[Unidentified recording], Oxford

0:00 Ladies and gentlemen, welcome to the weekly seminar. And before we begin, how would you like to make the... Well, thank you very much. I would just like to introduce and to welcome Professor Shumei Cheng from Shenzhen University in China, who is a philosopher of science and a philosopher of physics, who will be with us in the Philosophy Center at Wilson College until April. It is a great pleasure to talk about China, to have a local hero from theoretical physics give our weekly seminar today. So thank you John very much for coming and giving us a talk on S-matrix theory. Thank you Jeremy. I'm afraid this talk is probably going to be unusual from what you normally have in this. For completely different reasons, it's actually quite different from the normal sort of seminar I would give over in theoretical physics as well, so we'll see if it works. It's got a kind of an historical perspective to it. A lot of the things I'll be talking about will not be, so we'll just, this being a theoretical... On a personal note, I'm actually quite old now, and I became a graduate student in the Department of Applied Maths and Theoretical Physics in Cambridge in 1968, and at that time my supervisor thrust into my hand a copy of a book called The Analytic S-Matrix.

2:30 And he told me that was what they were working on. I should read this book and everything about it. Well, I spent some time trying to read the book. It turned out to be extremely hard. It was mostly about the theory of functions of more than one complex variable, which is nasty stuff. And I'm glad that I didn't spend a great deal of time reading it, because it turned out that that book was the last gasp, as it were, of S-matrix, a theory of the strong interactions which had been being developed in the 1960s and 1970s. There is an inherent inability of quantum field theory to describe the strong interactions because although quantum field theory has been very successful at describing quantum electrodynamics basically because the coupling constant which is the firm structure constant is small, when the Analogous parameter was sort of experimentally measured in nuclear cross-sections for the strong interactions the large therefore therefore usual analysis of Feynman diagram perturbative expansion to try to understand well explain is some maybe you can sort of

5:00 Just think about the S-matrix and throw the quantum field theory away. That was the idea, anyway, that the S-matrix contains all the observable data. This was the idea of the mention. This engaged us. And so, why is that? Well, first of all, let me explain what the S-matrix is. Heisenberg, I'm wrong and it was to give us an easy way of calculating so if you think about the scattering of some quantum mechanical particle by a by some object for example the scattering of an electron by a Here, of course, from a quantum mechanical point of view, what we're supposed to be doing is thinking of the electron as some kind of a wave packet, which arrives from the right.

7:30 We should integrate the Schrödinger equation forwards in time, and we will find that... This wave will, this wave packet will evolve into some linear combination of things which hopefully for large times we can we can think of as some superposition but obviously solving the time dependent Schrodinger equation is hard if not impossible so the idea of Heisenberg was to think of When we put the system in a box, we switch off the interaction at time t equals minus infinity, and then we start with a, so that the eigenstates of the Hamiltonian are just plain waves, with a wave number k, we then switch on the interaction of instate, the interaction, and then we switch it. Switch it off again and we look at time t equals plus infinity, what basically the smudge bit at the bottom is supposed to indicate that in the original scattering problem if you want to calculate the differential Cross-section is something experimentally measurable, then that's proportional to the transition matrix, which is basically S-1, the modulus squared.

10:00 So this isn't the time or the place to discuss why the wave packet picture and this plane wave picture are... These are our equivalents, but let's assume that that's the right thing to actually do. Now, this particular object, the S-matrix, has some very beautiful properties. For example, if you look at the, first of all in a non-relativistic theory, if you think of the S-matrix as a point of the energy of the center of mass, And of course, it only makes sense physically if the energy is positive, but as a function of it, it enjoys various analyticity properties, and so you can think about continuing it. Which occur at negative values of E from the non-relativistic Houn space for property linking. From a relativistic point of view, notation here, four vector notation for relation, then if we have a scattering process, for example, just two to two, two-body scattering process,

12:30 For memento are P1 and P2, P1, P3 and A4. Because of Lorentz invariants, you can show that these S-matrix elements depend only on quantities which themselves are Lorentz invariants, and these are called the Mandelstam variables. So, in terms of a relativistic theory, what is verified is by studying Hyman diagrams and the perturbative expansion that the S matrix elements enjoy analyticity in these variables S, T, and U. One of the very beautiful Consequences of analyticity in a relativistic theory, which you don't see in a non-relativistic theory, is this property of crossing, so this diagram here is complicated, so this is supposed to be the complex S-plane here, and so, basically, remember that S was P1 plus, plus, So, if you're actually looking at a physical process, that tells you that s has to be larger than 4n squared. So, the physical region is the positive s-axis of 4n squared. Scatter may produce a real state that impacts this of this asymmetric mattery and immatery part, which is to do with the fact that for those experts, the T matrix of the branch cut here along the positive real.

15:00 Now the nice thing about crossing is that if you, it turns out there is a, just a kinematic relation between s and t and u which is just written there, so if for example you fix t, imagine t as being fixed, then negative s corresponds to positive u, and that means that if you channel, so the s, at least on the sheet anyway, Now, the beautiful thing, as you function, is that if you know them exactly in any very, very small region of the plane, you know them everywhere. That's the beautiful property. So, it tells you that in the S-channel, by analytic continuation, you can tell.

17:30 And so, for example, if you were, if you were capturing, say, a book, then, by crossing, we could say... Something about an antiproton and a neutron going... Of course, from a practical point of view, that's a bit hard, because one of the things about analytic continuation is that you really do have... It's a nice idea, anyway, and it was applied to all sorts of... Okay, so that's the S matrix talked about, but trying to calculate things alone and throwing away the underlying quantum... That needed one other observation, which is bootstrap, and the idea here is precisely to do with this bound state property, that for example, if you're considering the scattering, continue this atrium to the bound state in this channel here, which corresponds to the residue of this.

20:00 All of these will be proportional to the S-matrix that you would get, which involves the particles. So you can relate S-matrix on processes. That's fine, except if you use crossing as well, then you can equally well think that B-bar is a bound state of K-bar order. So you can also turn this whole thing around. This goes under the name of nuclear thermocracy, that in the S-matrix for just one process, by looking at its poles and its bounds, so there is one S-matrix. So this was the idea of, of course, theories of, this wasn't a theory of, and so Chu, consistent with all these principles that are in Bootstrap, And then, of course, because they convinced people that true may be there, so this was a nice idea, but it sort of died, as I said at the beginning, and there are various reasons why it died.

22:30 First of all, this analyticity was never completely understood. On the first Riemann sheet, which I showed you a picture of there, it's clear what the If you dive down below one of those branch cuts onto the other sheets, then all sorts of stuff can, in fact, emerge. You're not in the anomalous thresholds. People found these by studying Hyman diagrams. Basically, by considering complicated enough Hyman diagrams, they showed that these then... Of course, there are physical questions. Matchless particles fit into this. This presupposes that some articles initially lay far enough away from each other so that you can switch the colon, interaction, and that. There was a kill bit. I'm going to talk about S-metrics. It's returned from the wilderness, or reborn.

25:00 It's reduct, but only in 1 plus 1 dimensions. In three-space and one-time dimensions. Archikov brothers actually beginning in the 1960s or so. So why study physics? Why study estimate from the particle physicists view that since three plus one is what is much too hard why not just physical reason that's an admission of failure. So the real reason is that there are real physical systems that can be described by one plus one dimensional relativistic and these occur in condensed matter and there are many condensed matter systems, many solid state systems where the crystalline structure is such that it consists essentially in a very long So we can see that the chains and the interactions along the chain are very, very much stronger than the interactions between the chains, so to a very good approximation, we can treat them as one-dimensional quantum systems. Moreover, these systems often have massive excitations, excitation gap, and they're only effective excitations, that is, at sufficiently high speeds.

27:30 They are composed ultimately of excitations of electrons and phonemes and things, but they're clearly a relativistic matter. When you look at the dispersion, in some cases, it's not sex, this, if for me, velocity. And the other important thing is that there are often strong coupling systems, which are very large, so we're back in the situation of hadronic physics. The Kaplan constant is a large perturbation theory. Why is 1 plus 1 dimensions easier than 3 plus 1? And that's largely because of 1 plus 1 dimensional kinematics. So this is pretty elementary, but it's important. So once again, let's look at the 2 to 2 scattering process in the center of mass frame. Okay, so two particles, one with energy e, momentum p, now this is this one component now because we're in one space dimension, the other with minus p because we're scatters into with energy e prime and momentum p prime.

30:00 So since energy is p, so the momentum, the energy momentum of the particles in the final state is always the same as the energy. And that's basically because you're in one dimension, so the particles either scatter forward or they scatter back with nothing else to do. So basically, at least in two-to-two scattering, just identical particles. One very useful parameterization in 1 plus 1 dimensions is this sort of parameterization. Field theory to normalize the single particle states of your field theory in terms of their and to normalize them in quantum field theory. Okay, so once again, if we focus on 2 to 2 scattering, we now have an S matrix scattering of 2 to 2.

32:30 What do the various principles tell us about? First of all, it turns out that later on, Lorentz transformations on this system just correspond to shifting the rapidities by the same number. So, that's unitarity of the response to probability conservation, but what you can show is that S dagger unitarity equation crossing, well, once again, I've just written this assuming that we have one type of particle in theory, It's a bit more complicated, but it relates what's going on in so-and-so theta. One of the things about parameterization here is that the Mandelstam variable S, and now it turns out that although there was this nasty branch cut in terms of theta,

35:00 so of course there are still things like that, and they still correspond. And that's where the story would end, because we have solved some of the analyticity problems, but we still have the basic problem that the analyticity completely controlled. So what brings it under control is this additional assumption that the theory is integrable. There are some infinite set, some set of charges, not like electric charges, but some kind of obvious Q with each other and Hamiltonian. These charges have particular Lorentz transformation properties. It's useful to actually talk about the left-moving and the right-moving momenta. So actually these are just in terms of that probability variable.

37:30 Lorentz transformation, I told you that theta PR gets to be. And these charges have the particular property that under Lorentz transformation, they behave like so that acting on some particular, they give the sum of the lots of examples. And it has a remarkable scattering because we not only have the sum of the momenta or the end momenta, but we have sum of powers. All of the energy memento is also conserved and track down what the consequences of that are if you have a scattering process which is labeled have quantum number so so we once you have an integral model it means that if you start with a state with n particles in it you can never have anything different there's no particle production there unlike there is three plus one

40:00 You can bang two protons together at sufficiently high, and you'll get all sorts of junk. One plus one dimensions, all you could ever get would be two particles of the same mass. Okay, so this, then, factorization. So now you have three scattering maps. We start with a state which has... If these particles started off such that they can all sort of go to the limit, And now, because of analyticity, you can show that it factorizes everything into two steps.

42:30 And that really means that this vague notion of the bootstrap can really be... Let's imagine that we go to a particle sphere, and the residue of your bootstrap can pull yourself up. I've never tried it though. When I started reading the Zomologikov's papers, I was confused, so with my student, with my postdoc, Giuseppe Mazzardo, we simple, suppose we have a universe which consists of one type of particle which is its own antiparticle, and let's suppose, just to make it a little bit non-trivial, that the particle A occurs itself as a constraint cell that First of all, analyticity. Now, this is one important consequence of factorization that I didn't tell you about before we talked about integrability and factorization, as there were still these branch cuts, but since 3 to 3 scattering is related to 2 to 2 scattering, those branch cuts, S2 now, and unitarity, that's a very simple example just to show you that not everything that you get, but you can do it anyway.

45:00 This is all in one just one image. So, I'm sort of coming to the end of my talk because the interesting question quasi-philosophical point of, if you write down these S matrices, they're not unique. We have to make an assumption of course, given that they're the sort of question of what theory is. Well, first of all you can ask, is this a meaningful question? So, this is a very...

47:30 And if we were talking about the S-matrix of the universe in 3 plus 1, I think it would be a meaningful question because if 2 had been right and there was one unique S-matrix which describes the universe, we don't have to ask, but in the context of some condensed matter systems, some quantum spin chain, what chemical composition does it have? On a more formal level, we can ask, is this the estimator? Are you going to write down some Lagrangian for that quantum field theory? Or some lag? We can start with the turbid, too.

50:00 And so, there's lots of very sort of, there's a lot of questions. I mean, basically, you want to write down what they estimate. And the only way that people will do this is to guess what they estimate. Try to understand what they might be able to show themselves. So, depending on what those QNs are, that places all sorts of constraints on what the possible S could be. So, this is a guess. But then, given that S matrix, the way is to actually, from this S matrix, try to deduce the option.

52:30 Try to understand how this theory will behave, because once one knows the S matrix, or the guess of these points, this goes on not to be assigned. And then once one has that, one can try to compare either of these things with either sort of things that you expect to be true model or... So I want to stress that the steps from here to here are there. So really, the starting point in... And so what I'm trying to say is that you should think about dimensions of them as... On-shell quantities, the particles that do this scattering are real particles, on-shell, to things like Green's functions of local operators. And the way that one thinks about doing this in this sort of approach is to go through, first of all, calculating the form factors,

55:00 And one of these aspects of sum. Assume that is easily computed because all you do is to insert a complete set of states here and then keep high space here. So the big question is how to use form factors. And it turns out that within quantum field theory you expect that there are equations that determine the form. Form factors. For example, this is the two-part tool. Determine the form factor in terms of what happens when you interchange these two parts. Crossing. So, these are sometimes called Watson equations. Overlooked, and I've studied. And in 1 plus 1 dimension, you have a form. So these are the analog of the equations which specify the S-matrix. Except now notice that they're linear. That means that they, that the solutions span a linear space, and in the few examples that we've actually, Smirnoff has actually, I mean, he sort of assumes these, from that you can show that the, so, so it gives us, so, conclusions, it has predictive power.

57:30 Could something turn on the lights? Let's see. No, I don't think so. I mean, the point is that the string is a 1 plus 1 dimensional object, but it lives in the space of however many dimensions. 10 dimensions or whatever. So you can calculate, from the knowledge of the string amplitudes, you can calculate S-metrics on 10 dimensions or whatever.

1:00:00 But you can't think of string theory in terms of a field theory, a worksheet of conformal fields. You were mentioning about these chains, you see, which are one-dimensional objects which are in a three-dimensional space. Yeah, but they're not strings. They're fixed in a crystal. They're very definite, rigid objects. They're one-dimensional just in the sense that they don't to the other chain. The axiom system is a very natural question. Similarly, in your 1 plus 1 dimension, you suddenly sprang interoperability on it. Isn't it reasonable to ask whether there is some physical underlying reason for that? Those are two separate questions, right? I don't see why the axioms of S-metrics theory are any more unreasonable than, say, the Whiteman axioms or something like that. In the end, they started to become a bit artificial, I agree, and that was connected with the fact that it doesn't really work. If you look at the kind of axioms of S-metrics theory that you need, then they have simple...

1:02:30 Simple. The second part of the question was yes, of integrability. And so the answer is that, I mean, I don't know how to go outside that. There is, there are some arguments that say continuum field theory that you call simple continuous, one couplet, renormalizable in 1 plus 1 dimensions, that does correspond to some kind of, but there are exceptions. The condensed matter systems are, of course, only going to be approximately integrable and so to the extent that our integral model is sort of close to the real physics than the predictions that we make and experimental data. The matrix has predictive power. Now, my impression is from your presentation is that you sort of force this, that you start with experimental results, then you create a model, then you construct a matrix.

1:05:00 Today, aren't you sort of forcing this predictive power? Well, we're not assuming, well, what I mean by something having predictive power is that we take There's one lot of experimental data from that we predict further experimental data. So if we study a particular quantum spin chain or something and we measure its interactions and we find that they're of a particular form by doing some kind of... and we find that they're... and then we go ahead and we solve this one. Yeah, well, the probability is just one. You're absolutely right. So actually, if you did find, but the S-matrix itself is a pure, is a phase shift. There's a time delay.

1:07:30 Is there a 2 plus 1 question? One of these homologics, and they actually apply to the scattering not of particles but of rods moving about in the plane. So you imagine infinite rods at different angles and they move about in the plane. Of course, I never quite understand this because they always intersect each other, but they seem to describe these kind of weird objects. And presumably, somehow, planes in 3 plus 1, though I'm not sure that anyone's actually working out. David? I'm just slightly confused. It's a quick clarification question. What is it in the 1 plus 1 dimensional case that prevents there being two particles coming in, four particles going out? That's the integrability of the model, that if you have, say, and it's not, also, it's not allowed for two particles.

1:10:00 So effectively, the integrability is taking away a lot of the field theoretic character. Yeah, it's taking, oh, okay, you could say this, that this is a very special, there's no interpretive way of non-integrable one-plus-one-dimensional theory, it's just... Don't give it... John, could I ask two vague questions? One is just a clarification, really. Your slide which said this is a meaningful question and what defines a QF... When you had your list of approaches, were you saying one just needs to... So were you saying that Faber's one of those... I was being brave there. These are telling another theoretical physicist, not a mathematician, but another theoretical physicist, which he is talking about.

1:12:30 So, for example, that S-matrix quantized that Lagrangian, but it removed the regulator from... This is... Is that true for the other items in the list, like... I mean, and that's why we'll get into the next slide about basic games. Well, at least with a lattice, you might think... Yeah, yeah, sorry, but quite a lot of the theories that we...

1:15:00 Tell me what that question... It may be that the theory has a classical... The problem with these S matrices states with many particles in them, or something, if they're bosonic. This is very weird within the... whether they... they go towards the semi-classical... Do you not use conservation laws? Oh yeah, but we use them, we assume that conservation laws have come from somewhere. Yeah, we don't necessarily know where. I mean in some cases we can, as was said, we can sort of write down the theory. The classical theory has classical conservation laws. The cross on brackets vanish and then there's a consistent way We could start from some, as I said, start from some conformal field theory which has an enormous number of conservation laws and then switch on the mass somehow and then argue that some of those conservation laws. Well, not these kinds of conservation laws, because these are very funny, local, I mean...

1:17:30 Yeah, energy and momentum conservation is not a problem, that's all I said. Oh, well, it's not a justification, I think it's an axiom of... There is something in the translational invariance of the S-matrix and the notion that the S-matrix is an operator and you just check it commutes with spatial translation. Well, spatial translations don't act on that. I mean, translational invariance just means that the momentum All of these are commutes with Hamiltonians. Yes, so, I mean, and I was wondering, Tyler, the role, what role do symmetries play in the scientific theory? Right. Well, they play a great deal, right? Because they... Okay, there are global symmetries, which maybe is what you're referring to, right? The global symmetries, you know, like SU or something, The particles transform according to irreducible symmetries, and that really comes in when you're trying to commute the flavors of these particles. So the space-time symmetries, these are really generalizations.

1:20:00 I think one theme of the recent discussions is that in a way, during the second phase, yes, one example, So this is one reason why a particle physicist justifies granting A and C.

1:22:30 And they had, if there were, you said some of the, in the late 60s, the four or five we saw are the only ones taken. Once you have the integrability, then saying that there's things, I'm surprised that in the context of estimate, maybe there might be some, there's certainly some non-integral theories that we can still sum very limited. One of my theoretical physics colleagues here has...

1:25:00 Yes, so even those equations which I wrote down in that very simple, I found those equations and I said that was the solution. Nobody asked me whether the answer is it's zero. You go through this process of an empirical science. Yeah. So there needs to be another principle there, one of your extra principles that you take the...

1:27:30 Yes, I mean on this it's in no way embarrassing. You seem to and you want to know. But the only real nature is free space.

1:30:00 So if you need to know the function in a relevant way, rather than on an expert, just walk out the door. So, it's the exact same thing. Yeah, I guess that's what triggered the departure. I seem to remember from my undergraduate studies that you actually just need to know it with an interval. Right, so then it's all right. You see how I was wondering about knowing n. Thank you very much. I think we're kind of satisfied and we wish you very well for the next duration. Right, well, I should say, once again, that there are many other people who are probably more exposed to this than me on this, but it's something that I've sort of kept an interest in. I hope there weren't too many questions. Perfect for us, so thank you very much.

1:32:30 Thank you for your attention. As infinite sums of homo-physiological space, there are people involved in the media from Monday through Friday through Saturday through Sunday through Sunday through Sunday through Sunday through Sunday through Sunday through Sunday through Sunday through Sunday through Sunday through Sunday. Well, yes, I mean, I think, even if he's not going to tweet me, I'm sure. Thank you for your attention. I just want to give one more look at that stuff. Last week it was a lot. It's been a happy few days to go past each other, so I just wanted to throw in a few things to you. I don't know if you want to do it, but I'm just going to have a discussion about it tonight. This is what I think is going on. There are a couple of them. One of them is . That's a good talk, a good discussion.

1:35:00 Well, I mean, in mathematics theory, I mean, the thing with this is that if the three plus one plus one plus one plus one plus one plus one plus one plus one plus one plus one plus one plus one plus one plus Many of the questions you've expressed have actually been discussed in one of the comments that we've had in the last year and a half or so. But I found that there are many principles that I find interesting. In his own way, in a particular way, from a physical, natural, analytical point of view, talking to a second scientist at a loss about the fact that he is a scientist, I wish I could make a change to that. It's not what you throw at the board. It's not what I was going to do. You weren't going to do it. Yeah. You weren't going to do it. It's not what I was going to do. It's not what I was going to do. It's not what I was going to do. It's not what I was going to do. It's not what I was going to do. It's not what I was going to do. It's not what I was going to do. It's not what I was going to do. It's not what I was going to do. It's not what I was going to do. It's not what I was going to do. It's not what I was going to do. It's not what I was going to do. It's not what I was going to do. What are you going to use your education in my conference? Oh, because we're happy to be using the standard approach, but the standard one is that it's all kind of a waste, OK? We're not. I mean, that's where we are, but I mean, we're not there. No, we're not. There's no such thing as a standard approach. It's been around for every single one of you. I don't know if you've heard of it, but it's been around for every single one of you. And, and, and regard it as a pick-up point. You've got to approach this like a pick-up point. And I think it's repetition. Well, I think it's repetition. I'm sure it's repetition. And that's much better than mathematics, because it's a few sectors in it.

1:37:30 In fact, it's not quite as good for the entire history of mathematics, which is a part of the history of mathematics, especially algebra, physics, and many other parts of it. It's clever, isn't it? It's not clever. You know, it's so, so, so, so, so, so, so, so, so, But we do have quantum physics, and we haven't checked it yet. Which means there's a true way to do it, in some sense, not very much, but it's a little more proximate at the same time. And that's the hard part, isn't it? Yes, that's the hard part. Otherwise, if you want to, it's not very much the entire thing you need. There's a lot of rubbish that doesn't even track the system to do it. Well, there is. I mean, try it. What I wanted to do, I think, was a starting up. I think it's a good way to do it, because then they find the model, and so do you. Yeah. I think that's a good way to do it, because I want to do it, and so do you. Thank you very much for your time. It's always here at 4 on a Thursday. It will also be a probability next week though.