Broken Bootstraps — Rise & Fall of a Research Programme / Dirac on Mathematical Beauty (contd.)

0:00 So, you'll see that at best what Dirac's argument establishes is that i times k, or never mind i, he threw that in because he knows it's going to show up in the end, just k, is in the commutant of all commutators, that is to say, it commutes with all commutators. If you go through the logic of his argument, you'll see that that's all you really get out of the argument is that k has to commute with all commutators. But that's because the situation is even a little bit worse than that. Because you can define algebras, remember we have this very generic conception of what a quantum algebra is, it's any non-substantial algebra, you can define algebras in which, to put it loosely, products of arbitrary elements with commutators are special. They do funny things that will make this equation true. I'll just give you a very trivial example. Three or higher are equal. Then anything times a commutator is zero, both sides of that equation are true, and that's just the most trivial example. I've worked on this for a while and then sidetracked into other things. As far as I know, there's no completely general account of the cases where Dirac's argument actually works. So one question you might ask is, okay, let's not go with any non-commutative algebra whatsoever. Tear down what we mean by quantum algebra and try to get just that class of algebra in which Dirac's argument does work. I'll come back to that point in just a moment. But first, let's just note that everybody, you will find that people take as their starting point, that is to say, people take as their starting point something like what Woodhouse has written down here.

2:30 It's very fine and very difficult. This one here, this is where he's saying, I say that first of all just to be clear that I point out the ingenuity of the argument. I'm not going to go into the discussion that I'm about to have about how one might fix the argument because I disagree with it, which is obviously correct, but what I want to know is why is it correct. Well, as I see it, there are two strategies for revising Dirac's argument. They're mirror images of each other, and in some sense they're equivalent mathematically, but I think they're two machine problems. First, you could try to characterize the algebras in which Dirac's argument fails. You might, on the other hand, characterize your algebra in which it does work and then argue that those are the only ones that are interesting, or logical names for that strategy. I'm going to give you an example of each of these strategies. They're not the best examples I have in terms of the most powerful theorems I have. If I had completely general theorems, then I would just give you talks about them. But I don't have complete general theorems. The most general theorems that I have, because I just want to give you a flavor of how one can make arguments.

5:00 So I'll start first with the negative. Let's just find one where it fails. We're going to define quantum lead brackets, z times a commutator, where z is some element from the easiest way to construct a particle by a degree of three or higher equals zero. So z is something in the sense of the algebra, that is to say, commutes with everything. So this is the quantum leap racket that looks a lot like the rack. And so the question then is, and I think you could make a case, I don't want to, and for the following reasons, pick any Hamiltonian you like, any Hamiltonian you like. So what I mean is, imagine any interaction you want. What's the time derivative of this particular element of the algebra z? How do they change in time? They don't. Why would you think of this thing as being an observable? Why is this a physical quantity if nothing you do whatsoever could ever possibly affect the child, or even the probabilities for a child?

7:30 What I mean by nothing you could ever do is encoded in the idea of taking any Hamiltonian elect. So if you imagine this thing as a property of some physical system, this particle right here, whatever you do to that particle, you slap it as hard as you want, it will never change the value for z on that particle. Not only that, but even on its own it doesn't change in time, it doesn't evolve naturally either. It's kind of a super constant of the motion, if you like. It's a constant of every motion. And my claim, again, I don't want to call myself too strong on this, but my claim is that's not a physically interesting algebra. If you found yourself working in algebra like that, I bet what you'd do is you'd take all these Z's, which are going to form a subalgebra, and you'd divide out by them, work with the algebra of things that can be changed. Oh, one quick objection that someone might have... People have made this objection, so I'll just say it right now to get it out of the way. Someone might say, well, what about the Hamiltonian itself? That doesn't change in time. Well, the answer is, yeah, it does. I mean, if you fix the Hamiltonian, say, this is my Hamiltonian, I can find other Hamiltonians such that the value of the thing I fixed will change. The proof of this theorem is a little bit longer than I really want to take, so I'm not going to give you the sake of time, but let f be the free polynomial object. Generated by two elements x and y. So what is the free polynomial algebra? It's all formal linear expressions, all formal linear combinations of formal products of x. So elements of that free polynomial algebra, if you let k be r, would look something like whatever. Imagine any combination of x's and y's you like, turn that into a product. Put any number from the field you like out in front, take any number of those that you like, and make linear combinations of them.

10:00 And that's the pre-algebra, the more elegant way to describe it, categorically and theoretically. So what is the theorem? The theorem is the only leave bracket derivation on the pre-algebra by two elements. The definition of what we mean by the leave bracket, and k is an element of the underlying field. That's the result of direct one. And so now, if I'm following through this strategy, I'm going to argue for you that the free polynomial of the Grover field is of some physical interest. And what I mean by some physical interest, of course, is not one that applies to the world we live in. But what I mean is that there's not some objection to even thinking about it as being the one that applies to the world. So in the previous case, my objection was that you wouldn't have a physical theory that has elements that cannot be affected by any interaction whatsoever. And so one might try to make the argument, and I'm not going to hear, I'll say a couple words and support the idea, that the free algebra over two elements is potentially of physical interest, even though it's not actually the one that we use when we do physics. I think it's the one that Dirac often had in mind, by the way. In his case, x and y are positions and momentums, so he often had in mind, and I think implicitly you can see in him when he writes down equations, he takes into account members of an arbitrary algebra, I think what he had in mind by an arbitrary algebra is actually just a free algebra, even though he didn't... Write down the definition of the algebra concept explicitly in life. I think when he says to himself, let me be as arbitrary as possible, he writes down stuff like this. So I think what he had in mind. And he doesn't allow himself, if you don't integrate algebra, to reduce any of these things to any of the others. No term, no formally distinct term is equal to any other form of distinction. So I think he had in mind the free algebra, but the point is that if you thought that, let's say, position and momentum were just the fundamental properties of physical objects, and that's an idea that you might possibly get from classical physics, then the free algebra over those things, if you think that algebras of observables are of interest anyway, the free algebra over those might be something to labor that point.

12:30 Let me pause here, oh no, I have it on the slide, never mind. What was Dirac's attitude towards this argument that he himself made in his textbook, the argument that quantum lead rackets had to be a constant time machine? Well, in 1925, he proved it in the context of matrices, and it is true in the context of matrices, and later he generalized it in his It appears in every edition of the textbook, unlike many, many other things that are in that book. It appears in every edition more or less unadulterated from one edition to the next. It's false, but it appears in the textbook in this very generalized form that I just showed you. It doesn't appear as a discussion about mathematics in general. But the language that he uses earlier in his career about this argument is the language of necessity. I think I can just show you that in one case. So if you look, he says, these conditions are already sufficient to determine the form of the quantum Poisson record. That's the kind of language he uses. It follows from, it implies, so on and so forth. The language that he uses, this is a dodgy historical argument, especially given that I'm not

15:00 There's a noticeable change in the language that he uses from early to late about this argument, and later he uses more language like this. It turns out that this bracket expression corresponds, it's very closely analogous to, so no longer is he using the language of determination or sufficiency of implications. He's using the language of analogy, correspondence. I think he did that because he realized that the argument, I think he came to understand that in fact, the argument as he presented it in great generality did not actually, and so the question then arises, well why did he continue to insist on it? Why did he continue to make the argument if he eventually realized it? Or, let's not make the argument, I mean, make the argument in this transformal way, I didn't say give him the language of correspondence and analogy, but why did he continue to refer to the argument itself, right, he doesn't just say, in later life, he doesn't just say that there's an analogy or correspondence between Einstein's commentator and the classical class on practice, he says that, and he refers the reader, or more often in these cases, listener, to The argument that he gives is, well, let's switch gears a little bit and think about beauty and generality, in my view, and then at the end, very briefly, we'll lay it back to the discussion about quantization. So let me pause here for just a moment, actually, and say one more thing about quantization. This is a bit technical, so I know many of you will, but I think it's worth saying. One way of thinking about quantization in a much more generic way than I just spelled it out is to think of it as a functor, in category theoretic terms, from the symplectic manifolds where the morphisms are canonical transformations to the category of Hilbert spaces where the morphisms are isomorphisms.

17:30 And the question is, well, does there exist such a functor? One's intuition, if you know the stuff by Van Gogh and Grunewald and so forth, is to say, no, there isn't one. But there is one. There is a function that does that for you. What you do is you just let the Hilbert space be, well, let your symplectic manifolds be of a color x, okay? And let your Hilbert space, L2, is defined with respect to, and that works. And in fact, what people in the, by pre-quantization, the problem with it, the reason that's not the end of the story, the reason why it's free, The problem is that a one-parameter group, a symplectic generated by a positive, will get mapped, but it won't be generated by a positive. So for physical reasons, you have to do something to fix that problem. But from a purely mathematical point of view, the thing exists. I don't know, so here's a conjecture that I made to myself about a year ago, but I've since learned it's been made by other people. Conjecture is that you can prove that there isn't, if you require that one parameter groups of symplectic maps generated by positive Hamiltonian in the theta base and the symplectic manifolds get mapped to one parameter, you can prove that there's no such function. So that's a conjecture. If he made a conjecture, he doesn't know whether it's true.

20:00 As time goes on it becomes clear that paper is full. So here's in 1963, I think it is a peculiarity of myself that I like to play about with equations just looking for beautiful mathematical relations which maybe don't have any physical meaning at all in theoretical physics. It is simply a search for pretty mathematics. It may turn out later that the work does not have an application. If the work does have an application, then one has good luck.

22:30 He's talking about the Galilean group, and he makes a remark earlier on that the Lorentz group is more beautiful than the Galilean group, but never mind. He says the latter, the Lorentz group, is a much more beautiful thing than the former, the Galilean group. In fact, the former would be called mathematically a degenerate special case of the latter, that beauty has something to do with generality. All of these will probably require a more drastic revision of our fundamental concepts than any other that have gone before. It's quite likely that these changes will be so great that it will be beyond the power of human intelligence to get the necessary new ideas by direct attempts to formulate the experimental data in mathematical terms. Alright, so here's a history question. Who's he talking about? In this last sentence I just read. Yeah, yeah, Heisenberg. So what he's saying here is, I mean, what he frequently says is, Heisenberg was a genius. Theoretical worker in the future will therefore have to proceed in a more direct way. That's very striking, the words that he says is more direct, this way that he's about to articulate is more direct. The most powerful method of advance that can be suggested at present is to employ all the resources of pure mathematics in an attempt to perfect and generalize the mathematical formalism that forms the existing basis of theoretical physics. It's more direct than the Heisenberg way, which seems pretty direct.

25:00 Again, I'm putting this before you just to give you some additional hints about what Dirac might have had in mind by beauty and, in this case, ugliness of mathematical theory. And here he's talking about quantum theory and why he thinks it's ugly. It says, the increase thus arising in the non-mathematical part of the description of the universe, and what he means by that is, every time there's a jump, every time there's a collapse of the wave function, you have to eject from the community outside, and he calls that the non-mathematical part, provides a philosophical objection to quantum mechanics. Now, pause here. He doesn't mean, it's not that he objects to collapse, okay, per se. He's not going to make the argument that many people make, well, it's distant and what have you. That's not his objection to it. The objection, rather, for him is the fact that there is an increase in the non-mathematical part, by which he means, it says the objection does show the foundations of the dislocation and the material surrounding it. Simplicity, he says, is something like the minimization of boundary conditions. And those of you who are familiar with Dirac's work in other areas will be fascinated by the way they assess parameters. There is a tension between these two views. I think just to be clear, I want to point out that Dirac often himself conflates beauty in a very broad sense, encompassing what I would think of as generality and simplicity. And other times, he's more, beauty is more like, that's the usage he has in this location.

27:30 There is a kind of tension which Dirac recognizes between beauty and generality. And the point is just this, generality often produces wider scope. This is highly general theory and then you want to describe the actual world, you're going to have to put in lots of information that isn't in the theory, precisely because it's highly general theory. If you want to get from all those possibilities to the ones that we actually see in this world, you might actually have to introduce lots of extra. And that's going to go against the grain in a somewhat vague sense of, well, not too vague, of simplicity as, what does Dirac say about this tension? If a professor of fundamental law should strive mainly for mathematical beauty, he should take simplicity into consideration in a subordinate way to beauty. It often happens that the requirements of simplicity and beauty are the same, or they clash in a lateral way to each other. Okay, so let me brush this all up and clearly think that mathematical beauty plays an important role in the development of theoretical physics. And if beauty is generality, then what he thinks is that mathematical generality plays an important role in the progress of theoretical physics. How are we going to make progress in theoretical physics given the mess we're in? And so the question then is, how does mathematical generality increase the likelihood of scientific? Well, the suggestion that I want to make to you is that it does that by showing you what the possibilities are. That is to say, a more general theory shows you what is possible. What the possible theories are. So, let me go back to the issue of quantum theory. Why did Dirac want to prove, try to show, or seem to prove, if I restrict my attention,

30:00 that if my theory was verbal and represented, working in quantum theory itself? Well, I don't think anyone doubts that. That's not the issue. I mean, we can all accept that claim and move on and try to do physics. If you say to a physicist, I want to try to prove that claim, they might say, well, that's great, but whether you have a proof or not isn't going to stop me. So where does generality help to a much more general concept? Well, it will help when we get into Trump. It will help when we find out that we can't actually make progress anymore by simply taking this as an assumption and working in the same old context that we've been working in. I'm not sure I'm persuaded that there's a tension between generality and simplicity in your work.

32:30 It's not clear to me that that's what kind of generality that we're confronted with when we compare, say, the Lorentz group to the Galilean group. And that was the kind of generality that Dirac seemed to be talking about. If I focus my attention on that kind of generality that contains, you know, the comparisons between the Galilean group and the Lorentz group, Then I find it a lot easier to persuade myself that generality in that sense really just is the other face of one coin of punctuated matrices. Well, I guess I would say that Lorentz groups can be seen as, because it was definitely defined as groups, but can be seen as... Genus species isn't quite the right word that I want to use either. I'd rather use the language of containment. Because the point about the example for algebra is simply what Dirac did was move away from considering just algebra of quantum observables to considering a class of algebra, one of which is the algebra of quantum observables. And since you can think of the Poincare group as such, the Poincare of a rest group is clearly not a class of groups, at least that's how you normally think of it. You sort of can think of it that way if you think, well, what am I going to do? Well, I guess you can think of it that way. On the other hand, I certainly... Simplicity, where he tried to define it as, you know, the simpler theory is a theory with the minimum number of arbitraries.

35:00 That's the way to think about simplicity here. It varies the kind of generality that we have. One picture of the Lorentz group, which is a project from the Galilean group, is one of symmetry. I think that would be another, obviously, very brief point to make. You also said you can't push it all that far, that the time dimension is still difficult. But I think that would be another aspect of beauty. I haven't said anything about something like that. You were talking about the idea of negativism. So it's all for, it seems, you're saying, when you're going to rule out a non-physical elder person, is that a non-physical center? Well, I don't think you can see this. Well, what I really want to rule out is not quite that. It's true that in my example, my example amounts to that, but the reason my example amounts to that was that I defined the Lie bracket, the Lie bracket contained a commutator. And so what happened in that case was I did have L and 2. Since the Lie bracket contained a commutator, which is a product of a commutator with some other stuff, then it's true that the algebra has a non-tree. Everything in the center doesn't change in time. But I wouldn't want to say, generally, that I rule out any algebra with a non-trivial center, because you might define a Liebrach in such a way that the things in the center of the algebra don't necessarily have non-zero time regimes. Sorry, don't necessarily have zero time regimes. And you would do that by defining a Liebrach if it wasn't just the product of .

37:30 Well, in that case, I mean, isn't it right that the stuff, certainly those things have non-trivial centers, but don't you in the end physically divide out by the center and just think of the factors themselves as being physically relevant? Or do you think that... Well, there's two approaches to it, and some people do it that way, and other people say, oh, if they're doing these things, they'll make it better and connect with the center. In other words, you've got to break down the superposition. Well I mean I would say in my example as well you certainly could say well I want to work with I want to work with this particular algebra that has this non-stereo sensor with the legal definition that I'm doing and maybe have good reasons for doing that, maybe have good mathematical reasons for doing it, maybe make the calculation easier, whatever. But if you then go on to say And that Z in the center, right, that I used, and that thing is a physical property of objects, then I get a little bit nervous because it's a physical property of objects that nothing can ever affect, and that bothers me. So in the same context, there's still the case of the Super Selecting Company that would like to say, well, I don't want to divide it out by the center. I want to work with the big space and just forbid there to be observable consequences of various things. That's fine. But I guess I do think that if you're going to go on to say, but, give me two vectors that are in the same sector theory, but, but nonetheless there's a real physical claim.

40:00 I didn't understand what you were saying. What do you mean by that? In fact, you want... Oh, oh, that's of course what you're asking, yeah. Quantum, quantum... Yeah, yeah, no, that's, that's clearly what you're asking. I guess I think the more fundamental interest in that is... I mean, it's not an ecology question. I just wondered what you were going to say about the dynamism of the theory. I don't know if it would signal that when it comes to generality, rather, when it comes to the main possible theory, if you get that, it encompasses lots of five-dimensional theories in a sense. And you find, and you know, you roll out all those planets, but you will not come here for a reason. I'm sorry, I'm not quite following you. Yeah, of course, but I'm not following you.

42:30 The Poisson bracket, in general, is not equal to the case as it is, and you're not going to form the case as it is. With x and t? Yeah. But Witten did it now. I mean, we don't want to do finite-dimensional quantum mechanics with x and t. No, not quite. You can't do anything in finite dimensions if you have a thing. It's lacking in generality already. We're looking for an analog of the capacity constant of x which is going to be equated to a constant. But why does it always have to be equated? I mean, I agree it's equated to a constant if I plug x and t in. Does that help me well if I'm looking for a period of x and t? The introduction of the desideratum of finite time to be a quantum inherent variable for requiring that when the Poisson bracket is a constant, that it will under quantization become a constant and so forth, those are specializations to Various cases in which one might have quite a legitimate interest. And then when you ask yourself questions like, well, what about the case, isn't that very generic point of view that you can say something helpful to yourself? Ah, well, in that case, I have to shift my class. I don't know. And as you probably can guess, I'm actually more sympathetic than that people in the field. But I don't think there are thoughts.

45:00 I don't believe the Dirac problem is still the problem. Right, so... Yes, you quoted Dirac in saying that the things that the mathematicians found interesting and useful can often turn out to be the things we've chosen to work the best. And I certainly agree with that. And I think for that reason that if you find some sort of logical incoherence in a theory you believe to be fundamental, Then it's reasonable to think that if you investigate that, you might discover something important. I mean, I would say that is the reason, the foundation. Yeah, you'll find no argument for that. Quantization, the process of quantization, getting a quantum theory from a classical theory, is in a slightly curious position from that point of view. Because I would tend to think of that as more of a practical heuristic. I mean, after all... If you think of it from, I don't know, a golf ball mechanical world, classical mechanics in that form of view is just pragmatically developed. You wouldn't particularly expect this view, and so I wouldn't myself be particularly worried by any adherence in the process by which you get the other. I agree with everything you just said. The point about worrying about quantization in the end is actually to worry about why this. I mean, the point is here. There is then a question worrying about, is it worthwhile to ask, to examine my starting point?

47:30 And so then the question is, well, I think at that stage, not in the hopes that you will find a complete, but in the hopes that it's a higher level and tries to derive those principles and so on and so on and so on and so on.

50:00 And so this attempt to derive something, and so what you achieve by that is greater and greater generality and weaker and weaker assumptions, perhaps. Although I'm not sure I want to say weaker and weaker assumptions, but certainly greater and greater generalities. And so that's the connection with Aristotle. The idea here being that we start off by assuming, let's say, quantum mechanics. And we say, why this? Well, I can give you an answer to that question. Then I say, well, I want to understand why I'm using that algebra of the rules and not some other one. Jump up to a higher level so you can try to derive and try to see that. We'll try to see why it isn't working in that context and not in some other context. Yeah, but the point here is that there's also a connection with scientific... and we're not trying to make progress in classical physics because any theory is doubted about this sort of theory. There's a lot of interesting questions to ask about it. What we're seeking is a quantum theory. I would say the answer is that we're not worried about classical physics. You lose whatever reason you have to a quantum science if you think that...