Conversation on further European integration incl Sie C Powell & R Lubbems

0:00 Are you stupid? I'm down there. So it is a real privilege to have somebody who started this whole thing off to come and talk to us and to talk about the history of it, to talk about the plan of heading for, to talk about the future, to talk about the significance of this, and then I encourage that to be done. I have far more transparency than I need because I'm not quite sure what I'm going to say here. I'm not quite sure of the expertise of the audience. I've been told that you are not expert but not afraid of technical matters. So I assume you know nothing but will learn very quickly. And, as Professor Laird had said, it will be about simple string theory and its ups and its ups. I don't know it now! The subject is difficult to describe partly because it's, actually, it's not a big subject. It has origins in a lot of the work on quantum field theory and particle physics, which I can't possibly, obviously, review in this talk.

2:30 And also it's an exploding subject. What I'd like to convey in the latter half of the talk is that there is a tremendous excitement in the community. This dates back now maybe a couple of years. It's an ongoing generation of ideas which come out every day. We all use email archives. We all communicate. All papers on the subject and related subjects get positive opinions. And an electronic archive, which then gets sent to us each morning, and you wake up feeling worse, you log on and you sell them on 15 or 20 papers every day, several of which might well be doing just what we thought you were doing. And it's not, this is actually not just a lot of people repeating the same work, but a lot of incredibly interesting results, and in fact, just today, there seems to me to be something fairly stunning. So this is a subject that's certainly moving, and one of the reasons why it's so fascinating is because although it's obviously moving, we all know it's moving, we can see the developments, everything looks very interesting, we don't actually know where it's going. The aim of the subject is to try and, in some sense, to understand the underlying physical laws, of course, to find a theory which some people have called, third, everything, which is part of the reason for my title, because I'm even mistaken. The other thing we quote here obviously makes some allusion in front of this audience is to present what is known in the way in which we, rather than maybe deeper, I mean, I will give you essentially a set of rules for talking about this kind of physics as it's developed in 3D without making, without placing too great an emphasis on what they mean rather than deep.

5:00 More than that, even if what they meant was staring us in the face, I would like to spend a couple of transparent days just filling in some background just to show where the subject is in the structure of particle physics. So this summarizes basically what particle physics is about, what it has been about for the last 20 years or so. And there are basically two motivations for looking at a theory like the string theory. One is that we would like to have a unified theory of the forces and particles. That has developed an extremely accurate and ever more accurately tested model of all the forces if you ignore gravity. That's the so-called standard model. And the standard model builds in a theory of the electromagnetic weak forces and strong forces. Build them in a framework, put them together, and if you put in the particles that we see, namely the quarks, the electrons, the muons, the phenol, etc., all particles which are detected in particle accelerators, then this combination of theories, which I hesitate to call a unified theory, is an incredibly accurate description of what's seen. In fact, in the last year or so... The top quark has been discovered. The top quark was discovered partly because the accuracy with which this model was now tested was so great that one knew exactly what mass the top quark had to be. And this wasn't based on vague approximations of the theory. This involved doing the kind of quantum loop calculations that used to be done in electrodynamics, which were very accurate in that theory. Now these theories, the electroweak theory in particular, is really well established for that, which is very accurate.

7:30 So these are beautifully established theories which 25 years ago didn't exist, but in their own right, of course, it's a huge understanding of the subject. However, this isn't really what we're after in terms of fundamental theory, because to start, in order to fit a large amount of data, one has to input the species of the law, the masses, charges, and so on, and therefore it doesn't really give a fundamental understanding of These are the kinds of things we want to know, and furthermore, it doesn't even unify these two forces together, so it puts them in the same framework. Attempts to unify those forces go under the grand unifying theory, and I may mention that a bit more later. So I call that the aesthetic problem. One of the problems with the standard model, in fact, the major problem at the moment with the standard model, is that there isn't a single piece of experimental data which suggests that it's wrong, and until we have such a suggestion, we won't know how to go beyond it. We won't have experimental tools as to how to go beyond it. So in a sense, it works too well, although there are ideas, for example, if one could only measure a mass of a neutrino, one would then have a key to what goes beyond the standard model. But, quite apart from that, most of us feel that there's a problem because this model ignores mathematics, and the central problem in this kind of theory of physics seems to be, and has seemed to be for the last 60 years, the question of how you can build a quantum mechanical theory which encompasses the ability of quantum mechanics. And, luckily speaking, most of you do. Realize there appears to be a rather fundamental conflict between the usual way in which quantum mechanics is talked about and the usual way in which quantum field theory is discussed amongst mathematical physicists. Mainly, if you try and just apply the rules of quantum field theory to general relativity, you end up with disastrous divergences. Those divergences in some sense originate from the fact that in quantum theory of gravity you should not be treating the space-time background As something that's non-dynamical, that's part of the moon, in some sense, there has to be some dynamical, some theory which includes the dynamics of space-time, so that if you just ignore that, you are led to divergences at scales, because of ultraviolet fluctuations, at scales which are on the Planck scale.

10:00 And of course, string theory is only one approach. I can't imagine myself if any of the other approaches are going to make much headway. Because, as I'll show you, at least in string theory, the indications are that there's a huge, deep underlying geometric structure that is needlessly understood in order to understand how to discuss physics in the context of that study. So, that's just to tell you what kind of physics string theory is meant to... What is string theory? Well, I gave this talk ten years ago, and I think probably I did give this talk ten years ago to the same community of people, and I probably started by defining string theory and its transparency. This transparency defines the whole of string theory in a certain sense. Very simple transparency, because the idea behind string theory is very simple. But this is string theory as it was ten years ago, or as it was more than two years ago. In this way of thinking, it's just a generalization of quantum field theory, trying to think of quantum field theory as a quantum mechanical version, relativistic, of point particles, and string theory is sort of the obvious generalization, where instead of talking about fundamental objects and points, you think of them as having extension in one dimension, so that you get string-like objects. And then you can see immediately that there's a magical difference in the nature of the theory that I've described, because strings can vibrate whereas points cannot vibrate, and so that infinite number of possible modes of vibration in a string, every one of these modes has some frequency which determines its energy and just about the usual devoid of energy, so that there are, a single string carries an infinite number of possible, so that a single string looks like an infinite number of particles reclined by the masses.

12:30 The reason such an obvious-sounding idea hadn't been implemented many, many years before it was, is that it's not that simple to construct an extended theory of a verbivisible culture. This string is a verbivisible string in the sense that it's set aside. And that means that you have to worry about the fact that it's on a time frame and then you have to discuss what we mean by an extended object. And the theory has to be a very special theory in order for it to make sense as a theory related to this very object, but it does. And so the classical theory can be discussed and its notions solved exactly. And then you can pass the quantum mechanics in the standard way. So I'm telling you all this because I know that in the end that's probably not the right way of thinking string theory. And it's probably not the right way of thinking quantum mechanics. And it's certainly a strong feeling amongst people doing this subject that where we're heading, especially in view of what's happening this year and last year, where we're heading is towards a statement that the theory without quantum mechanics probably doesn't make sense. So the hope is to understand string theory, but also perhaps to understand something more about what it means to have a quantum mechanics theory in the context of having extended learning. But still, following my notes, I will just illustrate what I meant by saying there are infinite number of states. So here is a plot of the spectrum that you get by just taking a string-like object and asking for the system, formulating the system, the specialness of it.

15:00 And there's one kind that can be described. Now I put in parentheses here the word super. The theories which seem to be consistent and the overwhelmingly most important aspect of the recent developments in this subject, as well as in earlier ones, is the fact that the theories I'm going to talk about have a certain symmetry called supersymmetry. And supersymmetry is a symmetry which is a generalization of space-time symmetries as we know them, so the Poincaré symmetry. You add join, which is essentially about the square root of a thermionic symmetry, which transforms bosons into thermions and thermions into bosons. Aesthetically, we want such a symmetry. It gives us an origin for keeping the bosons. And supersymmetry was in the 70s precisely because everybody wanted it. As time has gone by, supersymmetry has become more and more desirable. We have no wish to require supersymmetry. In string theory, it seems to be a super-simple decision in nature would have to be detected in relation. We also have, there are units, what I'm telling you is that there are states, there are states of the string, so the theory, that characteristic scale, I said this was like a spin, or in other words, if you're not resolving this very short distance, then the string will actually look like that. What that means is that these masses are effectively infinite, and the theory reduces to...

20:00 Those masses are no longer important, you never excise them, and it's a theory which only has these massive states. But that is the first apparent miracle of the theory. You see, I didn't tell you anything about what forces I intend to describe. I didn't tell you anything about the mention of the word force. All I have mentioned is that I am doing a quantum mechanical treatment of an extended object. And lo and behold, you discover without asking for it. That it naturally contains, so this object is moving around in space-time, and it can exist in all these different states, but amongst the states it can exist in, is this set of states here, I keep forgetting this label, it can exist in any one of these spin states, but nothing higher, and as I say, there are a variety, like all of them, without any exception, that they have a massless spin-2 particle in the spectrum. In a supersymmetric theory, there's a part of that guy called the gravitino, which has But that wouldn't be expected, although I've drawn it here in math. In fact, what I've drawn here is the spectrum in the theory without supersymmetry being broken, as I mentioned earlier, in nature is that particle we have now. In fact, the mass scale is so enormous here that every particle we've ever seen in nature, like the heaviest particle we've ever seen, top core, those particles on this scale of 10 to the 19 GV would actually be massless anyway.

22:30 Could you actually weight with area after area? The Planck mass, I think, is 10 to the minus 5 grams. Pretty high. The particles of spin-1 are particles that we would like to have in our theories, because all our theories have our base theories, in which there are photons and Ws, that those are the same. And then we have spin-1 particles in our theories, like the quarks and electrons. There are a bunch of spin-zero particles, the Higgs particles, and this theory also contains a particle called the dimetron. So the kinds of particles that crop up here are just the kinds of particles that we want to have in our theories of nature. But they haven't been put in here, they have been, they are fundamentally there by virtue, for reasons that are a bit mysterious, of the fact that we are looking at quantum accounts that are extended on them. Excuse me, what are you saying, partly following on this question, what are you saying, the setting of the unit of mass, one being so high you won't ever see it, comes from the theory without... No, sorry, sorry, any theory can't contain the unit of mass. But what I'm saying is that... So you are pushing the string... The other way of saying that is that it doesn't matter what I put in the string tension, I think one can be given. I mean, I... Right. I'm just telling you about the interaction between strings, which is actually what determines precisely why these classes come to be. At the moment, I'm just giving you an overview of the spectrum of the string and telling you that it turns out that because of the existence of these massive particles, this theory contains, without any choice, both gravity and electromagnetism or something else involving spin-1 particles, like we've done with the bosons.

25:00 And when I say does it without having any choice, I should perhaps interject a little and describe it a little differently. String theories originated in the late 60s and early 70s, but brought together different terms. They were not invented to describe the force of gravity at all. They were actually, strings then were supposed to be strings of electric flux joining a quark and an anti-quark in a meson. Which had quantified force and the strings joining them were meant to be described by this kind of theory. However, that hit a problem, and the problem was that those theories that people constructed always had strings doing the maths as a particle. And that was a catastrophe, because in that context, which is trying to describe a strong force in force, you definitely do not want to have a vanadar. And people try to migrate it throughout the early 70s to destroy that aspect of the theory. They try to change string theory in many different ways, and they always fail, and in a sense, that's probably the most profound reason for thinking this is trying to be a theory of gravity. There's no consistent string theory that I know of. So that's a picture of the spectrum. So what I said was that at low energies, that you can ignore the massive states. And then the theory looks like a theory only involving the massless particles, and that theory necessarily turns out to look like general relativity combined with the Maxwell's theory and the Ertel-Weiss theory, perhaps, or whatever. I mean, this is a... I'm not making a precise statement about which theory... It looks like, in the end, because people are trying to make contact with the standard model, but at this point I'm just saying it looks something like a standard quantum field theory. But it's a theory which also involves general relativity, and there's absolutely no way of switching off general relativity to get the right theory about it. That's the unifying aspect of this, but more than that the theory contains

27:30 Deviations, for mine, at scales, at the Planck scale and beyond, at energy scales at least, it deviates. We want to get a discrepancy because it's that scale, and if you do the kinds of calculus, always, any thesis where I describe it, let me say a bit more about what I mean by that description, because you shouldn't be fooled into thinking of quantum gravity. So let me illustrate what I mean by comparing Feynman diagrams in conventional quantum field theories. So at the top of the page there are a couple of Feynman diagrams which describe the scattering of point particles, you can think of these, you can think of Feynman diagrams either as an expansion of quantum field theory in small fluctuations of the field around some classical background or the complementary way of thinking is to think of those small fluctuations. Particles, point-like particles, propagating between interactions. So there's two particles coming in, joining, propagating, skipping.

30:00 Here we have a radiated correction, where you have two particles coming in, exchanging a particle, propagating, exchanging another particle. And these are well known, of course. And it is well known that such a diagram will typically be infinite. In a renormalizable theory, like electrodynamics, that infinity can be treated, at least it can be. The infinity comes from the fact that in calculating this diagram, there is a region. The diagram illustrates a sum of all possible histories for these particles moving between these points, but in that sum there are histories in which the four points come very close together. In string theory, we define the theory essentially by these diagrams, which are the stringy analogues of these diagrams. They can join together at this time. In this time sequence, you can see a difference. If I could choose a different length frame, and I would move this along here, so the time at which two strings join, and it doesn't affect, it's not needed to calculate an amplitude. This describes an amplitude, scattering amplitude, two particles coming in, two particles going out. They join at some time into one particle, which then splits. It's a very geometric interaction. This spans a sheet, a world sheet, a world surface, just like this is a set of lines. Unlike this network here, there are no points picked out as special points on this surface. That is the most important. In particular, if I look at the next diagram up, this is a world sheet actually like a torus. It represents the re-scattering of strings.

32:30 I referred to it earlier that it's finite. You never get... One way of saying that is that there are no marked points. This is a surface with no singular points on it. It's extremely straightforward. It's highly complicated. You automatically spawn large numbers of diagrams at each order. In string theory, there's just one diagram. It's classified simply by the number of holes there. And the only topological distinction between different diagrams is the number of holes, which goes in which order you are. So there, in a nutshell, is the description of string theory in terms of particles moving. Now, presenting it that way, which is the way in which string theory is formulated, it should become immediately apparent to you that this is fine in the fact that it is beautiful. However, it cannot be the whole story, because this is the string version of this, but this is just the thing about string theory. The major developments of our last year have been to do nothing of the old ones.

35:00 And going beyond this is, in fact, the holy grail of the subject, which is general relativity. If you only had an expansion of this type in general relativity, what that would mean was that you didn't have Einstein's equation, you didn't have the Einstein-Hilbert action, all you knew were expansions, small oscillations around some given... And in the same sense, there is actually a lurking behind this that we haven't yet really discovered, which will somehow be much, much bigger than general relativity. Maybe I'll say a couple of words about how I calculated such diagrams. I'll say this as quickly as I can. By making reference to the way you would calculate, one way of calculating Feynman diagrams, if you had point particles, would be to use Feynman's way of, Feynman's partistry description of quantum mechanics. Imagine calculating an amplitude for a particle to move from y to z. And you draw a world line dramatized by a term called tau. So X mu of tau is the position of some point on this world line. X mu is its position in space-time. So this is like a sequence of pictures of a point moving from Y to Z. And that the amplitude is given by summing over all possible world lines, all possible curves going y and z, weighted by e to the i s, where s is an action that is independent of the parameter tau, the parameterization independent fraction, which you could take, which is typically the length of this world line. Now, summing over all such curves, It's easy to say, it's less easy to do, of course, because all the infinity parts between two points, and one way of doing that is to chop them up into sections, straight section links, make them chain, evaluate this sum when you have a discrete set of degrees of freedom, and then take the limit in which the number of degrees of freedom comes into it. That's a standard way of doing that kind of part integral.

37:30 And he used the Feynman property in some of his cases, and people generalize this procedure in quantum field theory to do quantum field theories by discrete approximation. That's the way much more mathematical mechanics can be described. There's an exactly similar picture for describing string theory. So this is the same picture, but now we're talking about a string moving from some initial string to some final string configuration. Strings reach down to a sheet, as it were, and therefore the generalization of the Feynman approach is to sum over all possible histories between all possible sheets, joining y and z. Any point on this worldsheet, which is labeled by sigma and tau, corresponds to some point on the string at some time, and xµ is the spacetime point on the worldsheet. This world sheet has to be an independent, has to be the action of the world sheet. If we're going to think of this string, we might describe it according to this level.

40:00 How am I doing string coding? So I'm thinking of a single string. That single string is described in terms of an associated with two. One can generalize. I said the S is the area. Of course, if we're interested in all possible action, general string theory, then we wouldn't just write a particular. So I'm going to write down the most general two-dimensional recalculations and actions. And this is it. I was told in another paper to formulate, so here's a little formula. So this is an action, which one would write down in string theory, in the place where I had it in Tegerspan-Spanish. And it's a two-dimensional action involving the coordinates x, or their scalar fields, coupled in a way which involves a metric, a two-dimensional metric. The fields on the world sheet with the metric on the world sheet, little g, are the fields on the world sheet but they're also interpreted as the coordinates, and then capital G is metric of the space in which the string is, and this first term in relativity, however it's not unique, you can add many many other terms, in particular the metric of the symmetric of capital B, all possible two-dimensional theories, string theories, by solving two dimensions, to do the sum of the world sheets in the

42:30 Now, when you sum up the worldsheets, summing over histories involves summing over the x's, which are the embeddings of the worldsheet in the target space, but you also have to sum over all possible metrics on the worldsheet. This worldsheet, this two-dimensional sheet can have curvature, intrinsic curvature, whereas the worldline couldn't. So you have to sum over all possible metrics, which means summing over all worldsheet curvatures, and also the worldsheet can have non-cubic topology. So one is basically doing this sort of sum, and I'm not going to describe in detail how one does it, but roughly speaking, one has to replace, just as I replaced the continuous curve to the point particle by a chain of links, you can always replace a surface, a continuous curved surface, two-dimensional surface, by tessellating it with a graphical triangle. All of these terms are approximating the curved surface, so that if you have six of them meeting at a point of the surface, however, if one is missing, if you have five, then you have an angular deficit, and that describes the motion of curvature, and if you had seven, it would be negative curvature, and so by putting these triangles down in random ways, you can describe an arbitrary methodology of self-consummation, and in that sense, the x's are like spins.

45:00 Which is a function of wave-world-sheet is now a function of these discrete positions. Now I'm saying all this simply to illustrate that the amplitude at this point becomes a sum, a summing over all histories now becomes a sum over all possible waves in which you can triangulate the world-sheet, and a sum over all possible spins, or the interval over all possible values of x, and the sum of its apologies, and roughly speaking what I'm saying is this now looks exactly like a two-dimensional statistical-mechanical system. This is in fact a two-dimensional statistical mechanical system on a random lattice where you put spins on the lattice sites and you sum them all across all those random lattice. So there's a one-to-one correspondence between the string theory and the statistical mechanical system. Now in the end we want to go to the continuum theory, the well-achieved continuum. And there's a standard way in field theory of understanding how to go to the continuum by tuning the lattice theory so that you sit at a phase transition. Continuum quantum field theory, we tuned, are sitting right at a phase transition in correlation then. What I'm saying then is that the consistent string theories are those which correspond, they are one-to-one correspondence with conformal field, with the statistical mechanical systems. And the space of all possible theories of this type is the space of string theory. What I'm talking about here is string theory in correlation with field theory.

47:30 In field theory, it's a semi-classical theory in small fluctuations. Semi-classical means that the expanding around is a classical theory. A classical theory, in some sense, is a theory of this type. So in other words, as I said, the classical solutions of string theory Which, in fact, this subject of conformal field theory has become a very, a hard then, a hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, hard then, I said the string was moving through the space-time. In space-time there was some metric, g, and other fields. If I were talking about point particles moving through, that would be it. I would just choose to let them move the way I wanted. In string theory however, and this is the second most important point so far, it is not consistent for the string to move through an arbitrary space-time factor. There is a consistency condition on the background itself Without which, quantum mechanics and the string would not make sense. What would happen if you would get negative probabilities? There's a consistency condition on the theory which requires that total attention is very vulnerable in the direction of the mass square, so that no more returns in this series don't matter at low energies.

50:00 What I'm saying is that this equation, this is, as Armin knew, is the Vichy curvature of the space-time curve that the string is moving, it's a function of the metric. And roughly speaking, there's a scalar field, I wrote about earlier, there's a bunch of fields living in the space in which the string is moving, and they have to satisfy an equation, which is in fact Einstein's equation, and what I found is that low energies, string theory, only makes sense if the background to which the string is satisfied is Einstein's theory. Coupled to other fields and other equations I have written down for the anti-symmetric tensor field and the scalar field, which all make sense as equations of motion for these fields. This had to be true if I had been correct in saying that string theory describes gravity. Because it had to be true that the metric then, some version of Einstein's equation, it's a nice consistency. What occurs in the space-time fluid is the very self-same metric, whose small fluctuations give you the a-pi-rhodony, there are two different things, on the one hand there's the metric, capital G, of the space in which it's moving, on the other hand it's spin-2. The statement is that when you solve this equation, now this equation is very difficult to solve exactly, it's an expansion in 1 over t, and it's only in very special cases, but there are special reasons why we know, The key thing is that this equation is almost unique. This equation defines what I mean by a given string theory, and as of ten years ago or so, there were five kinds of string theories, and that's almost unique, and it would be pretty good for us physicists if we only had the choice between five different theories.

52:30 People use this. I guess the balance that we talked about before may be that this, although the equations are unique, there are incredibly many known solutions to these equations. Many of them are solutions in which the standard model sort of sits very nicely, but many less solutions. It's a pretty unique theory, but with huge numbers of solutions, which criticizes theories for that reason. I mean, nobody went out with a theory and said, you've got a nice theory, but you've got too many solutions. In that case, it is very nice to have a theory which has one solution in particular inside the universe, even though we don't know why the universe selects that particular solution. People are much more disturbed in this situation, I guess for psychological reasons, because when I say this theory has many solutions, a different solution here means a completely different set of particles and forces. So the different solutions here are not simply different cultures or different notions of bodies in the universe. There are actually different laws of physics and therefore one may be more disturbed than one might be in other ways by having that situation. Let me just very briefly tell you what I mean by these different solutions. In fact, let me only describe them on a more critical screen for you. The most interesting solutions that we know about are ones in which space-time has a very peculiar dimension. In the original non-supersymmetric string theory, that's 26, in superstrings the dimension of space-time appears to be 10, which is 6 more than we're used to. But not only have we learned to live with those extra 6, we've learned to love them as time has gone by. The way the extra 6 function in solutions that we think make a difference in physics is, of course, that the 10 wakes up into 4. Ordinaries based on dimensions are possibly flat. And the six extra ones represent extra structures and theories, and there are many different ways in which you can describe that extra structure, and in fact I don't have to say that they're extra dimensions. If you don't like the concept of extra dimensions, you never have to use that word. It's just a very useful shorthand for describing a structure.

55:00 The point is that there are solutions to those equations that I had before, in which, because of the theory of general relativity, space can curve up, and there are solutions in which six dimensions can become very very small and not observable, in other words, the scale of this extra structure is very very small. And, in particular, when that happens, there are extremely strong constraints on what solutions can look like. And they naturally, although not uniquely, lead to precisely the kinds of theories in four dimensions that people have been looking for when they try to unify the qualities in a strong force, so-called grand unified theory. So, and this is all now ten years old, there were a number of very, very compelling features that were shown to be true, which suggested strongly that this is a promising way of describing physics in four dimensions. The message that I want to convey here is simply that although that's very nice, and it was very nice as far as it went, there has been a block to going further, which is due to the fact that all we know is the population expansion of the theory. We know, not just for theoretical reasons, but for purely pragmatic reasons, that in order to really understand the prediction of the theory, we're going to have to know something about the non-determinative structure. We need to know, for example, how supersymmetry is broken, because only by knowing that can you possibly hope to fit the particle masses that we see in nature. Also, non-protein information is needed to describe black holes, banglades, and cosmological implications of the field. So, knowing the perturbation expansion has already revealed some very nice features, but it's not enough to understand the integral field. This is in regards to non-perturbative structure. They got illustrated with one or two very simple examples from them.

57:30 So we don't know anything about non-perturbative string theory to start with. And we don't know even in what space this perturbation theory is sitting. So, for example, we know that we have the minimum, we have the analogs of the minimum of a potential, the minimum of the potential of the primary capacity. So what can we do? Well, even not knowing any of that, we can detect certain very peculiar features of spin theory with some inspired guesswork discovered in the late 19th century. So look, these tricky peculiarities have come to be called duality symmetries, and they come in three kinds. The first is called t-duality, and I'll just illustrate this one because it's the simplest one. Imagine that space is flat, but one dimension is a circle. That's like having a system in a box with periodic boundary conditions, and you're knowing quantum mechanics. You then get a quantization of momentum in that direction, because the wave function has to be periodic, quantized in units of 1 over r, and there's an integer m associated with that. That's one integer that comes up in ordinary quantum mechanics. However, in string theory, something entirely new happens. In addition to that integer, if you have a circular dimension, a string can get stuck around it as a winding number. That is something entirely stringy. If you look at the matter that I described earlier for the string, but now in this compact space with radius r, then in addition to the ordinary excitations of the string, it gets a contribution of u to n squared, because when the string winds around and gets stuck, it has an energy. And that illustrates a fundamental feature of the string theory. There is a symmetry here, which has no analog in it. It's a symmetry under argon.

1:00:00 This is going to 1 over r if you also interchange m and n, the winding number and momentum. This is a duality symmetry and actually technically it looks just like any other duality. It's an interchange. What it's telling you is that when you let r go to 0, which in normal physics, in string theory, instead of this symmetry, let the r go to 0, and there's an exact symmetry between them, so that one string theory is actually precisely the same as another string theory at radius 1 over r. And in a sense, there's no objective meaning to a distance less than the string theory at that distance, that the string theory contains. This generalizes, this simple example generalizes, instead of just being a circle, strings moving in backgrounds. And this kind of duality works in all of them. It does startling things. It relates, it tells you that string theory moving in backgrounds, which may be singular, is the same as string theory moving in some other background. In other words, the nature of what we mean by string theory.

1:02:30 What is happening is there's some sort of symmetry which is very different beyond the coordinate transformation of general relativity. Remember, in general relativity, you have invariance under arbitrary redefinition of the coordinates. So x goes to x-prime, which is some general local function of x, the general coordinate symmetry. The duality symmetry I just described is a symmetry in which it's not x which is being transformed, but the derivative of x on the string. It's some non-local transformation of x in which x is replaced by x', which is one symmetry, which is that kind of symmetry, symmetries which work not only on an ordinary program, but which work on the whole, on an ordinary program. The strong coupling and the weak coupling. The time is really running out and I will just summarize what's been done with a chart. And you won't really probably understand the meaning of this chart, but I may as well put it up. These are all theories with closed strings. This one actually has strings which can have open ends as well as closed.

1:05:00 And they look totally different as theories in perturbation theory. But these were not really different. Using the symmetry I just described, for example, you could take one, invert the radius. These have no internal symmetry. These internal symmetries are quite different, except once again, is that the, even though those look completely different, we now know that they only look different. One is the weak coupling. Although these look different in weak coupling, we now know that the strong coupling limit of this theory is this theory, and the strong coupling limit of this theory is this theory. I mean, it's an old fact in quantum field theory that two-thirds can lack the quantum field theory, though I think there's a very old, oddly-sign-born theory about them, too, though. And what I'm saying is that now, in general matters, for this far more sophisticated subject, the two-thirds that are completely different in perturbation theory are actually identical. We know they're identical by some very clever arguments based on supersymmetry, which circumvent the problem of taking...

1:07:30 As extended solitons. Magnetism, in certain cases, which allows one to replace the electric charges by magnetic charges, is a whole symmetry. That kind of symmetry is at work here. In other words, this string is really the solitonic, represents the solitonic sector of this string, so that in fact there's only one, we don't know what it is, but we know there's only one. We don't know what n-theory is the name given to this thing. We don't know what it is. Even though we don't know what it is, it is related to the strong coupling... You see, in third dimension theory, the statement is that the strong coupling limit is the limit in which an even more amazing... Then you get another one in extreme English, that you claim around the idea, not knowing why, of not the 11th dimension. So this morning, there was a very interesting paper in the paper entitled, Evidence for F-theory.

1:10:00 The F-field is another conjecture field, which, in terms of many explanations, there seems to be a deep...

1:12:30 ...a unique idea, a rational position, a unique answer, a good answer, a great answer, but now we've got N-field, we've got F-field, we've got P-field, we've got P-field, we've got P-field, we've got P-field, we've got P-field, we've got P-field, we've got P-field, we've got P-field, we've got P-field, we've got P-field, we've got P-field. Well, that kind of thing is not understood, because, it isn't understood at the moment, since we don't really understand how to obtain, we don't understand what it is that is given by the perturbation, so we don't know yet, systematically, whether one particular perturbation, so we don't really know how to... The kind of advance that has been made...

1:15:00 This is one which has painted what were thought of as being different solutions before, are really not different solutions. So not only are the theories not different, but the kind of thing that's in general and quantum evolution, some of them are different. It seems very likely that spaces with differences are coming about. But the fact that the big blurring of the pieces is not something that's about the matter, it's about the value. So the question that arises... I believe in my own mind, I'm sort of mistaken to think it's about that, regardless of the level of detail, but I'll assume that you're being vague with the concept of a man-in-person course, in which turn up a sort of, let's call it an act of description, that sort of thing. I was wondering how much of that is necessary for getting this kind of theory out.

1:17:30 Well, the problem is, you see, you're using a manual which is utterly non-string. And the problem is, you know, in string theory, you are not, we don't know what the right way of thinking is, what is clearly not true. But what we think of as being a description of a manifold in ordinary geometry, which is relevant to ordinary general relativity, that is certainly not the right language to be used to do that. Let me illustrate that point again by an example. There are these spaces which I call Calabi-Yau spaces. These are six-dimensional manifolds, which correspond to particular ways in which six exponentials can curve up. And they play a central role in understanding why classically string theory matters. If you're in four dimensions of super string theory, you have six extra dimensions, which you can describe, in some circumstances, in the semi-classical limit, as being a six-dimensional manifold. That same six-dimensional manifold, however, arises in a completely other description. So mu is 1 to 10, and then I said 10 was 4 plus 6, and I said that this 6 was 6 values, say, i equals 1, 5 to 10, 5 to 10, in the internal direction. However, when I described what we actually do, I showed you that x can be thought of as a spin signal like this. And instead of using six columns, you can actually replace this by any other spin degree of freedom you like. So, for example, you can have spin-1⁄2, for instance. You need more of them to make it very consistent. You can replace these six by anything else you wish, which may come in a song you've got quite numbers of. You can still describe the same. By describing two-dimensional systems, you actually mimic or generalize this.

1:20:00 Somehow, the notion of a manifold, a space diamond, is something intrinsically, is intrinsically wrong to say about, I mean, we have solutions in string theory that look entirely different, and that's why we say there are 10,000 of them, actually. We don't really know that they are different because of the physics of string theory. So if you're doing this theory, you're going to find a handful of facts, a handful of beliefs and so forth, and under that, we will spill out all this information and start to do some more research into it. But there's a lot of other programming that we must put together right now. So we're coming up to the bottom of this theory, this is the time, I think, that you've made, and we want to let you know that there's a huge level of power in that, and I think you're going to get there. And maybe you don't reason or fear that you could come down explaining the whole card as it goes on, rather than you just go on expanding up to these people's interests and so on. It's a different approach to doing theoretical physics to effective people. It seems to be a different approach. Well, I think, especially when I think around the date, I think that there's been a tremendous convergence in viewpoints because the people who believe in the Broca-Hugger approach, a large number, not all of them, perhaps not the main people, but certainly a large number of them, are trying to see how the standard model can extrapolate up to the so-called grand unifiers.

1:22:30 And those would be the bottom-up people. However, those people have discovered they can't do anything without supersymmetry. And furthermore, once they're up near the grand unified scale, they're so close to the Planck scale that they can't do anything around gravity. Now, there is only one theory that you can use to calculate it, because there's only one theory that you can actually, you know, dive around and get actual answers to, and that's string theory. So there's been a convergence in view. People going from the bottom up have become much more ambitious about what they mean by up. I mean, they've really gone up. Even experimentalists, for example, would like to say that their experimental data, when extrapolated in the right way, meets at some granularity, the coupling points that meet at some granularity. But in order to make that statement, they firstly meet supersymmetrically, and secondly, where that point is, So one answer to your question, I think, would have been some time ago for those people. The other point is that there have been, the advances that I've been alluding to here and speaking of, are actually based on, they come rather close to the kinds of observations made about two years ago in ordinary field theory by Seiberg and Witten, Seiberg and Witten. There are a number of people who have worked in America who have transformed certain aspects of four-dimensional quantum field theory. So for the first time, to my knowledge, we now have a non-trivial quantum field theory of four dimensions, whose spectrum we know something about. And it's a non-trivial spectrum. And that's a supersymmetric quantum field theory, which is a supersymmetric version of QCD. That's one of the objectives, and that's a whole way of life in quantum field physics. In QCD, as we know it, in real life, the only way we know how to solve that theory, so to speak, is to put it on a computer. On a bigger computer, but extremely powerful.

1:25:00 It's not reflective, but it is the same as the other two. All the low energy physics, the set of all possible... We're heading towards the low energy behavior, so instead of asking questions, I don't think that's effective. Do string theories satisfy something like what happened in Spinster? Can anyone write down Lagrangian? Well, you had an action. That Lagrangian has the same status as you would have for a point particle, say, where you say that the action to a point particle is the length of its world.

1:27:30 That's what you would put in if you were doing quantum mechanics, that's true, in the first quantized framework. So you can write down an action, a mechanics action if you like, for a single string, but what we lack is the analog of quantum field theory, or field theory, classical field theory, we don't have a field theory of strings in the sense that we would like to have, perhaps, you know, a generalization of a quantum field where instead of creating and destroying particles at points, we would create and destroy string lines. Objects, and sub-objects, and so on. Is it the case that you can't actually get a theory out of variation of this action? We can get a theory, we can get a classical theory of a single string. And then you can... Sorry, that is how quantum mechanics of a single string is described. When I showed you a spectrum, it was precisely my using Amazon's principle for a single string. But in a quantum field theory of a string, this is what we need to have. That principle may not in the end be all that relevant. So for example, the most striking example of that is, it is well known in field theory, that theories which are not that far from electrodynamics, some elaboration of electrodynamics, theories of electrically charged particles, have of course electrically charged excitations which are in some sense created and destroyed by the quantum field. But if you just had those particles If that's all you knew about, you'd miss the fact that some of these theories have these extended solutions, they're field configurations. I mean, particularly, the one which has set the scene for everything that's happened in the last couple of years is an old conjecture, made in the late 70s, which is the following. It's called, it's embedded in Yang-Mills, which has a gauge in SU2, and embedded in SU2, there's U1, so U1 is electromagnetic phase symmetry.

1:30:00 And in the theory of this type, which also has supersymmetry, it's been known since the mid-70s in fact, but the theory of this type, as a classical field theory, has extended solutions which are magnetically charged. So there are magnetic monopoles in this theory, in which the fundamental particles carry electric charges and not magnetic charges. And this is, in fact, a good prototype to illustrate everything I was trying to say at the end about it. Here is a theory which, if you only knew about its fundamental expectations, which are the fields which go in, that theory, the only particles you could have would have electric charges. However, we know that the theory also, a classical theory of extended solutions which have magnetic charge, non-singular solutions which have magnetic charge. That was known about in the mid-70s. In the late 70s, it was shown, in the case where you make it super-symmetric, There is very good reason to believe that that theory has a symmetry, which is an exact electromagnetic symmetry. It really is, you know, an angel between perhaps Max and whoever. It's a theory which really is, and now is really believed to be, exactly symmetric between the interchange of electricity and magnetism, where you are exchanging what were the fundamental fields of one theory with what were the solitons of that theory. So what this means in fundamental education is that you miss that kind of symmetry, especially in the lecture about the fields and missing the solitons. The solitons typically have very much higher mass. The mass of a soliton, the mass of a magnetic monopole in this theory, goes roughly like one over G, or one over alpha.

1:32:30 So in perturbation theory, the masses of these solitonic states are extremely heavy. They're not seen in perturbation theory, they're non-perturbative.