Twistor string theory / Q&A

0:00 I'm not sure if he's going to answer his own question. Well, I can say that with that title, you know, usually it wants to wait until we enter the talk to find out what the answer to the question is. In this case, I'm going to give you the answer at the beginning. Can we leave then? No, I'll give you the answer at the beginning. Those, I don't know. But nevertheless, I'm daring to talk about this subject. It does make me think that many years ago, I talked around about ...1970 or so, I can't remember exactly, but with Chris Isham here and wherever you were, you were here at that stage a long time ago, and at Birkbeck College, where I was, and we used to have wonderful sessions, I seem to remember, talking about these things, and we were just in time to start talking about this issue here. We didn't really know what we were talking about, but all sorts of wonderful things came out of it, and I certainly look back to this. And maybe forwards to future occasions when there are those. So, it's a really great pleasure and honor for me to be able to pay my respects to you, Chris, on this occasion, and I apologize in advance that I can't be here tomorrow for family reasons, but I'm very pleased that I could be here today and talk about something I don't really understand, unless the sort of thing that our conversations would have been very useful for. Now, twistor-string theory. I do know something about twistor theory, so I'll start sometime about that. In fact, this is a low-transparency book that I've been using for about 20 years. Nevertheless, it feels quite handy. Here we have twistor time. And here we have a library, twistor space, and we can think of more or less the points of the libraries. Now, Heidi wrote some points, so we think of the family of libraries through the point over here, and that is described by a sphere over here. And the remarkable thing is that in three spaces, one in which I haven't mentioned, the celestial sphere, which this really is, It happens to have a conformal structure. Well, that's not special to a three-space and one-time dimension, but for all dimensions, or for instance, spacetimes, or spacetimes, if you like, what's remarkable is that the conformal structure that you get is also equivalent to a complex structure, and we can regard this as a one-dimensional complex space, in other words, a human sphere.

2:30 In order to make sense of this, well this picture over here is just showing you what that depends upon if you take two observers looking out at the universe and transform from one to the other. These two observers are passing very close to each other, but at large relative speed, the transformation from one field of vision to the other is a conformal one, which preserves circles. That's what allows you to call this a Riemann sphere or a complex space. And the algebraic twistor theory is somehow to regard the whole space here as a complex space. And this is a complex one-dimensional submanifold of this complex space. Now... The developments that have been happening recently, which I want to give some indication of, are developments in string theory, this is a work primarily done by Ed Witten, based on some other work that occurred earlier and in which, roughly speaking, the target space of your strings is not space-time anymore, or not some space-time to which you attach extra dimensions, This projected twistor space is referred to it as. I should say just a comment, which those of you who are very quick to calculating a company, I should say, need to realize that one of the dimensions of the space of light rays is actually five, and for a complex manifold, you need to have an even number of new dimensions. You actually get another dimension by considering these things have a helicity, not just a spiral that's going on here. So this light ray is not really... And I'll be releasing in space time something with a bit more structure, which comes about from the specificity, and also the fact that the energy is involved, so that gives you another parameter, so you actually get six dimensions over here. Six real dimensions, which give you a complex manifold, in fact, which is a somewhat striking fact. Now, you can say this is just a reformulation, it's a way of re-expressing your physics in a different way, so you can say, okay, there isn't any new physics here, we're just saying, okay, let's read it off from this space rather than from this space, and indeed that's the case, it is a sort of translation from one geometric picture to another one, and you might say this is useful or it's not useful, and so it depends on the details whether this is a good thing to do. It does have certain implications.

5:00 Later on, say you were trying to do quantum gravity, and you might say, in a more conventional picture, your metric in some sense becomes quantum operator or something, which means that your light current, or at least your plasma, your light current is positive in some sense. Subject to quantum uncertainties, but the picture that one has from twistor theory is something more like this, where the light rays being primary objects, you don't lose them, but nevertheless whether two of these intersect or not is something which is subject to quantum uncertainties, quantum operator. And so these are the kind of fuzzy points. So the idea is that it gives you a different perspective on what your quantum geometry would be likely to look like depending upon which point of view you start from. So it could certainly lead to physical differences, but they'd have to be dependent on what developments go on from the geometry you start from. Now, one of the troubles that I've always had with string theory as its... Now, one of the problems that has developed is this problem about the extra dimensions. Now, at this article which took place at another famous birthday conference, the one at Stephen Hawking's 60th birthday, and I had an argument there which has to do with the problems that I have with extra spatial dimensions. And it seems to me that they're liable to be extremely unstable. One of the problems is you have all these extra degrees of freedom in the extra dimensions, and to argue that you can't excite them because there's not enough energy available, that doesn't seem plausible to me. Because, okay, you may think you're trying to excite them in part to accelerate or something, and you might need blank energies for that or something like that. Non-linear instabilities are something like that, but geometry as a whole, that amount of energy is trivial on a cosmological scale, so it's very hard to see, at least I find it hard to see, the material. It's going to be immune from exciting electric dimensions. The argument is much more involved than just this, but this is part of it, and in case you can peel through singularity theorems, which tells you the classical theory is likely to be extremely unstable.

7:30 So, let me just leave that. Here, this is one of the reasons I have difficulties with looking at higher dimensions, and so that although I have nothing against string theory as such, in fact I rather like the ideas from the first order initially a long time ago, the problems I've had with the subject have to do with the extra dimensions. And here, the idea is, in essence, in regard to this space here, is the target space for your... String, so in other words, the room of surfaces, as you see we already have room of surfaces sitting there, and I'd be concerned that the more general ones are not just the room of spheres that represent points, but other kinds of room of surfaces, which represent structures over here, which would be rather hard to understand directly, to see much more easily what may be over here than over here. So that's very exciting to me. I hope the people who've heard me talk on this before don't mind me telling the same joke over and over again. People who've commented to me say it's very refreshing and exciting and thrilling and all this to see the string theory ideas now in twistor theory. And it's giving them what's likely to be a boost. And I say, yes indeed, that is true, but there is something that worries me. And I describe this in terms of, well, it's a bit like being on safari. In Africa, you can look through your binoculars and you see all the buffaloes rampaging all over the place. I don't remember everything. Then you look over there and you suddenly see a kind of straight horse. And this time you're running out of time. This is the best time to flatten that recognition. But still. It gives a new meaning to target space. Sorry? It gives a new meaning to target space. Target space. Well, yes, I've heard it. It's a nifty footwork to me. Let me be a little bit more specific about what's going on in twistor theory. This is more or less the picture I showed you before, but let's see a little bit better on this. I'm not going to assume that everybody here knows about twistor theory, although the subject is being around all the time.

10:00 Over here, we have this space, which we think of as a complex projected free space, Cp , and the coordinates here are four complex numbers, the ratios of which give you the conformance in that space, the projected and the complex numbers. And the basic relationship which relates these two spaces is this incidence relation here. There's a matrix of these things here, there's a mission matrix, and this relationship is called the incidence relationship, and you can read this either in here or here, you can fix the blue ones, the first coordinates, and ask for the, which points correspond to, which points are incident with that. For a moment, you can imagine this competition has to unfold, which I'll explain in a moment. You get a light ray over here, you get it that way around, you fix a point here, that one fix the orange ones, and then the blue ones run around. Then you find a little piece over here, which represents the point, and that's this projected line in this free space, which is the sphere. Sometimes I draw it as a line, sometimes as a sphere. It's just complex lines, but it's got two dimensions. Where does this relationship come from? This comes from the fact that you may want your points in space-time to be real, and you look at the incidence relationship and see what that gives you. That's just what I've got down here, the relative condition. This is the incidence relation, that is this over here. So this is the twistor condition for the reality or reality condition of your life, which I like to write like this, by defining a twistor as an upstairs object here, and the complex conjugate of it involves making complex conjugates of all these numbers, and interchanging the first two with the second two. So the conjugate twistor is that thing, and it's a dual object, and this is the, say, the product between the twistor and the conjugate twistor, and this condition here is the, as we wrote down on the previous transparency, the bottom, which if you set it, roughly speaking, is the reality condition, which is the real space down here. If you've got this condition here, you can make complex and complex space going here. And that's the basic thing.

12:30 This is the correspondence. I'll say a little bit more about some of these things. First of all, we go to a spinner notation and write this thing. This is a spinner with one primary and one primary index. And we have the incidence relation, which is this corner here, omega equals IR pi, something we just found a minute ago. Over here, that's the omega, this is the IR and this is the pi. These things now, the first two components of upstairs and the prime spinner, the last two components of the downstairs prime spinner, one of the slight irritations of this thing, not exactly like the account of death, but it's... The first paper I wrote on this subject, I had my conventions all wrong. I had this one upstairs and this downstairs and used different letters and all sorts of things. And then I sort of... I've used that to improve the notation and I've used that to improve what I call an improved notation of the sentence, the problem is that Witten always refers to the original paper, it's the only one I wrote when the conventions were, I said to you all the wrong way around. This is only the beginning of it. I've several problems I have with that. Let me just say a bit more. Interpretation of these objects is two spinors, the omega and the pi. It's not just where the indices are, but the letters also have to do with what they mean. You see, one has momentum and angular momentum to represent, in this term, the spinors. The momentum, if you have a null momentum, it's a massless particle we're talking about, then we translate that into two spinors, factorise it into a product.

15:00 So this P is for momentum, that's why the pi is over here, and the angular momentum, we represent that in terms of the symmetric product of the other one, pi, and we get this expression, and for a massive particle, there's a condition that the Harley-Wilansky spin vector, it is, This is proportionate to the momentum of the Mathis particle and this is automatic with this representation. So it's a very neat way of representing Mathis particles which is otherwise a little bit complicated because you not only have the condition that this should be a future point in an algebra which is automatic when you have this description here but you also have a relation between n and b which is also automatic with this description which is encompassed in this relationship here. And felicity, which is the S here, this is the factor of proportionality between the momentum and the spin vector, is simply this, basically it's a half of this, this particular product, so we'll get to that in a second. And it enables you to interpret the twistor space, not just the lightweight space, which is the string in the middle, but also the top and bottom halves in terms of spinning photons, well they're not specifically photons, they're mass of particles, it's a description of mass of particles, and the top half, of course, moves to the right hand and the bottom half left hand. So this is the basic physical interpretation of twistors. A few more points here. Okay, yes. Geometrically, if one wants to interpret a dual twister, that would be a plane in this subjective space, where the point is represented by the original twister, the contiguity of it would have to be a dual twister represented by the plane. So we can start from points and planes, but in particular cases where the point lies on the plane, that's the space Pn that represents the light rays of time. Okay, if you don't have this condition in the equation of Pn minus the lightweight space, if you don't have this equal to zero, and this was, well I should take it a little bit another way, let's say we don't have the condition of the option in real, one way of translating...

17:30 From twistor space to spacetime is to think of complex, fully complex structures, you don't worry about complex conjugation, and then you find that the incidence relation is satisfied by an entire plane whether or not the reality condition is satisfied. But you find that for any arbitrary system, we have a plane which points on an exact side of the isthmus. This is referred to as an alpha plane. So to do all, I did create what's called a beta plane. I didn't invent this terminology, it's actually very classical. In fact, it all comes from classical geometry, which was referred to as the Klein correspondence. The thing is that what Klein was interested in was how to represent lines in free space, the form of the projected free space, which is the space which represents lines, is this quadric surface in five projected dimensions, that's called the Klein representation of lines in free space. If this is a real space, then this will be a real quadric. These are some elements of what Witten is going to be doing, but he's going to do some complex space and some complex poetry, and the parts over here are represented by alpha planes, and the planes are represented by eta, so that's the geometry, and this relation is going to be... But that's the classical geometry which underlies twistor theory. It's just that in twistor theory we read the correspondence in the opposite way. So Klein was thinking that this is a nice way of representing lines in space, like points in this manifold, and what I'm doing is the other way around. So you think of this space here as space-time now, it's a four-dimensional space, and a complex space with four complex dimensions. The points here represent lines in this twistor space over here, and the alpha things represent points. So it's the plan of correspondence in reverse.

20:00 Okay, that's geometry. Now, I think at the top here is just to do what happens when you shift the origin. Remember that this is a nominant thing, so it's natural that it's not affected by a shift in origin. It exists as an agarominant thing, but we need to combine. If it is affected by a shift in origin, we go from the point O to the point of the position of the cube, the existing cube, and that's how the algorithm gets transformed very simply. And that translates into the correct way that the nominant and the agarominant get transformed. But here there's something new on this transparency, which is what happens when you go to the first quantized twistor theory, where here I'm imposing canonical quantization between the twistor, Z, and its complex conjugate Z bar. So, it's very neat because one just has these two things, it's the primary equivalency like, and the complex conjugate is the source of the canonical quantization. If you have this thing, these connotation relations, twistor can be to itself and canonically conjugate to its conjugate, then all these, everything I've said before is unchanged, except if you have to be a little careful about the helicity, it turns out that it is polarised, not just that, zz by z, and then you say, what's the analogue of a wave function? For a single particle, and twistor theory, well, in ordinary physics, you say you use either position representation or momentum representation, and to say your position representation, you say that your momentum variables are represented as operators in the wave functions, and it's a function that doesn't depend on it. These are the functions of the momentum coordinates of the lower arm, the function of the momentum coordinates, and it doesn't depend on the position coordinates, but here we're saying in the z representation, if you want your f to be independent of z bar, what does that mean? Well, that means the f of z bar is nought, in other words, f is homomorphic to z, so we're working with complex analytic objects, and this is one of the important underlying principles of twistor theory, is that one is

22:30 Going to cohomology for complex analytic structures and there is a lot of mathematical power in such structures and I'm just trying to harness that power in this approach. What you find is if you want f to represent a pure helicity state, that means that the number of operators, if I'm working in the z representation, instead of bars of operators, And S itself is this operator listening to the order homogeneity operator, unless it's shifted, this is minus two, the elicity is basically the homogeneity of the function, so if you want to represent the pure elicity state, it's not just homomorphic, it's homogenous as well, and the degree of homogeneity across most of the elicity to the elicity. And this is the very basic class of fields. This is the wave equation, this is zero, and that's the field equation for non-prime indices, there's the negative velocity and the prime indices for the positive velocity, this is here, this is the wave equation, this is the field equation for acids, half of this is zero, that's the Dirac-Watt equation, that's two and a half, for the neutrino, I assume that's this here, that's the photon, you get the left-handed path, that's the anti-cell fuel thing, and that's the... And you have homogeneity 0 and 4, where the left-handed part is 0, the right-handed part would be minus 4, in some conformity, which I'm going to show you the relationship between this part and this part, and linearized gravitons, you can see the plus 2 and minus 6, where the respectively left-handed and right-handed parts can graviton. So you have this very neat way of describing mass as particles in a pure felicity state in terms of homogeneous functions, but then you want to see how to get backwards and forwards from that to space-time, and basically what we do is we do a conger integral, and then we do a zero case, whose function is homogeneous to the degree minus two to the zero, and what you do is you basically think of this as a function of omega and pi,

25:00 You substitute the incidence equation here, so that's a function of pi, and then you integrate out the pi dependence, which is down here, so we end up with a function of x. And doing that, you automatically get a field equation, so this is the case of wave equations. For the other opportunities, you automatically get the fields that I just described in the last transparency, with all the different felicities coming out. Versions of this formula were known to Whitaker and Bateman many years ago, but this gives you the general expression for arbitrary spin, arbitrary velocity. And there's a point I'm trying to illustrate here, which is that if you make the quantum integral, the function you're integrating has to have some singularities that you integrate round. And the typical case, this is the simplest case you can write down, This is the homogeneity degree minus two, in this case the wave-equivalent case, so you have a product of two linear factors, and these will occur singular on you, on two planes, that's the A-plane and the B-plane, that's this magnitude or this magnitude, and if you arrange these intersects in the bottom half of the crystal space, then you find that in the top half, the singularity sets are always separated, so wherever your line is, It will always intersect these separated similarities, and you take a contour which goes between them, in the picture it's not a bias in the p-n, it's a real field, and it puts you to a non-singular real field. Perhaps I should bring something in which I was going to say later, but I think this probably is the point I'm going to make. In fact, one of the important aspects of all this comes from why it's useful to the... The twistor space that I've been talking about is divided into two halves by the space which represents the light rays, the top half and bottom half. And that was a very important early motivation for the twistor theory. Because it's like the way in which the ring and sphere from the top end of the cylinder divide into the top and bottom half. And the idea is that we have a function which is defined on the equator. Speakers also include quantum mechanics, geometry, algebra, mathematical physics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics,

27:30 Now, as I said before, a complex point in spacetime is represented by a line somewhere in this space that may cut across Pm or be somewhere in relation to Pm. But let's look at the imaginary part of the complex position vector of this complex point. An imaginary part could be... VRO could be space-like, it could be future-pointing, or past-pointing, or future-pointing, or past-pointing, and the different possibilities for the imaginary part of that complex position vector are expressed over here. And the imaginary part, this past point, that's what's called the forward tube, and what I'm really interested in quantum field theory is field altitudes which extend into the forward tube, that's the positive frequency of the condition, and what that means in twistor theory is that you're talking about something which is defined in the top half of the twistor space here, because it's the lines right here. So that was an important ingredient to... Twisted theory has a nice representation of this positive frequency condition, which is basic to quantum field theory, and going back to the transplanetary which I was explaining before, when this line applies to the top path, these planes are intersecting the bottom paths, you can see that the singularities are separated from whatever this line is in the top path, so you have a non-singular field in the top path, and therefore it's positive frequency. This is a nice way of generating positive frequency fields, and in fact it's indeed in general, if you phrase it almost appropriately, I'm not going to go into details of that, but it's still a little awkward, you say, what about singularities which are separated, what does that mean? Well, they could be more general, they could be great patches of singularities, and again, if they're separated, then it's not singular in the top part, but that's good to know for frequencies.

30:00 It took a long time to appreciate what this was all about, and the answer, after some re-education from Michael Atiyah, which was very helpful, is essentially what is called sheet cohomology, and, although I don't particularly want to go into all this here, it has to do with the covering of your top part of the twistor space, and the simple case that we're looking at here, the covering would be Say in the top half we leave out one patch of singularities, and in the other patches we leave out the other patches, so we leave out the other singularities in the union of the two covers of the whole object, which is the space. So you're moving the covering of your space wherever it sits, and then you define the cohomology in terms of what's in there. I won't talk about this in detail because that's just... If you know about it, that's fine. If you don't, we'll take too long to explain what it's all about, but I think what's really helpful is to think of an underlying point to all this, which is that if you think of a space which is covered with a lot of patches and sets, And you want to somehow to deform something which was initially, say, flat into a curved space in some sense, what you might want to do is slide one patch over the next, and the next patch over the next, and so on, and build up a curved manifold in this way. And in fact this is exactly a sort of what you call a non-linear version of cohomology, the vector fields that you can attempt to slide one patch over another with respect to the fields going along. If you just think of this as a pentagonal, then you have the first term on the other. And that's exactly what I mean. So as before, if you imagine performing this in a final amount, then you have nonlinear terminology. And this is the way you start talking about general relativity and so on, but I don't want to go into the details of that, this is just to show you for like if there's a good reason for these cohomology ideas, it's something that has the potential to generalize into a nonlinear theory where you aren't simply talking about something modular or something else, you're really imagining shifting actual factors across each other in an empty world. Okay, well that's basic twistor theory. I think that's pretty well said. Now what are these new ideas that have come about relating, well, they actually have to do with work that was done previously which didn't apparently have anything to do with twistors.

32:30 These are blue arms. You try to represent your blue arms in terms of helicity states, and some of them are positive helicity represented by arrows that mean that you are perfect, and the opposite of this is by arrows coming out. And if all the arrows come in, you get zero for your amplitude. If all but one come in, then you will get zero, or I'd probably rather say you get zero again. Then you get interesting attitudes which can be worked at exquisitely, and this is what these people did here, and Mayer did in fact notice that there was something to do with twistor theory, I'm not totally sure about the relationship between all these things, but he certainly did point out the connection with twistor theory, and Witten then... We can argue that these have a string theory, which is the space from which these things would flow, and it's a very attractive thing in many respects, like to me, in the first place, when I was talking about honest interactions, which have to do with physics that we know about and believe to be out there in the world. The space-time is four-dimensional. That's nice, from my point of view. And finally, it actually does make use of twistor theory, which is nice for me, and obviously not nice for people who don't know about twistor theory. Let me start by showing you something that did. If you think of a function of your momentum, or say a single momentum, and you can write that as a function of the pi and the chi bar. And then you can do a Fourier transform, not the normal one where you do a Fourier transform with respect to p and you get a function of x. So if you transform with respect to one of these, it's very much the same, it's asymmetrical. Say with respect to pi of r, and that puts you for a moment. Remember that only when you get pi of r are canonical conjugates. So it's not unreasonable that you go from one to the other by a Fourier transform.

35:00 So that is the way that you get to a twistor function. Now, I'm going to sort of interject various points of worry that I have over time. This very much underlies what people have been doing in the subject recently, but in a certain sense it's not quite right. The problem is that when you go from, I mean, what you can imagine, suppose you did it not just on one variable, you then did it on the other one. So the Fourier transform, sorry, I didn't say it on your own. If you have a function, twistor function. And then the Fourier transform on the omega, that gives you one of these functions, which is now function of the momentum. You could, on the other hand, Fourier transform on the other one. So you could do this on the other one. You could go the other way, where you could Fourier transform on this one, and then you get a function of the dual crystal. So you could say, this is the way of getting from the function of the dual crystal to the dual crystal by doing two Fourier transforms. But we know that's not right. When you Fourier transform, you do what's called a twistor transform, and you go from a twistor to a loom twister, and it's something which is not correctly represented by a Fourier transform, it's something else, it's different. You might use morally a Fourier transform, but in detail you find it's not good for that, and one's somehow throwing it overboard by using this transformation. And this does begin to show up in certain places and stuff. So I'm afraid, although I don't understand what's been done here, Every now and then I'm trying to read, and little things come into my mind, so I'm not going to understand what's involved. So, I think there's great potential here for trying to, well I would say, there's a lot of string baggage to make use of a lot of ideas from string theory, But the motivation for a lot of these things comes from thinking that we start in the space science theory, but we're not really doing that, and so I'm not quite sure I can follow that. I'll perhaps explain what I mean a little bit more in a moment. But let me make another comment here, which has to do with signatures. Now, in Witten's approach, he tends to use plus plus minus minus signatures.

37:30 Here you have this twistor space divided into two, top half and bottom half, and this is physically what you really do want, and this is certainly at the level of what one directly experiences, the way this twistor theory should be used, but pure mathematicians, if not so interested in physics They like positive definite space-time. Then there's of course other reasons which have to do with the fact that if you want a space-time in which the culture, the bio-culture, is anti-self-dual or self-dual, then you find it kind of complex if you start with one of these images. Whereas if you start with a positive definite signature, then you can have a relatively real space, which is certainly all-or-interceptive, and so the twistor correspondence actually turns out to be something, some space, but certainly something you can much more readily realize in geometry if you adopt a really positive space-time. And then you have a twistor-convex conjugation, which is not the kind that I refer to where the conjugate of a twistor is a dual twister, but where the conjugate of a twistor is another twister. And this case is one where you don't have any real twisters. That is, if you ever had z equals z bar, it would imply z equals 0. The case that Witten has been interested in is the plus, minus, minus signature, where I did describe this briefly as if you're considering a real projected free space and you're looking at groups in terms of lines in this real projected free space that are given by this real quadrant. It's the kind of thing that Witten has been working with. I think for various reasons that it's just sort of easy to see what's going on in a real projective free space, but the view I take it is rather like what would have been the case if you were working with a positive-definitive signature, where you've got some signature which you admit is wrong, and then you say, well, we're going to work all this out, and then we're going to get something analytic, which you then extend into the physically appropriate situation.

40:00 Now it seems to me this is in a certain sense going in the wrong way, because this is where you want to end up, so why not work with this? And it has structures like this, which seem to be just the sort of thing you want in quantum field theory. So, to me, it seems to me why not to look at all this again, and try to do it in the right symmetry case directly, rather than hoping to do some kind of pseudo-Hook rotation in the end, which tends to give you what you want. So these are kind of minor quibbles, if you like, but they have points of some relevance which might begin to be more and more important as it develops. Now what we can use is, this is in and out states, basically things which look like momentum states. And then you do this quick transform and you come back with things which are delta functions supporting them on alpha planes. This works fine if you're in a... This sort of real case that I was talking about, the signature plus plus minus minus, this case here, we get real alpha planes, but in the appropriate signature, there's something a bit funny about these things, because they can't be spoiled in a mathematical case, they have to be, I think these have to be singularities of some kind which stay in the complex in some ways. Anyway, let's not go into that, but I think that one needs to study this a bit more. The kind of string theory that Witten seems to be talking about, although here I couldn't understand any sketch to this degree, but one has this Chern-Simons action, which is used in twistor space, and, well, first of all I have a problem here, because what are actions for, and what is aeronautics for, and all that, thank you. However, in physics, through the use of grammatical differences in field equations and how they propagate in time and so on, but there's no time here, if you like, the time is all built into the twistor geometry, so somehow you don't have that. But, for this particular type of action, what you have is... Well, essentially, a topological quantum field theory is actually, in this context, what we might call a holomorphic quantum field theory, where it only depends on a complex structure and doesn't depend on particular connections. So it has this very remarkable property, the kind of reaction you're using, that it doesn't give you any...

42:30 I'm going to give you a few equations, and that's what you want, and with the topological quantum theory, you might say, well that's very interesting mathematically, but why do we want that in physics? Because you want a few equations in physics. But it did strike me that maybe twistor theory was the right place for this, and in fact I did write an article in the Twister Newsletter in 1988, suggesting that one might actually... A version of topological quantum field theory, which is really a holomorphic quantum field theory, I call this the holomorphic linking. I was concerned with the gas formula, which is typically the integral with the linking number for closed curves in R3. And this is one of the formulae with gas, which tells you the linking number in terms of the double integral. It's complex and complex, but it still makes sense. And then fiddling around with it, you read the things related to twistor diagram theory, and this is work which Andrew Hodges has carried forward, leading his students, and it's a way of writing quantum theory theory in terms of twistor theory. So it's very nice to see that it does relate to twistor diagram theory. So let me just mention one thing which is, for me personally, interesting, because if one is trying to represent these curves, these curves which maybe now are not straight lines, but are a heroic purpose, it actually goes back to the work that I did when I was trying to do algebraic geometry as a research student, and the problem I was concerned with had to do with how you represent spaces of non-maximal dimensions, not hyperspace, as you said. The first place you notice this is curves in free space. So how do you represent the curve in free space by equations? You just think of a K-deform and consider this as a curve, and instead of thinking that as a quite lovely world, particularly if it's a condition of the mind, and that is an equation of the mind world. So it seems to me this may well be the right way to represent curves in twisted space.

45:00 Now, one thing I haven't yet mentioned is that... I mentioned the higher dimensions, and I'm certainly pleased that I haven't thrown away the problems that I've had with higher dimensions in physics, but there's also supersymmetry. There's no evidence that says, yes, at least from observation, that supersymmetry really is a feature of adventure, whereas string theory as it's developed has very much been dependent on supersymmetric ideas. And in fact, I started cheating when I described what Witten and Co were doing. They really talked about not a Cp3, but a Cp4, where you've got a force that assumes the generators. Now, this is all related to the question of anomaly cancellation, which is originally a required force, strings, linear parameterization, variance, and so on. And it's always been regarded as a path or driving force in string theory. And it's carrying this over from space-time ideas to the dismissive ideas. Now do you really want the extra supersymmetry? Well, what we find actually is that usually you want the Calagri-Yau space, in order for this anomaly cancellation to work. Whereas here, what the Witten company have found is that if you regard CP3 as a, the core supersymmetry generators, then it is actually a super Calagri-Yau space. But to me, I would say, well, I'm not sure I like all these generators because I'm convinced that they're part of nature. Do we really need it here? Well, I'm really out of my depth here because I don't really know how it comes in. But it does seem to me that re-parameterization invariants may well not be something you want in this twistor theory. You really want it to be independent of the parameter here. In twistor theory, you see you can have a version of twistor theory which is completely conformant and invariant, or you can have a version of twistor theory where you actually know where infinity is, which is represented by this line here. And in fact, you can put the j of the plan from the twistor space down to what's called the pi space. So when we go pi, it goes down to pi. And this breaks conformant and invariance. Now is it the case that...

47:30 If you throw away your free concentration, In this case, we have this projection down here, where we know that this curve has a natural diameter all the way along it, which has to do with the planes which pass through this line, which have a sense of infinity, and there is a dynamical projection which breaks conformal invariance. We don't have conformal invariance in physics, we don't seem to have supersymmetry in physics, maybe these things are related and will have to do with somehow getting rid of the supersymmetry. All right. Well, no doubt this is all very controversial. People are telling me I don't understand what I'm talking about, which I would accept fully, though, so thank you very much. There's one sort of fundamental question that bothers me. The twistor space has to do with whole water functions, and you know to carry information you need non-analytical functions. So when you can't sort of generalize, what is this about? Well, the general comment I would make to that has to do with how you represent things which are not analytic, actually. I mean, I think it's called hyperfunctions, which I believe are the right way to talk about non-analytic objects in this sort of subject. See, one normally thinks of distributions, which are the duals of C-infinity functions, and you can talk about delta functions, not just distance communities, but delta functions themselves. But can you generalize that to duals of analytic functions? And indeed you can, and that's the theory of hyperfunctions. And theory, I mean, it's what I regard as one of the big miracles of complex analysis, if you like, that the hyperfunctions are defined fundamentally in terms of holomorphic functions. But it's basically the jump. If you take a holomorphic function which runs out to a binary somewhere, and you extend it to another one, the jump between one and the next is a hyperfunction. So hyperfunctions are in fact basically described in terms of holomorphic functions. Now it seems to me this is the framework, in twistor theory, this is the right framework for talking about things, as you say, which are not analytical, things which jump and so on.

50:00 And yes, I think you're absolutely right, you want to be able to handle such things, and it wants somehow the theory of hyperfunctions to be brought in to the subject, which is not something which has yet been done properly, but it's sort of marginally played a role in things that we've done, but we don't need that so far. I've got a couple of questions. First is, what is the physical problem that one is trying to solve here? Is there really anything that quaternary QCD people have not been able to solve that is being addressed using this new method? I believe that's the case, but I'm a bit out of my depth here. I mean, the original calculations had to do with these maximum helicity-violating amplitudes, which is where you have two, one helicity and the other one is all another helicity. Now as I understand it, or I don't understand it, but it seems that this string approach gives you a way of looking at all the, basically you start from something which may be an anti-self-dual solution and you perturb away from that to get the self-dual components. Now I don't know to what extent this is actually done by it. There seem to be some anomalies in what they talk about, and since some of the conjectures aren't exactly true as stated so far. So I don't know. I'm sure the status with regard to what's proved is pretty minimal at the moment. But what if you say people are perfectly happy without any of this? What is new? Where do they stop? Well, I would say two things here. One is something to me which isn't surprising. Okay, you're looking at conformity invariant, helicity, amplitudes, helicity states, and twistor theory is the right framework for making that look neat. So, that's just a mathematical form. It may not be physical. Now, it's not clear that this is a... to what kind of a theory is this? And I don't know. In fact, it seems that there are two different theories, at least. Because they have these D-brains, what they call almost space-filling D-brains, and I'm sort of bothered by that, I don't know what that means, but the strings, in one approach, they have homomorphic curves, which are supposed to have nice relations to these amplitudes and points of alignment and so on, and in one approach, these homomorphic curves are the strings, and in another approach, these are part of the D-brains or something. Well, that makes no sense to me until you've got a real theory, which

52:30 It tells you what you should really do and I'd like to see that. So I can't answer your question because I don't know it well enough. It's to some extent just, well, Witten has done things before which have had certainly mathematical implications which are often true and surprising. And so there's a reasonable chance that that's the sort of thing that's happening here. But I don't see it in what I've understood so far. I can't say I see it. But I would hope to see this as a theory in which you not just get these gluon attitudes and so on, but doesn't actually say anything about quantum gravity. Well, the claim isn't supposed to be going to, but the quantum gravity that is talked about so far, and this is only a sort of partial theory, this conformal supergravity, and I don't quite see, you know, how should I interpret that, that's not the gravity I know of. Is that the right thing to do? I'm a bit doubtful, but it's just a lead. That's all I see. This is a lead into, I mean, twistor theory for a long time has in some sense been stuck, that you've been able to translate amplitudes, there's a lot of stuff in twistor diagram theory, you've been able to take the answers that you know from Feynman theory and translate them into twistor theory and you get some nice diagrams and anthropologists have done a lot of work with that. And it looks like important stuff, but it's not motivated by some central principle, and here at least, although I don't see it yet, there's a reasonable chance of finding the central principle which will supply what's been missing from twistor theory for a long time. But I can only say this in a conjectural way, I don't see it. The second question was, the basic formula will be wrote down, and you deny it, and all these people... That all had to do with just zero-resonance fields. It had nothing to do with kind of deep twistor theory, which has to do with the twistor transform and so on. So, is there any connection between the deep twistor theory and the... Well, if you call it the deep, well, you actually solve the nonlinear problem. Right. So, it's not the magnetic integral of nonlinear. No, I agree. That's what my... I don't see any gravity connections yet. What do you want the Yang-Yu to see? All I see is a claim, and here you have to confess that I'm not very good at extracting information from these papers myself.

55:00 I stare at them for a long time. But I have a colleague, Lama Mason, who is much better at this sort of thing than I am, and he tells me that, in fact, You could read these amplitudes which you get out of this theory as perturbing away from an anti-self-dual starting point. So you basically use the nonlinear theory to represent the anti-self-dual solution, Yang-Hu's solution, and then you perturb away from that. So it's not representing the full Yang-Mills thing, it's a neat twistor expression, but you're sort of half using it, you're using the ward construction and then you're perturbing away from that, which I think is interesting, but it's not, you know, maybe it hasn't gone as far as one would like to see. Well the trouble is I can't tell that from what they've done because Witten works with this funny signature, what to me is a funny signature. So there's no mention of PT+. That's supposed to be at a stage later on when you can do some pseudo-rotation which translates the plus-plus-minus-minus signature work that they're doing into what I would regard as the plus-minus-minus-minus signature. So they don't talk about that at all, which to me is missing a big part of what twistor theory is all about. Take that picture and see, translate what they've done and see how much it can be translated into what I regard as the correct signature. But it's not what they're doing. Not yet. You mentioned the twistors that are not null, we thought they were spinning photons. Could that connect with Waldron's Holistic Quantum Theory of Life? I've got the clues. I'm assuming it's something I know a bit better. I'm sorry? He was professor of mathematics at the University of Ulster. I see, well that's interesting.

57:30 I was just going to say, when it starts with plus plus minus, it gets translated into a random signature. How important is it that this starts off with a random signature? Well, I don't know. I mean, it's basically the same philosophy as one had with quick rotations. I mean, in some respects it's obviously putting a path into goals or something where you might have some convergence. There's a chance of convergence between positive and significant shapes. Of course, that's not true with plus plus minus minus. You don't have that advantage. But there seem to be other advantages in this, what he does. And the intention is, at the end, is you pseudo-liquidate. That is to say, you can change one of your pluses into a minus, which are all your entities, by analytic continuation. And this could make sense. But it does seem to me that it's kind of going the wrong way. I mean, you're not going directly to what you want, you're going indirectly and hoping that it's going to get to you. And you don't have the advantage that there is with the positive-definite signature, the ordinary solution approach, where at least you have a chance that certain things will converge. So, I certainly don't see that advantage. I'm a little bit amused by what the advantages really are about using them, and I think it's mainly because it's easy to think that when you talk about real projective free space, somehow you can picture what's going on. And you can talk about these alpha planes as things of the sea in some sense, but it seems to be somewhat limited and no real advantage of them. That's my impression. Sorry if you answered this already in your talk, but I missed it. In his application to the... Well, you end up by having explicit expressions, which are in terms of these momenta. It's fairly conventional, that is to say, you have amplitudes that you would have written in momentum space, and then you have a delet function which expresses momentum.

1:00:00 But that's very conventional. And the twistor theory is just a way of getting it, as far as I can see. No. Well, Witten's perfectly aware of these things, and certainly relates to them when he can. But he makes some comments about you should really do this using D-bar cohomology, you see, well, the cohomology I use is Czech cohomology, which I find much easier to understand, whereas the kind of cohomology that you'd get out of this sort of culture and assignments approach, I think, is probably more related to D-bar cohomology, which is equivalent to that, but... It's equivalent to the things I've said, but in a rather roundabout and obscure way. It certainly can't be, when I say equivalent, it's not completely equivalent, because everything they do involves factoring out the momentum. You see, you have the functions of momentum, and then you have the del function, which expresses momentum concentration, you take that out, and what's left is a polymorphic object. Now, to a twistor theorist, that's not the right thing to do, because when you work with momentum states, you're working with non-normalizable states, and one of the programs in twistor theory is to translate everything into normalizable states, so the elementary states, I didn't use that term, but the things I had on the transparency of the wave equation were what we call elementary states, and those are completely non-singular normalizable states, and so that means that the amplitude you get out Are much more likely to be finite things, they only have problems when you have real honest divergences, but with the conventional approach you've got in a sense divergences already because you've got your momentum delta function, and in twistor theory you have to have somewhere representing that in terms of hyperfunctions or something, so you don't actually do it that way. I mean, the answer to your question is he hasn't actually used that, although Lange tells me that somewhere in the depths of one of these papers it is actually used. But I hope to show you myself.