String theory - towards a unification of physics? (contd.)

0:00 The action is simply the length of the line, and Planck's constant is the constant I mentioned earlier, which comes in the unsensitiveness. So what Hyman said, the rule is very simple. You get the quantum-galactic amplitude by assigning to every possible path the particle can take some factor, and then just adding up all the factors on each path. And it's easy to see that if Planck's constant is very small, which it is, Which S, the length of the line, is the minimum. And that is where the classical physics, the minimum length of the path, being geodesic, comes into being the path of the path of the path. That's how classical physics emerges from this path of history. So whether or not you like that description, it's a fact that it works. And given that fact, you'll be quite grateful. All string theory does is extend the same principle to strings. So now we have a string, we have a string here, moving from one place to another, and as it moves, it sweeps out the surface of the world sheet. You can imagine slicing through the surface at any time, and you'll see that the string at that time, so as the string moves, it sweeps out the surface, which is a bit like a circle. In fact, if time were really a fourth dimension like space, which it is in white, then this would indeed be a sort of circle. It would be the same picture of a sort of joint of things. And then the Feynman principle says that what you should do is to construct the amplitude, and therefore to get the probability that string one will reach string two, you have to sum over all possible parts of the tape, which now means you have to sum over all possible sheets. The obvious, consistent action to take, you have to choose, this action is determined by the fact that it has to be a property of the sheet, which doesn't depend on how you label points on the sheet.

2:30 The sheet is of course a fictitious sheet, and therefore I should be able to label its points any way I want, and no physics should depend on it. So there has to be some intrinsic property of this sheet, which doesn't depend on how you label it, and the simplest of properties is an error. That's obviously a generalization to a sheet of what is previously the length of the line of particles. So with this simple principle, you can set about discussing the quantum mechanics of strings. And lo and behold, what you discover is that all hell breaks loose, because this sum, I should emphasize, summing up all possible lines is bad enough. There are a number of possible lines and points. Summing up all possible sheets joining two curves is a slightly tricky thing to do, so inside this sum, sigma means sum, I've hidden a great property of mathematics. It's actually very beautiful mathematics. It turns out that this principle, the fundamental principle, is simply that you should not be able to solve a problem depending on how you relate to the sheet. That principle breaks down unless there are some very special conditions. In particular, the superstring breaks down unless the string, which is flopping around in space-time, is flopping around in a nine-dimensional space at one time. That's rather extraordinary, because in particle physics, you can imagine your particle moving through any old space-time. You can discover that it is... Here you discover that you can only formulate quantum mechanics and strings consistently if certain conditions are met, and one of those conditions is that if the space with which the particle is moving is flat, then it has to be ten-dimensional. Space-time has to be ten-dimensional. Now you might think that's pretty awful, because one of the things I told you at the beginning was that space has to be three-dimensional, so what happens to the other six dimensions? Well, therein lies another story, but roughly speaking, what you have to do You see, imagine space can be curved from the general relativity of space, space-time in fact, can be curved, and then with the possibility that there are dimensions which are very curved, which curl up themselves, so that although you would like to move in the direction, you want to be down to six dimensions, you can't because they're too small. You try to point your finger in that direction, and you get back to where you started. So the notion of very small, complex dimensions.

5:00 It's important to appreciate that there are examples starting shortly after Einstein's theory, which was in terms of time in the early 20s, there are well known examples of how this could happen in any theory of gravity. So what one would want is if this theory really had nine space dimensions. And it's a fact to be true that when you try and solve a theory, six of these dimensions turn out to be very small, otherwise you wouldn't have got 12. Now that aspect cannot be well understood. However, what is well understood is that whatever the x-axis, whatever the nine spaces in one time dimension are like, if they are curved, they have to be curved in a way which satisfies Einstein's theory of general relativity. Now that by itself is a slightly remarkable result. Let me illustrate that with another picture. We're still talking about a single string moving through space and thinking of a quantum mechanical string. But space itself, at this point, is classical. I haven't, you may, some of you may already be worried because I'm doing something which in the end should not be consistent. I'm talking of the theory of gravity because I know, I know it's the theory of gravity because the graviton is contained in theory. And yet I'm talking of space as if it was, without talking about the quantum mechanics of space, I haven't yet faced the issue I mentioned earlier that space itself should be dynamic. Let's keep talking about a single particle moving through space, then try and contrast this with what happens to a point particle. I'll try and describe what this picture means. This is actually just another version, up here, this is just another version of the picture I drew on the last transparency. But remember, I was summing over, I have a string moving from here to here, and I was summing over all possible surfaces joining the initial and the final strings. But included in all the possible histories are histories in which, since I'm summing up all surfaces, I have to include weird surfaces like this. This is meant to be one continuous surface that's very important.

7:30 It's a single surface, but there are bits of it with long tube-like things coming out and stopping it. They're just ordinary surfaces. And it's remarkable that this kind of surface occurs. Where in ordinary point particle physics it wouldn't, or the analogue wouldn't occur, unless you did something rather special. Let me contrast this picture here, which I stress is a single surface, with what would happen for the relativistic point particles. The particle would move along the line, but if you wanted the analogue of these tentacles, which go off and disappear into nothingness, you would have to add them. You would have to allow the line to split. What is happening when these things split and go on into nothing, here or here for that matter, is that the particles are disappearing to take a cross section at some time, here. Now, when you cut them here, instead of just seeing a single string, you've actually got two strings. You thought you were describing the quantum mechanics of a single string. And you've actually got describing two strings, one of which is then subsequently just disappearing into nothing. That means that the vacuum is getting changed by the particle disappearing. There's an interaction between empty space and the particle. And it turns out that this effect, the fact that the quantum mechanics of a single string contains already the interaction between the two, places very strong constraints on what the vacuum could be like. We started off trying to describe a particle moving through space and time, and we discovered that there are conditions on space and time which have to be satisfied in order to get a consistent description of a particle. And one of those conditions, space and time, has to have ten dimensions. There are much more stringent conditions. If you allow it to be curved, it has to satisfy Einstein with modifications. The key to why string theory is profoundly different from particle theory. Particle theories, if you are considering a single particle, you don't get these interactions with the rest.

10:00 In view of what happens with time, I will in fact skip a few transpositions and just say that by considering other quantities, not just single strings moving through space-time, but strings colliding with each other and scattering off each other, there are stronger constraints yet on what kinds of theories you can have. And what you find is that there is a rather short list of possible theories, although the list is only short if you look at it in a certain way. Some other way of looking at it looks immensely big and complicated, so let me bias your way of looking at it by describing it this way. If you consider these kinds of theories, the quantum mechanics of an extended object like a string, the only consistent way in which it can work is if space-time is ten dimensions. And if you first allow that spacetime not to be curled up, in other words, they're really nine extra dimensions, they're nine dimensions, six extra ones, then you discover that all the other forces, apart from the gravitational force, have to be unified together in one of only two possible ways. There is a large unification, much, much bigger than the SU3 and the SU2 times E1 that I mentioned earlier in the standard model. There are symmetries for SO32 or E8 times E8. These are mnemonics, these are mathematical goods which summarize vast relationships between patterns of particles, most of which haven't yet been seen and never will be seen. So there's a theoretical symmetry which is much larger than we see in nature. Now, let's ask what happens if the extra six dimensions really are curled up and are small. What we would like to do in the end is discover why they have to be curled up. We don't understand that yet. There are some remarkable things that happen if you assume that they're curled up. What you've discovered is that six dimensions curl up and become small, so that we're left with a four-dimensional theory in three space dimensions. Symmetries that are much too large become much smaller. And in fact, there are many, many, many ways in which people have understood how these extra dimensions can curl up. And so at this point, there are many string theories that work in four dimensions with so-called realistic symmetries.

12:30 In other words, although no one can show any particularly reason for choosing one or other of these solutions, there are certainly solutions which come from, oddly speaking, what we see in nature. The classification of all possible theories has become a whole art in itself. It's mathematically a rather elegant subject, it's called the formal field theory, and it's caused a lot of interest, a lot of interaction between theoretical physicists and those who are interested in the same kinds of structures from their own point of view. So this has been a very fruitful period, actually, of interaction between physicists and mathematicians. From the point of view of this particular talk, the mathematical interest is in the classification of possible string theories. There is, however, a more general interest, even from the point of view of more general physics, than what I've just mentioned in understanding the global field. But the basic point is that these are possibly consistent theories of particles based on this picture of strings moving through classical space-time. I'm coming toward the very last bit of the talk, but let me summarize where I've come to. The first feature that makes string theory very interesting is that it might resolve the conflict between quantum mechanics and general relativity. At the level I've described, what is happening is that the superficial aspects of the conflict are not there in string theory. In other words, if you do the same calculations in traditional theories of quantum particles as has been done in string theory, you find all sorts of problems in traditional theories. Which just go away for the particular theories I'm talking about. And at the same time, you find that there is a restriction on the way that the courses other than the course of gravity can come in. And in a certain sense, there's only one or possibly two theories to start with. Although there are many, many possible solutions to those, and we don't understand yet which one of those solutions should be relevant to physics. But the trouble with all this is that there is no intuitive or logical basis for the subject.

15:00 I've told you that the theory contains general relativity, but general relativity is a very beautiful logical theory based on physical insight, firstly, and also rather elegant geometrical notions. In string theory, all we've done is we've asked that we just try to describe a particle, a string. Now, for various reasons it's clear that until one really understands such a principle, then there are some very important questions that cannot even be asked, and these are questions that we really want to know the answers to. So that even if you take the pragmatic view, that you just use the kind of formulation we have at the moment, that isn't good enough, because there are certain obvious questions, one of them I've mentioned here in particular. At the beginning of my talk, I told you the universe was big. That was very obvious. About 10 to the 30 centimeters. And I told you that anything particle physics naturally misses, in this way of writing it, it misses by a minus sign. But it's a minus sign in the rather important place. The actual scale that you get, in fact, in any particle physics theory, is rather more like 10 to the minus 30 than 10 to the plus 30 for the size of the universe. Until you have a theory that claims to be able to cope with gravity, then of course you can't even begin to ask the question, what is the size of the universe or the scale of the curvature of the universe? But when you have a theory that makes some pretension about being a theory of gravity, You have to face this problem, but in the present formulation of string theory, this isn't a problem you can even begin to face. An associated problem is, I've told you before, in an earlier transparency I showed you that the particles that we see in string theory, the ones that we should, are predicted to be seen, are actually massless. It's of course true that the ones we see in nature are massless when compared, or approximately massless, when compared to the huge mass scale, the plane mass of 10 to the 19th that occurs in the theory.

17:30 But nevertheless, the particles we see in nature are not really masses. The people who discovered the W bosons, for example, thought they were discovering the most massive particles ever produced. So that we would really like to understand the non-zero-ness of the very small masses of these particles. And that's again an aspect of the theory that can't be described, and I couldn't understand it. Quantum mechanics that I was describing earlier, in the sense that on very short distance scales, the notion of a fixed classical space-time background should not be a good approximation. So here we have a string moving through space-time that we've taken to be approximately fixed, or we've taken to be fixed in some approximation. But if the string side is the blank side, then we shouldn't perhaps be taking the background space-time to be fixed. It's a combination of relativity and quantum mechanics in the approximation that you expand, if you take general relativity, as I said before, and expand it in an infinite power series around, in some approximations, by assuming that space-time is approximately classical, and then ask questions about it, we're not taking the full quantum gravity, at the level I've described it, the full structure of quantum gravity is not taken into account. It's not really quantum-gravity in the full sense. It's not a theory of quantum space-time, but that's why I was very careful before with everything I've said, to say that if you do the corresponding calculation, what would have been called quantum-gravity based on Einstein, then the problems which arise in every order there just don't arise here. There's a miraculous cancellation of all the non-renewable Einstein-critical problems and all the other problems associated with interpretative quantum-gravity. No, no, no, no, you get perfectly calculable and sensible numbers. In fact, every order that's ever been calculated, in this approximation, there is no problem with the scale of the universe. String theory says that the universe, that the problem with the scale of the universe is sometimes called the problem of the cosmological constant.

20:00 When Einstein first formulated general relativity, he included a non-zero parameter of the cosmological constant. Later on, he regretted this bitterly because... Experimentally, the cosmological constant is known to be approximately zero to one part in 10 to the power 120. When a physicist sees a number like that, he says, that's zero. And there must be a fundamental reason why it's zero. So what one would want is a theory in which there was a fundamental reason for the cosmological constant to vanish in order to solve this problem for the scale of the universe. String theory is not properly understood yet. To the extent that it is understood, it's in this approximation. The cosmological constant has to vanish. It just has to be zero. So when I said, I should have emphasized that, when I said that string theory contains general relativity as a small piece of it, it only contains that version of general relativity in which the cosmological constant vanishes, as far as it's been understood. The problem, however, is that in order for these two points to actually link, as far as it's been understood, particles are also matters. But particles are not really matters. So what hasn't been understood yet is the small thing that gives particles mass. And it's difficult to imagine a mechanism in the theory in which particles get a small mass which will not also screw up this other part. So they're linked, but to the extent that it's been understood, approximations allow assuming that a classical spacetime is a good approximation. To that extent, there is no problem. But let me get on to my last transparency because, I mean, I do not think string theory is understood. For the reasons you've mentioned yourself, it is not satisfying to have a theory of gravity in which you approximate things by having particles moving through a fixed space-time background. And for those of you who are not experts, let me illustrate that. Let me go back to conventional point particle physics. In ordinary quantum mechanics, there is a dual way of looking at things. You can think of particles or you can think of fields. So, for example, the particle I call the photon, the particle of light, can be thought of as a vibration, a small oscillation, in the electromagnetic field.

22:30 So you have a particle whose position is x at some point, or you can have a field throughout space-time, so there's a field, a function of x. And there's a complementary way of thinking of these, and of course quantum mechanics resolved the age-old argument about whether light was a particle or a wave, and it embodies both of these aspects. And it's not just photons. In modern particle physics, of course, any particle that you like to think of, like the electron or any of the other particles, you can shove that into your theory and think of it as particles moving through space-time at position x, or you can think of these particles as oscillating fields. Now the particle associated with the force of gravity, a particle I call the graviton, has a rather special role. Because in general relativity, as I mentioned at the beginning, the force of gravity is intertwined inextricably with the structure of space and time, so the particle we call the graviton is in fact a small oscillation in the field associated with the geometry of space and time, so that in order to describe the graviton in a field-like way, you have to think of oscillations, small oscillations, in the whole structure of space and time. But you can do that at least If you ask too many questions, you get nonsense, as I described earlier. But at least you can imagine that you can do that, and you would have some sort of complementary relationship like this. But I want to emphasize that in conventional physics, there is a disjointness between all the particles except the graviton and the graviton itself. And that's because for all the other particles, space-time plays no role whatsoever. It's an inert medium through which particles are moving. And then, if you choose to quantize gravity, you then quantize gravity and you run into trouble. But you may choose not to quantize gravity, and then you won't run into trouble, at least as far as we think, or seriously know about it. String theory is very different. In string theory you have no choice. You have to include the graviton, if you include anything. The graviton comes in on the same footing as everything else. The photon, the electron, the neutrino, everything else. These are all, if you remember my picture, these are all modes of a string and the graviton is nothing special. So the notion of a particle is replaced by a single notion of a string. That's the unified thing.

25:00 The problem in the way I'm describing things, in the way I've been describing things, is that I haven't got the right-hand side of this equation. Strings should correspond to vibrations in something which I call a stringy field, in the same way as gravitons correspond to vibrations of the metric, the protons are vibrations of electromagnetic fields. Nobody has yet managed to formulate a theory based on a field-like starting problem. And as any of you know, and as you have commented on me, in general relativity, you lose most of the elegance of the theory if you emphasize the particle aspects too much. Geometrically, the logic of general relativity resides in its formulation in terms of geometry of space-time, and that is lost if you only consider small oscillations around some fixed geometry. So in string theory, we really need to have this side of this equation in order to understand it really properly. I think that now there's been a lot of very important work in the last few months on trying to find some more fundamental viewpoint in string theory, Which somehow or other fills in the right hand side of this equation. The answer isn't yet understood, but it has to be, what we do know of course, but it has to be something slightly bizarre. Because the result of doing this is going to be that you can no longer separate the notion of a particle from the notion of the space-time in which the particles move. So in this fundamental reformulation, whatever it is, the notion of particles and forces will be unified with the notion of space-time geometry into some bigger structure. So that general relativity, which is this aspect here, will emerge as some small part of whatever it is. Now, the reason people are optimistic is because we have all this circumstantial evidence that string theory contains general relativity within it, as some small piece of it. And therefore, the notion of geometry of space-time should be contained inside some natural generalization of geometry to some degree. Some more, some rigid kinds of geometry. And that's, of course, why mathematicians are interested. This is some generalization of the kinds of geometry that they think about. From the point of view of physicists, this is going to look very peculiar, whatever it is, because it's going to make the graviton on the same footing as all the other particles, which means that there'll be nothing special about the metric of space-time.

27:30 So you ask the question, well what can you put into it? If you don't put that into it, what can your theory look like? It's a theory in which there can be no notion of distance. If you want the metric space-time to emerge from your theory, you can't put it in. And without a metric, you can't have a notion of distance. So you can't have a notion of signals. So here we have a theory of physics which has to be formulated in terms of something which has no notion of signals moving through it. And that is unlike any theory of physics that we're used to, and it's very difficult to see what it is, but there have been some extremely intriguing suggestions, many by Ed Witten in Princeton in the last few months, about the relationship of this kind of theory, which is not understood or formulated, and the relationship of that to some very intriguing areas in pure mathematics. In particular, I've mentioned some of them down here, but the latest is... In non-theory, there seems to be a strong link between non-theory and the formulation of some fundamental re-formulation of string theory. I've mentioned this last transparency just to emphasize that there is a lot of exciting work going on at the moment. The subject is by no means understood, but we have enough circumstantial evidence that it's interesting. Thank you very much for a very clear introduction. It's a very difficult field. It was very nice of you. Many of us have been brought up on the notion that relativity and quantum mechanics are really different games. In that connection, then, I have two questions. Would it be fair to say that you really started, in a way, from quantum mechanics and incorporated relativity to some extent? We'll have to be come to terms with some time. The second minor question I have is, would it perhaps be true to say that it is a historic accident that Einstein occupied space with gravity if all this strong interaction and so on had been allowed?

30:00 Might he have occupied it with something else? One attitude to take towards the string theory, in fact the historical way in which the theory developed, as I mentioned, was that the theory was constructed, people thought about describing extended particles, and the reason they did that was because experimentally, strong interaction, it was realized that particles like the photon and the mesons and other, the many particles which we now think of as not being elementary particles, but we now think of as being composite particles, in those days, it wasn't known. So they were composite, but they were known to be extenders. We knew that they were known experimentally, they had a science, and that's what motivated people to try and formulate a theory of an extended object, and the simplest extended object you can think of is a string. So it is the historic origin you were hinting at. That's right. That was in the late 60s and early 70s. That whole program sort of failed for various reasons, and part of the failure was due to the fashion, but the perfectly beautiful theory of the strong fashion came along. However, part failed also. Every time you try, whatever you do, it seems that if you try to describe the quantum mechanics of a object, you are forced to the statement that there is a massless spin-to mode. In other words, you are forced to the statement that there is a graviton. And in strong interactions, you don't want that. There are no massless spin-to, strong interactions. That's one of the reasons. But that's also the reason why it's a good theory of gravitational interactions, and it isn't that gravity is incorporated in some ad hoc way, which is, it's true that one starts with quantum mechanics, naively implemented forward way of starting from, the naivest view of quantum mechanics, but what then happens is that you discover you have to have gravity, you have no choice, so, and also you have to have some other force. So what I didn't say...

32:30 So I don't want to go into too much, is that in the search for this more fundamental way of looking at it, there are all sorts of indications that things, the way quantum mechanics is implemented might change. I mean this is all very, very undissolved, I mean it's not really a well-discovered example. The way I describe things, what I was describing things as if one set up a theory of classical string and then takes a classical theory and quantizes it. There are some arguments that suggest that, they're actually rather easy arguments to describe perhaps, that the quantization procedure won't work with string theory, in other words that there will be no analogue of a classic string theory in its more fundamental distinction between classical and, the distinction between string theory and point-particle theory is such that, whereas in the point-particle theory you can sort of switch off quantum mechanics. There are some indications that, at least to the extent that string theory has been understood so far, that you cannot switch on, which is a good thing in a way, but that's just a rudimentary indication that something strange is going on with quantum mechanics compared to what we're used to. There are many other questions, sir. There's one thing that I'd just like to mention. You said that definitely in this theory you must have the cosmological constant zero. If so... Did you say that? I said in the approximation in which it's presently formulated, that's true. Because if you do have that, then you're committed then to, in generativity, a single form of expanding universe with an age which is two-thirds of the receptacle of the Hubble constant. And that, of course, has the worry that it may be less than the age of some stars and galaxies. Well, I... I'll give you two answers. Firstly, the problem of the cosmological constants is by no means resolved in string theory at the moment, and I didn't want to imply it was.

35:00 What I was really saying was that in our present understanding, there is no cosmological constant. But also, there are no masses for any particles, so that isn't very satisfactory. I misunderstood. I thought you said it was definitely zero. No, but the other point is that in understanding the very earliest moments of the evolution of the universe, there's clearly going to be a difference between string theory and general relativity. General relativity emerges as an approximate thing. Approximate on length scales much larger than this fundamental plankton. So in terms of the evolution of the universe, general relativity should be a good approximation sometime after the Big Bang. But when you get too close to that thing, string theory says things that we don't yet understand. Certainly early cosmology would be modified to agree to have a structure. But don't we think about diameter as a classical script? You've spoken about it being a one-dimension of Q. Well, one interpretation of the question might be, why strings and not members? Why, if you've gone along one direction, why not go along another direction? And the answer to that, again, is going to be an invasive answer. The mathematics that makes string theory so interesting is the mathematics actually associated with these two-dimensional surfaces and strings to be found. And there's a lot of interesting topology and other mathematics associated with it. Those kinds of two-dimensional surfaces that the strings sweep out. As soon as you go to an object with more dimensions, like a membrane, then all that beautiful mathematics disappears. That doesn't mean, of course, that the theory is not correct. It just means that it's extremely difficult to calculate anything at all. And in fact, in a certain sense, if you try and study the quantum mechanics of a single membrane moving through space and time. You are faced with problems which are almost as difficult to face as the original problems of quantum gravity, so from the pragmatic point of view, for the moment, although there are a certain number of people working on this subject, they're finding it very difficult to say anything at all about quantum mechanics. I would love to give you a logical, intuitive reason as to why string theory is correct and membrane theory isn't, but I can't do that because we don't yet have a logical basis.

37:30 You didn't say anything about interaction. Do you, in string theory, have strings being exchanged in spin-1 mode, something like that, in interaction? Yes, although they're exchanged in all modes at once, so you can't... Let me flash up this for you, I may start. You can calculate things like Feynman diagrams, for those of you who don't know Feynman diagrams. In particle physics, if you discuss the collision... The way in which one discusses the collision of particles is in terms of these so-called Feynman diagrams, and I've illustrated at the bottom here one contribution to the scattering of, say, an electron with an antielectron, so if an electron and an antielectron collide, they can annihilate to form a photon, which can then split up into an electron and an antielectron. And this diagram is one contribution. The description of the collision of an electron and a positron. Now in string theory, the equivalent of a diagram is this one here. Instead of the point-like line, the line is being swept out by points. There are these surfaces being swept out by strings moving. So if you think of time flowing to the left, we have two strings that come in here and here, they come together, they touch, they join. One intermediate string, so they join at this time here, two strings touch and join. I actually wanted to show you this diagram because it illustrates a very dramatic difference between string theory and point and particle theory. In this diagram, such a diagram would be the sort of thing you would draw to any theory based on Maxwell's theory of electromagnetism and Einstein's theory of gravity, or any of the conventional theories which have diagrams of this type, so to speak. Here and here, I mentioned something similar earlier on, that when particles join, there is a node on the diagram, which is mathematically rather difficult to deal with. The strings, on the contrary, sweep out the surface and are smooth, so that on this surface there is no singular point.

40:00 And it's this smoothness of the surface and the fact that there are no isolated points on this surface, which is at the heart of why the theories are consistent. In particular, if I go to the next, if you draw a five-minute diagram, you also draw a diagram like this one, this is the next, the five-minute diagram way of describing quantum mechanics adds together better and better approximations, and the zeroth order approximation that I just showed, the next order approximation is this diagram, where the two particles come in, but one of them, say, splits up into two other particles, this one scatters off this one, This one goes on, it splits up into a final particle, and this particle, which then scatters to this one. So this sort of Feynman diagram is a more complicated contribution to the scattering of two particles to form two other particles. It's this diagram which gives nonsensical answers in ordinary fields of gravity. And the reason for that is, if you think of these lines, these lines represent the sum over histories in Feynman's past history by doing quantum mechanics. Remember, I said that you had to sum over all the possible histories of all the particles. What included in those histories are histories in which these four points get arbitrarily close to each other. Because of the uncertainty principle, essentially, the reason I mentioned earlier, when these particles get very, very close to each other, there are large quantum fluctuations, and in that region, you get a divergence, which is the nonsensical infinity that I alluded to before. In string theory, the corresponding diagram is this diagram, which is a sheet, which is a torus. Notice that the sheet now, when I say it's a torus, what I mean is that there are lines on this sheet, like these red lines. ...which you cannot contract off the sheet. So here we have a world sheet. We have two particles coming in, joining this time to form one intermediate particle, which then splits into two, and then those two combine and then split again. This diagram is the string version of this one. However, in this case, there are no special points. There are no special singular points on the surface. And therefore, there is actually no possibility whatsoever for these divergences. It isn't simply string theory of the right type doesn't count. The so-called ultraviolet problem of nonsensical infinities. There's no way in which it could happen. There's nothing in the theory that allows them to occur. So it is an extremely different kind of quantum theory from the conventional theory that we're best appointed on.

42:30 A few years ago we were hearing a lot about the theory of supergravity. Now, I think you mentioned supergravity. What? Supergravity has now failed, but it's somehow operating. Well, it's a program which in a certain sense failed, and in the other, in the sense in which it hasn't failed, it's been incorporated into the superscript. It's, if I go back to the transparency, which I showed you the spectrum, the state of the string, I started off by covering up all the string units. This list of particles is the list of particles that are put into the supergravity system. The supergravity is a combination of Einstein's theory of general relativity, Maxwell's theory of electromagnetism, Yang-Yau's theory of weak and strong interactions, and so on. All put together to spin three-half particles, crucial as well, in theory. And they're all put together for reasons of aesthetics, if you like. It's really in order to unify the blue dots, which are the photons, with the red dots, the photons. So there was a vague hope that by unifying things, you'd get some more consistent theory. However, all the problems of quantum gravity... There are still there. I never understood, but I still don't understand how anybody could ever have thought that they would go away. I mean, the physics wasn't changed. There was no change in the short distance problems which arise in quantum gravity. These particles, in the string theory, these infinite, extra modes, excited modes of vibration in the string, which give you a short distance structure, which changes in the string. Certainly if you have the notion of an extended particle sitting in space, you have a bin and you can cut it up into smaller pieces. And there are string-like things that occur in physics elsewhere where that's true. Typically, if you can ask that question another way, you can say, let me take a system which produces a string and heat it up to a high temperature. Eventually it will dissociate into its constituents.

45:00 So you can ask the same question in string theory, and unfortunately, it's one of those questions that we can't give a thorough answer to, but nevertheless there are some extraordinary indications of what might happen. You see, the problem is that unlike in all other theories, when you heat things up, when you heat up a theory of this type, you are talking about the physics of what's going on in high and low energy systems, So we're talking about distance scopes, which is the whole notion of space. Now, although we haven't yet understood what the equation is...