that we call calculus, that is to say, these are networks, these are networks in the past or in the past, we have rational structures, and in which the main operation is the scaling of the networks. And exactly in the same way, at the physical level, we have the scaling of the operatives, the giral operations. That is to say, the multiplication of two by one of the two. That's what makes it interesting. So, there are a lot of things that are not very clear, but it suggests that the idea that we need to identify gravitation so that in a given space, theorism can recognize it as any energy is not a good idea. Because precisely, I think, I'm not sure, but I think it doesn't make any sense to talk about geometry at all. It's not that mathematics is good or bad, it's worse. It's that the notion of geometry itself doesn't make any sense, because there is the notion of wave, which was created by Eusebio Spontanei. I'm sure there are questions. I've already asked a lot of questions. The strange thing is that gravitation is much weaker than other interactions, but we sometimes interpret this as a kind of dilution, especially in theoreticals, in additional dimensions. Can you say that there is also an effect of this kind in your model? Well, what happens here is that it's a little different, it's that we don't know how to have a constant of short gravitation, we don't have a formula, okay?

2:30 We expect it to change very little, until the end of the education. But we will talk about it later. So here, the calculation we made of the concentration of the education was based on the idea that education had not changed much compared to the previous ones. That's it. If it had changed by a factor of 100, etc., it would have been fine. In the end, it would have changed by a factor of 5. But that's it. What I have to ask you is, what makes it different? What makes mathematics so much easier than other theories is the fact that the function of mathematics does not change at the level of physics. It does not change. So, to answer that, I mean, are we sure? That's what we think. In fact, we have no questions. For the variation of the constant of the variation, here there are already Reuter, etc. who have made calculations, so in fact these calculations there, they have a very interesting side, they come in contact with that, a free hand is a type of calculation, it is a type of calculation, it is a type of calculation, well there are already people who have made calculations, but it is not difficult to be sure. On the other hand, there is something they found, which is not very interesting, they found that when they calculated, they realized that the geometric dimension of space 1 becomes 2 after 1. Okay? Calculating things, work... So what this suggests is that, in fact, the dimension of space F is not 6, but it is 2. As it defines the modulo 8, and that space is a kind of empty space, there is a sense in quantitative geometry that space is a complex notion. In fact, when we look at dimensional regulation in concrete terms, we find that there is a possibility. I think we have to be extremely open. I think we have to be extremely open and say that, on the basis of our understanding of the equation, what we see in most of the cases is that it changes very, very little until identification. And between identification and mathematical science, there is a big difference. We can say that it has changed in a very short time.

5:00 So what explains the difference is that, and of course the fact that the simplifications are very good. So there is another thing that needs to be discussed, which is the problem of hierarchy. And then there is something quite satisfactory that happens in relation to the problem of hierarchy, and you all know what the problem of hierarchy is. It is the fact that there are variables that are quadratic in convergence when compared to the quadratic theorem. And that therefore there are counterterms in the square wave, as we call it, in the way we do the theory of Wilson. What is remarkable is that in spectral mathematics, there is a nerve that is conceived and that appears in an entirely natural way in mathematics. That's it. Is it not a signifier? We had a signifier when we were thinking about physics, etc. It was also a signifier when we were thinking about... So that gives us a very interesting prediction. If we take that seriously, we should be able to do... There are a lot of ways to solve these problems, but I don't believe there will be a physical revolution. I'm sure there will be an energy revolution, but my question is whether this new public is formulating itself in a geometrical way with the central structure. I am very optimistic because between the first 10 years of the first article with Annie, there was the discovery of Carl Camden, we were discouraged, and I had my father who would say to me every time I saw him, you have to look at this, you have to leave this like that. And I told him, in my inner heart, it will never work, it was a coincidence, it will never work. And that means that this summer, by looking at the graph, not only did it work, but it was much better. That is to say that the physics of the two fields improved considerably in the field. So what we can hope for is that now the new experiments will tell us, yes, the x is too simple, it's not really the x, it's complicated and all that, and that if it works geometrically with that, we will realize that maybe we will be able to solve it. I think it's a theory that can't work without a theory. It's not a theory that we can guess and say, here's the theory that you've just realized. No, you can't do that. Organize... There are too many super-mixed types of models like that.

7:30 There are too many super-mixed types of models like that. I prefer to use what I said in my analysis, that is to say, for the first time, my conclusions are due to the new non-configurability that has occurred. These are the most complex theories, and I think they are open in this sense. I think that the super-sphere is made up of one of the main ideas of the super-sphere. The idea of the super-sphere is that, for example, if you have a large number of relations, you have the ability to have them all the time. And here, in this case, we have an initial value, and for the moment, it is equal to the value of the total. Because, as we said before, we don't see the initials, we just go down with the order. Did I get it right? If I'm not mistaken, is it an alternative to the grand unification? Absolutely. It's an alternative to the grand unification that doesn't have the decadent function. That is to say, there are the same values that exist without having the decadent function. That's it. That's already it. That's already it. You completely ignore the unification. If you completely ignore it, you have a grand unification theory in which it doesn't have the decadent function. Thank you for your attention. I would like to ask you about the mathematical field, which has a very good energy, you told us that there will be a state that is more or less symmetrical, for us you have not chosen geometry, but you think of a kind of functional integral. You are right, you are right. There is one thing that I would have said at the beginning, it is that the functional integral in this area, what does it look like? Well, we take the observable spectral, that is to say, it depends only on d. And we put the exposure of the functional graph with, say, fewer traces of D on the bottom, etc., and we put the graph on the bottom. And what do we put as integration or base integration? All this is called D of D. We must see that a graph operator has the same linearity as vectorial chambers. So, obviously, we will integrate on D of D. But what is D, then, people? It's not the same thing. It's not the same thing. And what is the role of the spectral axis? Well, the role of the spectral axis is to say that we don't care about the ferns that have an energy of a million times the base of Planck. We take them away. And in fact, we approximate the geometry because it is in the space of a finite dimension.

10:00 Which is what space is a space? Space is a truncated space. We are going to take them from there to there. What is remarkable is that we lose the symmetries, i.e. unlike people who say that space is made up of a multitude of points, the discretization. The discretization has a huge advantage, it calculates the frequency of the space. So we can say that from a space, we can do the same thing. Basically, space becomes a pointy space. That is to say that the space in front of us appears as a joint spectra of operators. They are punctual spectra that appear in front of us. This is what we have learned so far. From this point of view, we look at the system in which most of the work takes place. From this point of view, we look at the system in which most of the work takes place. Do you imagine that the transition will take place in the universe or will it take place in other universes? In the first phase, we are in the second phase, that is to say, we are simply in a continuous cascade of phases, which purifies the state in which we are in, in a state that is extremely stable, and which gradually, with the advent of mathematics, is purified. So that's it, if you can, that's my thesis for today. No problem, thank you very much. I don't know, I don't know, I don't know, I don't know, I don't know, I don't know, I don't know, I don't know, I don't know, I don't know, I don't know, I don't know, I don't know, I don't know, I don't know, I don't know, I don't know, I don't know, I don't know. The difference with string theory is that, well, we have a functionalization, but, for example, Amit Shantideen wrote an article in which he applied to string theory a spectral action. And he applied, if he could, to the space of the laces on the space. So he applied, as an operator of a graph, the space of the laces on the space of the space. And what did he achieve? He achieved the correct action in the field of the field, in the objective action in the field of the field. So, in fact, what does this mean? It means that this theory can be archived on anything.

12:30 It can be archived on the spaces given to us by the string theory, but it can also be archived on the spaces given to us by the quantum groups. And we must not have a class. My point of view is to say, for someone who is at the end of the day, to take the last space, she has to do something like that, she has to do something like that, she has to do something like that. Ok, but we must not stop at Penrose. I think that what we have to do is to start from space, go to space, go back to gasification and see if from there we can find a situation in space. If there is a situation in space, we can find a solution in space. But the dimension of chaotology, or two, is the dimension of physics, and that's where we exist. Well, there are spaces, like the space of a small case of a cube, which is wrong, and when the cube is not at its basic value, its dimension in the symmetric sense is zero, and its dimension in the cohomological sense is zero. So, these spaces are not the same. They are all the same. It's just that we can't do that. In my opinion, we can't do that. We have to try. And so, the theorem is correct. What is certain is that there is a very important connection with the idea of space-time. That is to say that the idea of space-time is more or less based on the idea that there is space-time at the level of space-time. And that, well, really, is the release of space-time. I think we have to be a little bit in the middle of everything, because we cannot combine and create reality. We have to wait to create space-time. I'm not doing all the work. I'm just throwing out ideas. You see what I meant by a quarter of philosophy, a quarter of speculation, and so on. I'm not sure which quarter it is. So, I think it's worth investigating. And what we should do is try to... You see, having a small interval of time is some kind of discrete process. It's not making the space time discrete.

15:00 There's even discrete chunks of information. So you keep the continuum, but you introduce something which is sort of semi-discrete into it. Now I should try to combine this with geometry, because geometry is very successful. The beautiful laws, the ones I don't quarrel with at all, are, you know, Maxwell's equations, Einstein's equations. These look very beautiful to me. Not that I don't like the quantum mechanics, which I share my insights and misgivings about. And so, what you have to do to develop this is, of course, you need to develop what you might call new subjects of retarded geometry. How do you do all the geometry you've done? After all, as I said, part of the original equations, 200 years of Zenith, was developed by Lagrange, Laplace, all these big peasants of the past, because it was a technique to understand mathematical physics. And the difference in geometry went hand in hand with part of the original equations, and so we had this math. Now, I'm suggesting maybe that wasn't big enough, maybe you need to open the door a bit further and do a new subject for it. Generalization, which involves retarded geometry, and I put underneath it, and also advanced, advanced means you change the sign backwards and forwards. That may worry you philosophically. It did worry me philosophically, namely you need the future to create the future. The point of view, when you write out a differential equation, say you need an expression of time interval, it depends on where you think you are in that interval, the middle or the end of the beginning. Philosophically, it's not that difficult to understand that, in the same principle, advanced equations are also truly part of the story. People have looked at retarded equations, as I said, in electrical engineering. But one fundamental, ambitious, ambitious in the sense of long term work, ambitious in terms of the aims of what you want to achieve, if you can't develop a theory which goes all the way to developing relativistic theory and quantum mechanics and quantum field theory and general relativity and gravity, it's not worth doing. We're trying to get everything, not just, I'm afraid quantum mechanics would be fine, but we want to get much beyond that. And the very first you fail has to do with relativity theory. Obviously,

17:30 How do you define the target equations relativistically, variantly? Because time is not an intrinsic notion in relativity theory, so say something depends on past time, what does it mean? Well, I think a little bit, there's an answer. See, if you did one variable, if you place time by time, shifting in time by distance minus r, you'd see that the integral generator is differentiation. So translation is the enunciation of differentiation. Perhaps there are. Now, that's one variable. When you go to the Minkowski space, you want to do the same thing, and you want to have here a third-order operator, because third-order operators may set equal exponentially, and they correspond to motion in some way, and there's only one invariant to define third-order differential operator in Minkowski, and that is a Dirac operator. So you have no choice. The only thing you can do is to expand the Dirac operator, and that corresponds to retardation. It is in fact that when Dirac discovered his Dirac operator, I told you it was really discovered by Hamilton, never mind, when he introduced the Dirac operator, he was searching, he really was led to it, he was wanting to look for a third order equation. Now, Sanderson books of physics will now tell you he got the right answer but for the wrong reason. It doesn't have to be a third order equation. The quantum mechanical theory of this type is the second order operators and Drac somehow stumbled at it by an accident. You know, he was led for the wrong reason to the right answer. Well, in some sense, I want to suggest that that's not correct. The right answer, what he found, was very fundamental because, for example, with this notion of retardation, you have to have a third order operator. You can't have a second order operator in terms of retardation. That's quite different. So, it's fortunate for us that Dirac, while thinking along these lines, got the third-order operator. So, this suggests that you could just use the Dirac equation in an operational spin, of course, and so here you take the first, away from a toy model, this is my physics, this I've done 20 minutes of physics, this is written by Craig Moore, you look at the Dirac equation. And then you add here the three tarnation terms, k times exponential minus rd, which is a modified Dirac equation, a retarded equation.

20:00 Now, actually, if you look at it carefully, you see that because spinors, as you know, come in positive and negative energy states in Dirac's discovery, that changes the sign. One whole spinnets are retarded, the other whole spinnets are advanced. So you get a combination of both. Now, when I work with physicists, they goodly write out exponentials of operators without blinking an eyelid. But if you're a mathematician, even if you're not really an analyst like me, you know, you have a conscience, and you say, what does it mean to exponentially, well, I know how to exponentially, the Laplace operator that I found, and I know what that means. But the direct operator in the cosmic space is not matter. He's a heat operator. So what does it mean? Well, again, if you go back to physics, it does make some sense. You interpret this operator as something you apply only to physical wave functions. A physical wave function is a linear combination of plane waves propagating with less than the speed of light. Things don't happen far from the speed of light, physically. And if you consider only plane waves which are propagating with lots of light, then each plane wave can be retarded by its own proper time. When you integrate over all the different components of the wave function, you'll get the definition of this. This operator is simply retarding each component of the wave in its own direction, its own time direction. So it makes very reasonable physical sense. They will give you the following sort of calculations. They will count all the dispersion relations. You take a stationary solution, carve the equation, you cut in a term like that, you decompose the spinners into two parts, and then you write down what's called the dispersion relation. And when you get an equation like that, with this term being new, normally you would just get e to the e squared. But now you get x terms. And this, interestingly enough, because the exponential has both imaginary and real parts, The energy is meant to be real. This would require the imaginary part to vanish. That would only happen if the appropriate quantization condition holds.

22:30 That leaves you with the condition that the parameter r, which we put in mathematically, not having any idea what it should mean, should be an integer multiple or something specific. And that integer multiple, or the thing of which it is an integer multiple, is something that is not physically well known. It's called the constant wavelength of the electron. And its order is about 10 to the minus 12 meters. All of these things are very important to us, and we need to be able to use them in real time, or in time, about 10 to the minus 20 seconds. This little bit of calculation that came out automatically, we didn't have much choice really, we had to use the Dirac operator and all that, tells you that the scale, called this hypothetical short-term memory, is 10 to the minus 20 seconds. Well, that's a pretty short-term memory. Even when you get old like me, you can remember 10 to the minus 20 seconds. And so on, and so forth, and so forth, and so forth, and so forth, and so forth, and so forth, and so forth. And those are the things that are initially at your choice. So we've got a quantization condition that sort of fixes the scale of the first one, the quantization parameter. The other one is still something that you would just wait around and compare with the experiment or something else. So it could be something extremely small. But they're independent. It's nothing to do with... the two have nothing to do with each other. So, at this time, we leave it as a parameter to be adjusted later. It could be so small that, again, you would notice it in many scales. Well, once you've got the Dirac operator, which is now common ground with Owen Codd, you can turn lots of handles and do lots of things. The Dirac operator is a marvelous piece of algebra to write because, first of all, the Dirac operator is geometrical. That means you can define it in curved space-time, so it makes no difficulties combining it with general gravitation that way. And you can write down the starting equations for spinners which couple to other gauge fields, for example, electromagnetism, and you can get Maxwell equations coming out, or curve space time, something like that. So, the fact that the Rack Operator is geometrical enables you to move rapidly from the flat case, the curve cases, or to the case where you have inchy fields.

25:00 So that's the advantage of having things geometrically, inch by inch. This, when I talk coupling to gravity, this just means having a fixed gravitational background doesn't tell you how to think about gravity. If you want the existence, you should ask how do you retire gravity? How do you retire Einstein's equations? I don't know. It's impossible. I mean, it's not all the equations. It's not clear how you do it. I don't think it's possible, because of the relation with spinors and so on and so forth. It's an interesting question. What is the right way to define or retire Einstein's equations? I'm just saying these are the kind of questions that this investigation turns up. I don't claim to have the answers, I'm just throwing out problems. Now, another thing which is very important in quantum field theory, at a fundamental level, is the notion of particle creation. In quantum field theory, there's the famous fluctuating vacuum. The vacuum that we live in is meant to be, you know, hands are shaking all the time, and particles are being created and annihilated all the time. Whether it's using conservation of charge or when you and I create one particle and you have to mine it, it's opposite. But is there a place where I can see particle creation emerging in the scenario I'm describing? Vaguely. Very vaguely. So this, I now do, I told you there will be 25% mathematics, this is the middle of that 25% mathematics. I recall the use of the term mathematics, the index theory. For example, in four-dimensional Riemannian manifolds, compact manifolds, with a spin structure, you have the drag operator, which acts on spinors, and the spinors come in two chiralities, and it maps one into the other one. This is an elliptic differential operator, and it has an index, which is the number of solutions on one side minus the number of solutions on the other side, and the beauty about the index is it's topological, and it's topological formula. So this is the sort of mathematical content. The way the index dracopera appears in topological context. Now, there is a three-dimensional Lorentz inversion of this result. It is not quite so well-known to people, but essentially, cobalt and physicists understand this in their own language. So if you consider a three-dimensional space now, then the dracopera there is itself an elliptic operator which is self-adjoining.

27:30 And if you imagine your space, three dimensions, varying in time, So you take space-time-time and you think of it as varying. Then you have one family of self-adjoined operators that vary in time. If the situation is periodic, so that outside the box you're in, you get back to the same position, or kind of back to the position, then the operator at time one can be equivalent to the operator at time zero. And when that's the case, you have a sort of periodic situation, then there can be what's called a spectral flow. Namely, the Sherbert-Hoff operator has a discrete spectrum, assuming your free space is compact. The eigenvalues move as you move time. But they are both positive and negative. As you move from one end to the other end, some eigenvalues can cross from being negative to being positive. But you get back to where you were, to the end of the operator's equivalence, the spectrum is the same. But it doesn't mean that the eigenvalues stay constant. Take as an example all the eigenvalues that move forward one step. Think of the operation of anybody x in a circle. Shift everything by one step or two steps. So you get a shift, or what's called a spectral flow. The amount of, the number of eigenvalues change sign. And of course if some eigenvalues are the other way, you cancel them out. So this number you get topologically based on the family. If you preserve the whole family in any way, this number does not change. And it is in fact closely related, essentially equivalent, to the four-dimensional index. And this has the advantage that it sounds like physics, whereas this sounds like geometry. This is about one parameter family of physical objects. And this number is interpreted, physically, as particle creation annihilation. So, for example, if your black operator is coupled to this engaged field, which is an outside force, then the spectral flow can be thought of as the number of particles treated by this external field. In the context of my retarded story, although I want to do things a little bit distinctly, the story is that I break symmetry, think of space, time, time, and I imagine that in the short interval between r, minus r, and zero, whatever happens there is an initial data, it's free. In that initial data, there's no reason why we can't have interesting topological features taking place, which means particle creation.

30:00 So, quantification would be something which is coming out of a mathematically known framework, and would fit in well into the idea of little bits of space, the initial data. And this initial data space could even include gravitation, because if the space-time itself has a qualitative variance, if the decalculation of space and time is not so defined, it has holes in it. Then, in the middle of that space, you would have a gravitational input, which would lead to the topological change in the index-like quantity, which is familiar in geometrical stories. So, there is space in this theory for particle creation, which is one of the characteristic features of quantum fielding. Now, if you can't explain that, you know, you don't fit with experimental results. I'm not claiming this is a new theory. I'm not claiming that this is the right way to go forward. These are initial steps to explore. And the first thing to point out, as I pointed out in the simplest equations, is the time flow for an entire equation is not unitary. Or, if you like, the spectrum is not purely magical. Same way the equations, the one I wrote down initially, it's not necessarily symmetrical between time and minus time. Well, that may or may not be a good thing. Your job may be to find the equations that are time symmetrical. Maybe they're not. The very small values of the parameter k, the deviation of these two things is given to be small and difficult to detect. So experimental tests along the line, the idea even at this very crude level, could be called experimental tests. You might be able to test some deviations from unitarity or quantification by... And then you might have to get some hold on the premises involved and so on. It's more likely you need some much more complicated models. I've written down the simplest kind of model and said we should explore it. But this is just to indicate that I'm not putting forward for you a theory in any sense. These are just hypothetical studies for investigation.

32:30 Let me finish with a remark. I said that you can do, we should develop studies of retarded geometry and the Dirac equation, the Dirac operator, is a very key part of that story. And similarly, we should go, what we call retarded mechanics, how are they mechanics, is a very important part of mechanics, the geometrical aspect of compositivity, and played a very important role. Can we develop, in a sensible way, the retarded mechanics? Well, what would it look like? Well, here's a picture. First, you have a convective manifold, and you have Hamiltonian function, which normally you write Hamiltonian flow, Hamiltonian vector field. And first, also, the manifold has a Riemannian metric. For example, it could be a Kähler manifold, which gives you both the vector structure and the Riemannian structure. And so if you wanted to look for paths which are solutions for what you might call the retarded Hamiltonian flow, take Hamiltonian flow, solve it in the equation, and retarget it, what would that mean? Well, if you take your path, that's the black line there, and then you take the point times t, you take the time t minus r, And don't say that the Hadley vector is equal to x to f, but that this Hadley vector is equal to that one of the parallel transport along the intervening path. Parallel transport is defined because you have a real-time matrix, so it makes sense to write down solutions of these equations and they will, if you think so, curve. And, of course, in all these situations, the parameter r, if it goes to zero, you recover the classical thing, because every, the classical limit, the complement r goes to zero, so you can study the classical limit, you can study what happens in the neighborhood of the classical limit, what information, but this is a mathematically obvious equation one could study, it's an interesting to look at, and irrespective of the application of physics, there's no reason why we shouldn't do these things. I've told you everything I know, really. I've been thinking about these ideas for a few years up and on. I nearly talk about them to physicists. And then, of course, I get shot down. Maybe there's a physicist in the audience that's come when they've come already.

35:00 But I found that physicists are always very helpful. They focus on certain problems, I go away and think more about them, I refine the ideas and so on. So I've benefited by talking to physicists. Some things are more expensive than others, but it's that way. So I think I'm going to stay mainly on my position with an interest in physics. And I think I'm just trying to pry this door a little bit so people can at least consider the possibility that there are avenues to explore which are not. All of these standards involving the idea of differential equations are not the only, without modification, not the only tool. By the way, that's what we've developed. You see, if you go back to my initial comment about, philosophical comment about, where mathematics is finished in the real world, you've asked, is the physical world built on these marvelous mathematical practices we've developed? And the answer is, well, of course. We don't technically think they're useful, but that's only true with God. The world is different. We don't have a tool for it. So here I'm suggesting maybe we need to enlarge our set of tools. Perhaps our set of tools is not quite well enough. Our future may or may not confirm that we need to go in this direction. And lastly, let me say that all the things I mentioned in the beginning about string theory, quantum geometry, and what not, Any interesting avenues to be explored to try to provide unification physics to them are not necessarily disjoint, nor are they necessarily disjoint from what I would say here. There could be links between all the ideas on a level. I don't think they're meant to be all wrong. If they're all meant to have some connection with the real world or model of the real world, they should have non-stripping connections. And overlaps and ideas going from one to the other could be very useful. So I offer this to you. If you're a young man, if you've got your PhD and you've got a job, you can work on it. Otherwise, tell us.

37:30 The primitive is defined by the devolution of space. What you are doing is that you are creating the devolution of space and space is what you are trying to do. You are trying to find the best thing in the universe, and that's what you are trying to do. You are creating the devolution of space and space is what you are trying to do. But we know, and we know this for sure, that one devolution of space is part of the devolution of space. In fact, you know that when you are going to pick up a baguette from a friend, you actually grab it right in front of you. So, you actually suck at this sort of thing. So, I've got to give you another example. Let's look at Iron Man at first. You're looking at a baguette. Then, you're actually going to put it straight on top. You're trying to take it off the floor, but you're trying to keep it straight, too. But you can't, uh, the whole part of it needs to be straight on top. You have to focus on the square, where it has to carry. And you will find that there are completely different types of formulas. So what you really want is directly to do. I am going to teach you how to do it. But first off, I think you will be able to understand that it is an abstract, unwisely concept of mathematics. And it is better, because I expect from a general class that you will be able to see exactly what we have been trying to do. Because I think that after this, then there is actually a different type of theory. All the way through to the end of the time, I ask that you throw away the hours and minutes you have left. Talk to the state of your data about how much of it you know about yourself, and who you are, and who you're trying to talk to about yourself, and who you are, and who you're going to talk to about yourself, and who you're going to talk to about yourself, and who you're going to talk to about yourself. There are a number of ways in which you will be able to understand the physics of mathematics and physics of mathematics. That's exactly what I'm going to talk about today. I'm going to talk about the physics of mathematics. I'm going to talk about the physics of mathematics. I'm going to talk about the physics of mathematics. I'm going to talk about the physics of mathematics. I'm going to talk about the physics of mathematics.

40:00 I'm going to talk about the physics of mathematics. I'm going to talk about the physics of mathematics. So it's easy to say that Einstein did not do the two of them together, but in the topology term, in the topology term, we sometimes know each of the terms wrong, whether each of them can be apparently the same, but at the same time, they don't seem to be the same. It's a good thing that we don't get into the topology of the two of them at the same time, but at the same time, we don't get into the time. I mean, this is a couple of people. I mean, there's a lot of young people here. So it's very clear. There are, you know, other than there's two types of stuff. You can do a little bit of contemplative stuff. And those are the types of stuff. So if you go to the classroom, you read some of these things, and those are the kinds of stuff you may be trying to come up with. That's part of what we want. So that way, I'm encouraged by that. I mean, I'm not surprised at all, given the fundamental role that the direct operator plays in the non-committalist story. So I don't think this is encouraging at all. Of course, the irodynamic toy model, I think, is animated and modified and generalized. On that, I wouldn't go back and say that I repeat what I said about Hawking and Grazer. Why don't you make a step forward to throw away a lot of old baggage and start again? This is just the ground level of things that I really care and then build up. So I've been trying to adopt that from this point of view, and it may turn out that that's entirely equivalent, or we can't put it into another language, which helps in a different way. We're quite familiar with it. It's different from the point of view of being much like in school. But I think there's some way in having a point which is A, mathematically, and B, philosophically, attractive as a starting point, without too much, you know... Not pretty much you can see mathematics behind it, but just one little idea and you see where it goes. But of course, a lot of work we developed to work out the mathematical consequences and how to modify it to fit data and fit other theories. Anyway, I'm encouraged by what you say about possible links, and therefore it's not inappropriate to talk about it. You know, it's not going to be done. More questions? Well, it's very naive of me to say that actually.

42:30 All right. And you can get into the specific type of ELR, and we've covered quite a lot of ELR, and that type of ELR, rather than, say, integrating over all sciences and certain functions of ELR. Sure, when I first started, I had ideas along that kind. I did think about the design, but the trouble was that it was very broad and you didn't quite know where to go. The idea was really difficult to find out. I then got from this chap, Raju, who did it. And then you see, very simply, a lot of things coming out not easily. Now, as a pro-model, that may not be at all right. You may need to integrate over a... All the other ideas... Yes, yes, yes, yes, that's right. Well, I think it might be other kinds of things. And I think we, I agree, I originally started off thinking much more generally. The fact that it wouldn't come into your way, although you would think about it philosophically, that gives you a little bit of trouble. Do you need to go back to the beginning of the universe to get all your data? So, I'm not sure, whatever you... All of these things you have should have very short-term effects, so therefore it's a very sharp cutoff, and therefore a short interval might have to be the first approximation, and as Alan said, maybe you could use today to get a better one, so, but it has merit, you're starting out with a radical idea, you want a ploy model where you can test a very simple thing, and this ploy model, you can see something coming out very simply, so, conceptually you can get to the, and then you can try to relativize, do it relatively, but it's a choice, because it's... Then it got easily to numbers. Before this, I was just weighing my hands and didn't know where to go. After this, I thought, well, I could see possible directions. Like I said, it didn't. It was just, you know, only one, probably one more, bigger range of possibilities. But, see, opening a new door, you do the simple thing first. No, real physics, ah, yeah. So, Michael, speaking of the simple things first, First, is the massless limit of your Dirac-Toy model regular or singular? And did you try to find a non-local version of Maxwell's equation? You know, by the way, historically, that Maxwell always wrote them with the Hamilton-Dirac operator, the Maxwell equation. Well, excuse me, I think the massless limit is no problem. I didn't begin any analytical difficulties with that.

45:00 No, but the question about the Maxwell equation, I did say you can couple spinners with anything you like, and the Maxwell equation is you spin a couple of spinners, so it comes out that that equation couples with something else, so you can do the Maxwell equation the same way too, but that, you remind me of something else I meant to say, that is a very, very deep philosophical relation. You go back... There are a lot of things that are related to the laws of gravity and the laws of gravity itself, and the laws of gravity itself, and the laws of gravity itself, and the laws of gravity itself, And you remember when Maxwell wrote down an equation, he didn't get the equation by saying, ah, two equations. He had a picture of rotating, vertically interacting, rather like Descartes. And he got the right equation. But when he got the right equation, he said, well, I don't really need the picture. It was a guy. He threw it away. Well, you know, I say, maybe these guys are not stupid. That's a small-scale user, not faulted. Maybe space-time itself is in some sense granular or ether-like. So maybe those pictures are not so crazy, because people quite often say that it's a great step forward. Well, you forgot about having the ether, whatever it is. And we got away from that, we wrote down differential equations, and we don't know what they mean, but it's fine. But again, maybe that was actually long-sensitive. Maybe that was actually a mistaken, going away with the old ideas, going away with non-philosophically based ideas of just mathematical work. Maybe that's too naive. I'm just saying we should re-examine some of our prejudices. And these are, although in one form discredited, some form of... The nature of space-time on a small scale could not be, might not be very different in a way from either of Maxwell's vortices or even Descartes' vortices. That's a good philosophical point to end on.

47:30 No, no, no. As I say, if you look at it mathematically, there is no difference between them. You know, usually you distinguish between the derivative and the value. You think the derivative is now, and we're trying to work out how we're going to go in that little step forward. And the path is where the influence comes in. You can change those rules around. The derivative could be the path, and the value could be now, you know, a mathematical equation to solve. It's still just a question whether you put yourself, where you place the origin of time. Time invariance. So that there's no philosophical difficulty about notions. And that's why these equations that Feynman wrote down, which he made in comics last time, and there's no mathematical difficulty. In fact, they have been studied in good statistical models, and in his case, there's even energy conservation. So it's something I thought was a philosophical obstacle, but it's not. You're going much too fast for me. First of all, I don't answer any of these questions. But I would like to say this. You start with a radically new point of view. So, you have to examine every question that was asked in the past to see whether it makes sense and is the right one to ask. You don't just make a fundamental change and then try to copy every step that happened before. So, all notions have to be redefined, reexamined, and so on. So, eventually, your theory has to be one which will reproduce the orthodox theory of the suitable limit. But the theory itself might be non-unitary. It might be all renormalized. It might be going throughout the window. It may come out. So, I think everything is up to the graph. That's what I think. That's why I don't think, for example, in a theory like this, that I would accept anything except Maxwell's theory, Einstein's theory, modified, and everything else has to be worked out from scratch, Gatesfield, whatever you like. Why should you assume anything? You start again from scratch to see what you're required to get. Hopefully you get, at a limit, you'll get some reproduced previous results.

50:00 All technical questions, I think, are on hold until you decide what the technical questions are you need to ask, and you're going to build this up on the ground slowly. So I'm not chickening out asking a question, because I think it's premature to ask a technical question before you have a framework.