 Mae'r jagwch yn dda, mae'n dweud hynny'n gweithio yma, yn gychwanedraethol, faith ein cyflwynth ar yr pensioedd cyfyddaeth gyda chi, fel rydych chi'n gweithio'n gweithio'n gweithio'n gweithio'n cyflwynth ym ni oedd eich cyflwynth arall â'r cynharu gymhau teidinig ychydig sy'n gallu gyrhau'n gweithio'n gweithio'n gweithio'n gweithio'n gweithio eich cyflwynth, but more broader than that, he's someone who has influenced larger areas of theoretical physics and corresponding areas of mathematics for more than 30 years. This ability to apply powerful methods £y Otherited Write-in-ğath Watch- With Modern Mathematics £y Otherока to solve problems in physics is quite unusual, but what makes him even more unusual is his makes him sort of unique. is his ability to take ideas in theoretical physics and develop profound new insights in mathematics from these ideas in theoretical physics, and that is really unusual. For those of you who are not physicists and don't know the details of his work, let me perhaps just quickly list a few of the things that he's played a role in. His work spans large areas of theoretical physics, quantum field theory, string theory, quantum gravity, and particle physics in particular, and a general feature of his achievements is his mastery of what's called the Feynman path integral that defines quantum mechanical systems in a quite general way, and this led him to develop mathematical ideas which also have importance in physics such as topological field theory as a tool for understanding the vacuum structure of quantum field theory and string theory. His ideas on Churn-Simon's theory have many physical implications but also profound impact on knot theory in particular for which he was awarded the Fields Medal in 1990. I think he's probably the only theoretical physicist who's been given this Premier Mathematics Award. Of course I call him a theoretical physicist, mathematicians would call him a mathematician, so it's a matter of point of view. He's been a pioneer in so many things, I can't possibly list them all, however overarching ideas are based on the use of supersymmetry in physics as a way perhaps of unifying the observed particles and their forces and also as a tool for understanding subtle properties of quantum field theories, so for example this led him and Natty Cyborg to develop what is called Cyborg-Widden theory as an extension of Donaldson's theory which is of great importance in topology. Despite these many contributions at the frontier between mathematics and theoretical physics, I think of him, and we can ask him what he thinks of himself as, but he's spiritually I think he's a physicist, and he's certainly contributed to many very concrete ideas in physics, in various areas of physics such as in particle physics, cosmonogy, and in fact his roots are way back in the mid-70s in important work on high energy scattering of photons for example. It's obvious that he's received many awards and I tried to make a list of these awards before this talk and I decided that, I mean you can all look him up on Wikipedia and you'll all find the same list, and it's such a long list that it's hardly worth repeating it. In addition to the Fields Medal, he's been awarded the direct medal of the International Centre of Vertical Physics, Danny Heinemann Prize, a National Medal of Science, the Hponcho Prize, and many, many others. So I won't repeat them all, but from what I've just said it's sort of obvious that it's appropriate that Edward should be awarded the Newton Prize because it's a prize in the name of a great physicist who was also somebody who had made huge contributions to mathematics and that makes it very appropriate that Edward should give this lecture. Thank you. Well, thanks Michael for the very kind remarks and thanks for the Institute of Physics for selecting me for this honour as well as for the invitation today. I'm very pleased to be here to have a chance to give this lecture and I'm also looking forward to changing the page because it contains a small misprint that's been pointed out. Now what I'll really be explaining in this talk is how over the last 40 years string theory has repeatedly forced physicists developing it into directions that they didn't anticipate and frequently didn't want. So as I've written here I'll be describing how string theory has grown beyond anything that anyone conceived at any step along the way and broadly speaking I'll be talking about three episodes. I've tried to distill it into three episodes where this has happened. My purpose isn't mainly a historical retrospective. The main point is that rather that I believe that this process in which a theory that physicists have stumbled on but only understand to a limited extent forces us into directions which we don't anticipate or really fully appreciate going in is far from finished. And we still can't really conceive of the end of the journey. I'm actually at best only semi-qualified for a historical account for at least one obvious reason and it is simply that there are a lot of the developments I didn't participate in. So a lot of the important stages which I'll tell you about are things that happened in some cases before I was even a graduate student but certainly before I was involved in developing the ideas. As a partial excuse I'll observe that few people have participated in all stages of this process as by now it's gone on for more than 40 years. In my case I wasn't directly involved until the early 80s or if you were generous a few years earlier and a lot of things had happened by then. And since I wasn't involved then I can't say what I'd have thought at the time of the idea of inventing a whole new theory by concocting a scattering amplitude out of whole cloth. So this scattering amplitude was the Veneziano amplitude and the details of the formula don't matter that much. S and T are basically relativistic versions of energy and scattering angle and Veneziano guessed a formula that was supposed to be an approximate formula for the scattering of two pions, elastic scattering, which had magical properties. It proved to be the starting point of a completely new theory although that was certainly hard to imagine at the time. I can't say what I'd have thought about it at the time. I can only say that looking at it in hindsight it seems like an utterly astonishing starting point for getting something significant. This is a rather complicated formula after all gamma is the Euler gamma function which is, I mean it's been known for centuries but it's still a rather esoteric function. And such a complicated formula as on the right hand side is usually the output of a calculation where you start with some simple equations and solve them and you get a complicated answer. You don't usually guess a complicated answer and get anything useful out of it. So I think it's surprising that that was useful. Now there's a reason that this formula was invented. It had to do with hadron physics. In other words the hadrons are the strongly interacting particles, the familiar ones like the proton and neutron and slightly less familiar ones like the pion, the kion, the delta and so on. So by the time this work was done in 1968 the hadron resonances had proliferated at particle accelerators. And this had actually made some physicists despair of describing the nuclear force that is the strong interactions within the known framework of local quantum field theory. But you can't get anywhere with only a negative assumption that there aren't going to be local fields. You need some kind of positive idea about what there is instead of local fields. And there was a positive idea that led to the Veneziano amplitude. But it's again a rather mysterious idea. The idea was that since there were so many hadron resonances when you scatter for example two pions or for example a pion and a proton to mention something which is more practical experimentally, there are many possible resonances contributing to the scattering process where you create an unstable state and then decays. There were so many resonances that it was at least imaginable that resonance scattering, the creation and decay of resonance was most or in the extreme case all of the scattering amplitude. So the following question was asked. Can a scattering amplitude that's given by a sum of resonant contributions only be the whole answer? So again this is a formula whose details don't matter that much. But A is the scattering amplitude that depends on the energy and the scattering angle. When you have a resonance you've got an energy denominator like in non-relativistic quantum mechanics where the scattering amplitude has a pole when the center of mass energy equals the mass of the resonance. And then the numerator is a Legendre polynomial. That's what pn is in the numerator. But anyway you can write a formula for scattering due to resonances and in normal physics it's important but it's never the whole story. In fact with only finally many resonances it can't be the whole story because that would contradict general principles of scattering theory. If you only have finally many resonances then the amplitude would actually be a polynomial in T but it's not a polynomial in S, it has poles. And that would contradict principles of relativistic quantum theory. But experiment showed that there were so many had-on resonances that people imagined there could be infinitely many which is more or less the still-the-view point today. And with infinitely many resonances you could at least imagine that resonance scattering might be the whole answer. So again since I wasn't there I can't say how well motivated the question was at the time. I'm curious about that but I really can't judge it. And it's, although certainly I know participants it's kind of hard to... It's hard to know from what people say well I would have thought at the time about how well motivated the question is. However in hindsight it certainly proved to be a fruitful one. So first of all the answer is yes. I didn't state all the technical details about the question but properly interpreted the answer is yes. That's what Finesiano showed with his formula. With infinitely many resonances, resonance scattering can essentially be the whole answer. And the theory that had that property turned out to be much more than just a funny formula that was written down out of thin air. For one thing it led to a physical picture. Trying to understand that formula led to a physical picture where a meson was understood as a little string with charges at its ends. So in this picture the meson resonances which are the poles in the Veneziano's formula are vibrational states of the string. Just like a violin string, this kind of string has got many states of excitation, many states of vibration. For a violin string those are the fundamental note and the higher overtones which fit together to give it the richness and beauty of music. Here the many hadron resonances, the many meson resonances which motivated Veneziano in the first place were ultimately understood as vibrational states of a string. So that physical picture that I just stated that a meson is a little string with charges at its end is now believed to be qualitatively correct as a description of strongly interacting particles. The charges are what we now understand as quarks. Or at least it's qualitatively correct as a description of the mesons. The story is a little bit difficult for the barions like the proton. But at least in its simple versions it's only qualitatively correct. Now about five years after the Veneziano amplitude a competing theory of the strong interactions emerged which was far more successful it overcome many problems that were not tractable from the point of view of the Veneziano amplitude. So much so that for some years string theory went into eclipse. Now later developments showed that what appeared to be the failure of string theory as a theory of strong interactions wasn't the last word. Ultimately string theory has taught us a lot about QCD most recently by shedding light on heavy ion collisions at the rick accelerator at Brookhaven and there's probably a lot more still to come. The modern view is actually that QCD may be equivalent to a not yet discovered string theory. But even though that's true it's not the main reason that string theory survived from this period when it appeared to go into eclipse as a theory of strong interactions. The main reason that string theory survived was not that it has some relevance to strong interactions it's that the idea that led to it was actually wrong for the strong interactions but it was right for something else. The physicists who started string theory had disparate of the applicability of conventional local field theory to the strong interactions but they were wrong with that despair. The success of quantum chromodynamics showed that the strong interactions can be described in the already standard framework of local relativistic field theory but there's another problem in physics perhaps a bigger problem for which this idea is right. That's the problem of quantum gravity. Both gravity and quantum mechanics are part of nature so there should be some way of making them work together but the nonlinear structure of Einstein's theory of gravity makes that very difficult. Quantum mechanics and gravity both exist but it's hard to make sense of quantum gravity by the usual algorithm of quantizing classical fields because if you take the classical field to be the gravitational field in the form described by Einstein you'll find out that you run into a maze of infinities when you try to do the quantization. That's a result of the nonlinear mathematics that Einstein's theory was based on. Now, in the context of gravity there can't be a gauge invariant local field 5x where phi is the field and x is the spacetime point for a reason that's so basic in elementary that it's actually easy to miss. The reason is that x itself isn't gauge invariant. Einstein's gauge symmetry, the principle of general covariance, invariance of the theory under diffiumorphisms of spacetime, the gauge symmetry exactly acts on x and therefore it's impossible for a local field 5 that depends on x to be a gauge invariant concept in general relativity. So a theory of quantum gravity is actually not going to have local fields that are functions of spacetime as we have in other branches of physics in electrodynamics or in the strong interactions. This was, as far as I know, first pointed out by Breistowit in 1968, which coincidentally was the same year that Financiano wrote down his amplitudes. Though again, as far as I know, it was decades before the two facts were related. In view of Breistowit's observation, a theory that has gauge invariant local fields cannot describe quantum gravity. Of course, there's no immediate converse to this statement. Just because you don't have gauge invariant local fields doesn't mean you're going to have quantum gravity. So the inventors of string theory wanted a theory without gauge invariant local fields because they mistakenly thought that that's what the Hadron residences were saying. But the fact that they did without gauge invariant local fields certainly didn't ensure that they were going to get quantum gravity, which in fact they didn't want to get because that wasn't part of the world of strong interactions. But it's what happened. So from rather early days, it was found that string theory generated massless being one particles from open strings, which had properties, I called them gauge-like properties, properties similar to what we have in the standard model of particle physics, and that for closed strings, it generated massless been two particles with graviton-like properties. This was considered bad. Well, the graviton-like aspect wasn't completely formulated in the days when this was considered bad, but it was considered bad because string theory was seen as a theory of mesons, and they're scattering, and in the strongly interacting world, there aren't any massless spin one or spin two particles. So, since the theory was supposed to describe the strong interactions in which all the particles are massive, the fact that calculations forced massless particles of spin one and spin two into the theory seemed like a drawback. So, if people tried hard to get rid of these massless particles, dozens or perhaps hundreds of papers were written, tried to get rid of them, and that effort failed. That's how we generated, developed eventually confidence that they really are a part of the theory. People who didn't want them tried very hard to get rid of them and failed. But eventually in the mid-70s, some physicists became daring enough to propose that string theory had been misinterpreted, that the strings were much smaller than had been assumed. Not just a little bit, but many, many orders of magnitude are smaller. And that the theory should be used to describe quantum gravity and other fundamental forces. Again, I'm only giving a second-hand summary since all this happened before I was in the field. So, and not for the first or last time, the theory veered in a direction that wasn't foreseen by those who had gotten the ball rolling. Now, there actually is a simple picture that helps explain why string theory has a mind of its own. It has to do with what happens to Feynman diagrams in going to string theory. Feynman diagrams are the pictures of spacetime processes that Feynman used to describe elementary interactions originally of photons and electrons in quantum electrodynamics. So textbooks will tell you how to technically get a number out of a Feynman diagram by a calculation in momentum space. But Feynman actually visualized it more intuitively as a spacetime history. Here, space runs from left to right and time runs vertically. A line here is an orbit in spacetime that represents the trajectory of a point particle. So at each time, the particle is somewhere. Later on, it's somewhere else. Classically, the particle would travel if it's a free particle in a straight line, according to Newton's laws. But quantum mechanically, according to Feynman, we have to sum over all possible histories. A completely generic history with the too messy to illustrate, but the fact that the lines in that picture are slightly wiggly just gives us a hint of quantum mechanical fluctuations. Then according to Feynman, there are spacetime events that here are labeled x, y, z and w, where one particle breaks into two or two recombine into one. And in this example, I've shown two particles coming into the past that interact in a certain way. It's called a one-loop diagram. And two go back out in the future. So assuming the particles are all of the same kind, I haven't labeled the diagram in enough detail to make clear if that's true or not, the diagram on the left is a Feynman diagram that contributes to elastic scattering. Just the same process that Veneziano had been trying to describe when he introduced his famous amplitude. On the right, I've indicated the corresponding picture in string theory. So for definiteness, I've indicated a closed string. So at a given point in time instead of a particle at a definite space, a point in space, there's a loop. Again, these loops are undergoing all kinds of quantum fluctuations, although I've drawn a relatively regular picture so that we can see it. But at one moment in time, the loop is here. At a later moment in time, it's here. The loops branch out and recombine just like the particles do. But there's a fundamental difference. Here's the string picture, and it has the property that what's called the world shade of the string, which I've drawn here. It's mathematically, it's completely smooth, two-dimensional surface. So any small piece of this picture looks like any other small piece. If you look at the whole picture, you can see that globally two strings came out in the past, branched and rejoined, and went out into the future. So that some kind of interaction happens. But if you look at any small piece, it looks like any other small piece. And locally, you can't say whether any branching or rejoining was happening. By contrast, in the traditional Feynman picture, there are definite spacetime events labeled x, y, z, w, where something definitely happened. If you draw a little circle around this point, it looks different from a little circle around a point where there wasn't a branching event. So those branching events are called the Feynman vertices. And those are the spacetime moments where something actually happens in conventional quantum field theory. While when we go to string theory, there's no definite spacetime moment at which anything happens. Globally, something happens, but you can't make any sense of the question of when and where anything happens. So this has a lot of consequences of which I'll give you a first list of three. The first is that the infinities of Feynman diagrams disappear. So Feynman diagrams undergo infinities, which arise when these distinguished spacetime events coincide. So for quantum electrodynamics, Feynman, Schwinger, Tomanaga, and Dyson, and so on, showed how to overcome the infinities by a process of renormalisation. One always wondered if one could get rid of the infinities a priori instead of renormalising them away. But anyway, pragmatically, renormalising them away gave a theory that worked well. It made sense and agreed with experiment. But it didn't work for gravity. If you take the Feynman vertices of general relativity, you find that the infinities that arise when x, y, z and w coincide are not renormalisable. They can't be removed by the procedure of Feynman and his contemporaries. So the first good thing that happens in going to string theory is that because we've got rid of x, y, z and w, we get rid of the infinities that arise when they coincide. So the infinities of Feynman diagrams disappear, and whatever it's going to be, the theory is not just renormalisable, but actually finite. The second important fact is that what the interactions are is not up to us. They're fixed once we pick the string. In other words, in this picture, well, in quantum field theory, you first pick the particles. For example, Feynman had electrons and photons since he was doing quantum electrodynamics. Then we pick a Lagrangian which determines the interaction vertices. So Feynman had a particular vertex, but if he believed that the electron had had a different magnetic moment, he would have used a more complicated vertex. Specifying the vertex as well as the particles is what it takes in this language to describe what the theory is. But for the string, since there's no vertex, once we know the law of what the string is, what a little piece in the picture means, the interactions are determined. The interactions occur globally when we put together a lot of little pieces which under the microscope look like they just describe locally propagation of a single string. So once we know what the string is, the interactions are up to us. We don't get to decide what they will be. So that's on my initial list of three consequences of that picture. The second one is that we don't get to pick the interactions. Once we've picked the string, the interactions are fixed. And the third point is that there's no definite notion of a space-time event. Well, what I'm telling you is really the beginning of that fact. So the prototype of a space-time event in the conventional picture is the interaction event of elementary particles. And that disappears when we go to the string picture. And that's a first hint of what turns out to be the situation that in conventional quantum field theory, you can make sense of a point in space-time, a space-time event in Einstein's sense. But in string theory, when you look closely, that concept isn't really there. So the last statement means what I've given you is only a hint of this fact, but deeper investigations from this starting point show that somehow the notion of what space and time is at short distances is quite different in string theory from what it is in familiar physics. So physicists, without originally having any intention of trying to get there, arrived at a theory that, first of all, is completely finite since the interaction vertices are missing. Secondly, it's got a massless spin two particle since no one could quantize the string without getting one, even though they wanted to and tried hard. And three, has its own mind concerning what the interactions will be. The interactions turned out to be the interactions the theory chooses are those of general relativity, plus corrections at short distances that are irrelevant for astronomy and impractical to measure in practice, but and preserve the general structure of Einstein's theory. They're needed for the quantum consistency and they're generated by the theory. So you get not literally general relativity, but you get a theory that looks like general relativity where general relativity is accessible, and it preserves the general structure of Einstein's theory. So that was a first set of consequences of replacing the Feynman diagram by a string picture, but there are many other consequences. I think I'll just mention two more, or three more, sorry. So the first one is that there are far fewer theories than in standard quantum field theory. So in standard quantum field theory, you decide what the particles will be and then what the interactions will be. In string theory, you only get to decide what the string is. And that's subject to very delicate consistency conditions that actually look impossible at first, but very delicately, they're obeyed, and when one learns how to do it, one finds that there are five ways to do it as counted in the mid-80s, but only one is understood today. Another thing that happens to you is that the theory picks the spacetime dimension. So Feynman took from experiment the fact that he wanted a four-dimensional theory. We see three dimensions of space, but since Einstein, we also count time. But a theory like Feynman's could have been developed in a different dimension. It works better in some dimensions than others. Going up in dimension causes trouble, going down makes it easier. But different dimensions are imaginable. Instead, in string theory, the spacetime dimension is picked. And I'll get to that more in a second, but I'll just say as a teaser that the answer didn't turn out to be anything that anybody wanted going in. And a third fact is that the theory picks its own symmetries. We don't pick them. Once we get going, one of these five-string theories, or only one is later understood, what the symmetries are going to be is out of our hands. We have to calculate one off to determine them. Well, the spacetime dimension turned out to be 10, which nobody wanted going in. In fact, at first, it probably sounded like a joke, although, again, this is a joke that played out before I was in the field. So I'm not describing that firsthand. But the joke turned out to be a blessing in disguise when the theory was reinterpreted as a candidate unified theory of all particles, the extra dimensions gave room to derive the complexity of the real world from a simple starting point. You see, a violin string has higher harmonics. They're labeled by a single integer, the number of half wavelengths that fit between the two ends of the string. That's interesting, but the real world of elementary particles is way more complicated. Electrons, muons, neutrinos of three kinds, teleparticles, many different flavors of quarks, W bosons and photons and so on. To get the real world of elementary particles by the vibrations of the string is actually only possible because the extra dimensions give room for more complicated forms of vibration of the string. So, I mean, nobody ever sat down in an ivory tower and said we're going to solve the problems of field theory by introducing strings and we'll have extra dimensions which will give us room to get the complexity of the real world. It just was forced upon people who didn't anticipate where they were going in the context of trying to make sense of the trail that was started with the Veneziano amplitude. Well, as for the symmetries in elusiating them, first of all, physicists rediscovered general covariance and gauge symmetry. General covariance is the fundamental concept in Einstein's theory of gravity and gauge symmetry is the bread and butter of the standard model of conventional particle physics. So, these are symmetries that were already known, although gauge symmetry in the sense of non-milion gauge symmetry was a comparatively new concept at the time. But in elusiating the symmetries, physicists also discovered a new type of symmetry, supersymmetry.