Twistor Diagrams

0:00 An analog to Lagrangian methods that are used in space-time. So that's rather unsatisfactory. It's not satisfactory. We can see a great principle and derive crystal scattering attitudes from it, but at the moment we can't do that. And again, of course, we've got many more things that come into the arena now than we thought of in Roger Fennery's thought of in about 1970 when all this started. I must emphasise that all the essential ideas I should describe were identified by Roger over 30 years ago, and he started off all these calculations. we didn't know, and no one knew then, I think, quite how important to list the amplitude would be, or indeed the fantastic success of this whole calculational approach, these miraculous things which are much simpler if one has any right to expect, and indeed the Feynman diagrams generally are much more hopeful of an approach than were thought 30 years ago when the strong interaction really seemed outside that scope. So that's all very positive from this point of view now let me start off with something that hasn't actually been mentioned yet which is very very simple thing about this representation to zero and resonance field and gives us a sort of elegant starting point for thinking about what linear functionals of these fields could and should be like and that's just the zero thought that when we've just got inner products we've got one in field one else field and we want the amplitude for it to go in as one thing and come out of something else, and that's just the inner products of two states. One of them understood here to be positive frequency, and one negative frequency. I'm not going to make much of the positive-negative thing, because it can be assumed that everything here is crossing symmetric. So you must just choose your channel and which allocation is positive and which is negative. And the whole point is that this very simple functional of two fields is given by an extremely simple contour integral where you just multiply because the representatives together, one functions together to a contour integral. Everything is gage invariant there. If these are spin one field, everything is gage invariant. Manifest is finite because it's a compact contour integral and that's the starting point. If everything were like that, we'd feel happy. That's the idea. And it's notable that contour integration comes in

2:30 But an important point here is that all the, and I'm going to write this as a little diagram in which that black glob, that black is in, and the circle becomes, it's thought of as a vertex, rather like in Simon diagrams, attached to an in and out field. But there's a lot of hidden abstract structure here, the point of those two lines coming out rather than one, to describe the external fields. that encompasses all the abstract structure described in first chronomology that Stephen Huggins introduced us to in that first lecture. So there's a massive amount of ideas in that simple picture, but nevertheless it is a simple picture. Where is the integral? Yes, all right. So that means, the wrap-up means integration over Z over Z. So the constant interval in Z space. That Z, I've left vague as to whether you think of it in the projected space or whether in the non-projective space. So that's all it is. Can you tell us what the point is? Well, it has to wrap around these, if you imagine these are simple poles, which they are in the simplest possible case. And if you use a non-projective space, there's simply one S1 around each of those lines. So that's the simplest way of thinking of it, and that's a good way of thinking of why there are four lines there. So Z0, Z1, Z2, Z3, if you like, and a simple pole in each of those. and it gives you, that gives you, that gives you an example of a concept. So you're using the fact that you think of each F and G's at two holes? So you do a four-dimensional. That's the, that's even called the low-tech view of what a twister function is. It's really a cohomology, but if you start building it up in the elementary way, then one of these is just two, just a rational function with two simple poles, and the sponsor just wraps around each of those poles, four altogether, taking out the four dimensions of the twister space. It's actually very simple, but of course, to do fields in general, then you need the machinery of cohomology to describe that properly. Right, well, the next thing we need, if we can do anything about momentum space, about capping apertures, is we have to represent what in momentum space you call the delta function on momentum, or momenta, but really in x space is just a question of integrating over the whole of space-time. So we've got these two different viewpoints. And probably everyone's very used to working in momentum space, so think of it as a delta function, but it's really completely equivalent to integrating over a real nephostic space.

5:00 Now, how are we going to do that? Well, I'm going to show you an adventure way of doing that, and then say that isn't actually quite the right thing to be doing from our point of view. Well, the first thing is just simply that the evaluation of a field amounts to doing the same type of contour integral, but with two simple poles onto dual twisters w and y, and then there's just a conformal factor which brings an infinity twister here. That's one way, it's a diagrammatic way of writing what the correspondence is between twister functions and spacetime fields. This is the scalar field, that brings in all the essential points just to talk about scalar fields. So, now these simple poles simply onto a line W-Y. That's what they do. They think of the simple poles here as being like a delta function, and they just enforce a certain position, and the condition is that the z, which is being integrated over, is full, it has to lie on the line W-Y. That's a way of thinking of it. But we're expressing it not as delta functions, but in terms of Cauchy's poles. Now, alright, we can do that many times through each field, each external field, and multiply them all together and then we just need another fact which is the integral over space-time that has a form here which corresponds to this form in W-Y space and then there has to be a contour corresponding to the extent of Minkowski space treating it properly and that again the whole point is I've put in compacted because it's so important that it's manifested primal. It's compactified that's the central point of this and there's a certain topology which is essentially whether to scale S1s and S3 times S1 form in Cosby's space. And so I can represent this larger thing if I use a new type of white vertex who corresponds to the Ws and Ys. So I can write that diagrammatically as this. This is the product of all our external fields all multiply together and integrate together in Cosby's space. I simply have loads and loads of Cauchy poles here functions pulling each of these external twisters onto a common line, W, Y, and then an integral over W, Y, which corresponds to those lines varying over the whole of the encostment space. And there's a conforming factor here, which is not very important, but it's necessary. It's not conforming topologically, but it's necessary for the thing to be correct.

7:30 If R is equal to 4, if there are just 4 external scaling fields, then everything's conforming invariance, and that you can forget when that factor isn't there. I won't say anything about crossing symmetry, you might follow, except this, that, you see, your choice of where Minkowski's phase is in relation to the twist of functions depends on which channel you're taking, and so you can say that there are many different contours for this, and for each allocation of positive and negative frequencies for the external functions, there's a sense of different contour in W-Y. So, the usual idea of crossing symmetry translates into the idea that you have a single form in a product crystal and dual crystal spaces, but a different concept, depending on which channel you're talking about. And obviously there's more detail on that, but I don't want to change more about that in this talk, because that's all our needs. Now we can go immediately from this to translate the pure gauge-theoretic scattering amplitudes. That's the essential ingredients we need. And so far I haven't really, really done anything. I've put things in a fairly superficial, twisted geometric language that you knew anyway. So what I'm going to do is to do an immediate translation of the fourth field, scattering amplitude, making use. I'm simply, I'm not deriving, I'm afraid, I'm just using the Park Taylor formula, which everyone's very familiar with, and we translate that into a corresponding twister integral. So the interior here corresponds to delta function, and think of these simple poles here, and a thing like delta functions, they just pull all the external twisters onto a common line, the integral over the right vertices in the integration over such lines, but there's another factors here, which are this is a temporary notation, here there's simple poles on the outside here, corresponding to those three factors, and then there's a cube numerator factor, which I left dashed at the top, corresponding to the 1, 2, cube. I mean, this is the, I've chosen the order of vertices here where you have a 1, 2 in the denominator so that it divides into the numerator factor in this particular case. now this so this is a straight transliteration of this formula it doesn't do anything else, that's all it is it is in terms of twisters

10:00 it has certain properties which you like and others which you don't one is that it is manifest engaging variance but there are many problems with it, one is it's nothing like the simplicity of a Feynman diagram representation here, but loads and loads of different elements here, numerators all sorts of denominators, all sorts of going on is horrible. And that's one problem. It doesn't suggest anything more fundamental. Of course, there are tremendous ways of going with this formalism, which everyone else has been talking about, with extremely powerful results, and we would like to make use of those more effectively. But the point of what I'm going to say is to see what happens if you go in a slightly different direction. That's the idea. And the second problem is is that, how do you know there are any contours to this, because I'm stuck in lots of new singularities here. And if I stick to my viewpoints wanting to be manifestly finite using compact contours, how do I know I can actually avoid these new singularities? And that's not a, when we impose this very strong idea of finiteness, which is much stronger than what you normally think of finiteness at the end of a calculation, that's actually problematic. So, I've got to do two different things here to show you how we sort of get going on this thing and then show where that leads to. Well, the first point is this, and it's a very important point. This only uses the most primitive idea in Twister geometry, really, which is the representation of the 0x-mass fields by one function of one Twister. But that's not the only idea in Twister theory, and right from the start, it's been seen that by representing them as functions of two twisters, or three twisters, or n twisters, not just of one twister, and in particular you can represent things which are not 0x mass fields by using functions of many twisters. So that's one thing that's very important. I've said several times that twisters give you 0x mass fields, or they're for handling 0x mass fields. I don't know what they are, but I don't want to give you the impression that all you can do is assume there is much more. So that's one thing, and that immediately, well, with the analysis of two cluster functions that was done in the 1970s, it really follows that we can represent, I can change this basic diagram, this internal diagram which I started with,

12:30 it's a straight transliteration of the idea of pulling all the clusters onto a common line. equal to this, and that's not at all obvious, that's not meant to be something which you just see, oh yes, that follows from deep properties of the two-twister internal symmetry group, the fact that you've made that replacement. So that's an ordinary symmetry, so I've made a choice here of which two, I've divided the four here into two pairs, and I've chosen and selected one pair for treatment as a two-twisted function rather than as a two-one-twisted function. So that I'm just asking you to believe that that's a worthwhile thing to consider. We're using a more sophisticated aspect of twister theory to do that. And if I do that, then I make a certain simplification in expression for the four-field of pure gauge theory scattering amplitude, which I've now got rid of, what, two singularities. Now, I should mention that's not quite the end of the story because in order to make the simplification, there actually have to be some boundaries in this new diagram. The boundaries are conformal breaking ones, and they correspond to looking at the edges of compactified microstatic space, i.e. the null cones infinity. And that means that actually you've got to introduce some new, into these new boundaries, which are boundaries on the, are typically, I mean, if X and Z are the twister variables, the W and Y are dual twister variables in the integral, you can introduce boundaries on subspaces of these, of this form. And those are important, but I'm not actually going to draw them into the diagrams that follow, they're considered implicit, just as simplification. But those are an important aspect of the So that replacement needs that new idea, and it follows in the fact that the two-quista function is not a conformal invariant idea. It naturally brings in conformal breaking. Now, why do I think that's an improvement? Because I've got rid of a lot of singularities here. Well, I can do an integration of my parts on what's left here because I've got three factors in the numerator and I've got three in the denominator and to buy a little bit of trickery well, not trickery, but rather important stuff

15:00 I can actually make those all cancel each other. Now, I just need a little bit more machinery for doing that. I've already brought in double lines for double poles I shall need triple lines for triple poles and quadruple lines for quadruple poles so I hope the notation is self-explanatory there single line just like a square root of a delta function just pulls the thing together these pull things together but they do derivatives as well but I need another thing and that's the most important which is an anti-derivative I need an object which I'm writing with a waving line which has the property that its derivative gives you a simple pole that's the most important aspect of it Formalism, really. That's been a creative part of it. And there again, there's a very natural choice for that. The only possible choice, really, is that this line indicates the condition that the concert we take isn't a closed concert, but it has a boundary, a boundary lying on the subspace where W, Z is north, rather than the Cauchy pole enforcing W dot Z is equal to north. Now, if I do that, I'm just going to give you a little bit of flavour working with swiss of diagrams is like, we start off with this one here, and I pull these numerator factors off, and by integration by parts, I can make them cancel these three unwanted similarities around the outside, at the cost of introducing these new boundary lines. So the first one, I pull the end of this numerator down here, and then down here, And lo and behold, it cancels the pole on the left, and it turns this diagram to this diagram. I can do that again, pulling this one down again, and down to there, and it turns into that. Then I can pull this one back up to here, and then down there, and it gives this one. And at the last stage, oh, next to the last stage, I pull this one down again, it gives this. And lo and behold, the last stage is conformally invariant. which has got rid of all the numerators and the unwanted denominators, and it gives this, and it gives this diagram, where it's quadruple poles, and these boundaries, they didn't mean an arm pole, they're zero poles, and they're not about poles at all, they're boundaries. And there's just a hidden part of it, there is some conformal breaking here, but conformal breaking lies in these boundary lines, which run across, horizontally across the,

17:30 across the diagram, which I've suppressed in my notation. Now you can see the effect of the asymmetry of choice here, I could even well have arrived at this. Just as good, these must be equal. And they're by no means obviously equal. They're equal by quite a deep argument. And that's the first example of the non-uniqueness of these diagrams, which is an important part of the question surrounding what they mean and what they're supposed to be and what we do with them. Now, here's another important point. I've chosen the external fields to be in a particular order, namely one, two, three, four, and you'll notice that they give rise to an internal polygon. I mean, now they've been so simplified with the emission of all these unwanted singularities that they just come down to a loop. and the order of the edges around that loop is in the correct order. It's one, two, three, four. And that's no coincidence. That follows from the form of the part failure, the formula which we used in doing this integration by parts. But it's not, I wouldn't say it's a bit obvious either, but it's always going to rise to that. So that's an important aspect of it, which I'll come back to. Now, the next point, if I haven't dealt with, it's a question of whether the concepts actually exist for these things. See, I've got rid of those poles, but at the cost of introducing something different. And that's only a formal derivation there, because I introduced these boundary concepts. How do I know that such boundary concepts actually exist? And so I ask, is there a genuine concept, a compact concept for the things that I just arrived at? And I can quite definitely say the answer to this. One thing I'm absolutely sure of is no. And the reason for that is now that many people have now referred to this problem of infrared divergences, or multi-particle singularities, which arise when you have a process which could take place through processes of lower order occurring independently of each other. And that's just the case here, in general anyway, not for every choice of the gauge elements, but in general that will be the case. And the underlying problem here is one with perturbation theory. It's not a problem with twist theory, it's a problem with perturbation theory. You can't, in this case, really separate the first order amplitude, which we're considering, from the zeroth order amplitude. But really the two things are confused, and it's because

20:00 the in-state and out-states aren't really properly defined into infinity when you have a Coulomb force acting. So there are all sorts of, and this is a problem that arises in this even at the tree diagram level, not at the loop level, because they're actually trying to evaluate these things on finite normed elements of the Hilbert space. Norming them to do that is quite happy with an answer which looks like 1 over k squared. And you don't worry about what happens when that k is 0. You grab that as a finite answer, but if you then try and take that answer back and express it as a function of the in-and-nounced space, which actually is the real finite normed one, difficulty which comes up at a higher level normally in the well now there are various things you could do you could say oh we won't worry about that we'll just treat these things formally we'll just say oh the boundary line is the thing which has the right derivative processes and so forth and that's a point of view you could take you could rather cheat and just say whatever you look at the exceptional ones where everything's fine I don't think that's really two people and the thing I favour is to modify the theory at this point and do a regularisation And normally, in fact, someone's already written down how you might, within a twister expression, do a dimensional regularization. So what I'm going to offer here is something which I think is more fundamental and quite different from what you could do in space-time. And the idea is this. First of all, you think what a regularized version of this amplitude will actually be, and then you find a twister way of achieving it. And basically, you've got to subtract off an infinite multiple, or a mandatory infinite multiple, order amplitude, and that will give you something which is completely finite. And that can be done in a limit way, as follows. You take the mediating particle at mass n rather than mass n, then the infrared divergence disappears. But you get it appearing in a logarithmic singularity in that mass 10 to 0. So I subtract off logarithm of that mass squared times the 0th order amplitude. but of course I have to introduce a fundamental reference unit mass in order to make that, you've sensed that logarithm but I can do that and then I get something which is finite as I let my varium m10 to 0 in momentum space this is indistinguishable from the original amplitude but it's now finite in that forward direction

22:30 in the zero momentum transfer case but of course it's ambiguous in that direction of the zero-sorder amplitude, and that ambiguity corresponds to your choice of the fundamental mass, n. So that's a reasonable way of describing what you mean by the physical amplitude here, which makes some sense, not complete sense, but some sense of it. So it's remarkable that there is a twisted geometric way of reproducing that regularisation, which is not the same as conventional regularisation, but seems to have something in common with it. And it's very simple, it can be completely geometric. I've described the simple poles, double poles, triple poles, quadruple poles and boundaries as the natural things you write down, which is where the inner product of a W and a Z is zero. You just change that to make it where they're not zero, but some definite number, K. What is, er, and then the actual values of K, namely, P to the minus gamma, where gamma in Euler's constant. And the reason for that is essentially the same reason that gamma comes into dimensional regularization. It's the derivative of the factorial function. And that's the same gamma that we're back to there. But the length scale, then, the reference mass, the constant mass M, has to come in somewhere as well. Well, that comes in through these important conformal breaking boundaries, which I've mentioned, but haven't drawn in the pictures. So those have to change too into these conditions here. These obviously Introduce a length scale into the whole formalism. Now, it's a very odd fact that this modification, which you might think would wreck everything, leaves completely unchanged. Everything that I've written down so far, which is properly defined, has a compact concept. It doesn't, you don't need an expansion. There is an expansion of the K and the N, but all the terms of the zeroth one are zero. It doesn't make any difference. And it's not the same as a cutoff. It's not the same as momentum. off at a certain momentum or a certain length and the k in the end is not to be thought of as small i mean k is to be thought of a definite number and the end is to be larger it doesn't matter what you don't know there's nothing to define what it should be but this important point is this uses the full twister space because now i've got a definitely i've got expressions here which are not projective twister expressions they're expressions in c4 and so i will just remind you

25:00 of the point that Roger made that hasn't been emphasised, I think, by anyone else, is that Twister theory should be thought of as taking place within the full Twister space and not confined to the projective Twister space, and in particular for the full description of gravity. But there may be other reasons as well. For instance, to the classical description of the particle in terms of Felicity, that brings in the full Twister space with a scale in it. But gravity certainly seems to need that. So there are good reasons for this, for allowing oneself to do this. And it brings in a feature which is not the same as what you do in copying space. It's not just copying something you could do in space-time. It's actually making use of some particular twister of geometry, which is now available. Now, I'd have a hope that having made this geometric modification, you can do it once and for all, and that it will help you with the ultraviolet diversity, too. but that's only a hope. And you might also hope that, of course, it isn't just a fiddle that you do just for the sake of scattering amplitudes, it's actually connected with everything else that's going on within the crystal program to do with a description of mass, a description of gravity, which, again, I'm not going to say anything about, but that's a hope in the background. It should have been made sense just as a fix for these concertings, of course. Alright, so, what's the form of these twisted diagrams, how does it appear? So, in the kind of diagrams I write down later, now and later, you should understand these lines now being redefined with the inhomogeneous phase in them. Otherwise they look just the same. Now, what's arising in these diagrams? Well, the crucial thing, we had the simple poles that are just to be thought of as pulling things into coincidence, so they're just enforcing all the external fields in a common space-line point. That's reasonably easy, and it's not much more difficult if there's a double or triple or quadruple, it's taking derivatives as well. But the boundary lines are the crucial bit because they're doing inverse derivatives, so what they're doing is they're pulling together, but they're also affecting, Well, the boundary line is like the square root of the Feynman propagator. You know, it does some inverse differential operator work, and so essentially it's solving differential equations. And the claim is that with this modification, we actually arrive at solutions to the differential equations which are expressed by Feynman propagator,

27:30 and those are the correct solutions. Well, that's a bit hopeful, because until you get to the whole loop level, you don't really know you're using the final promulgator at all. So that's a hope in the background. But that's the idea that boundary lines really carry the content, dynamical content, of the theory. They're terribly important. Now, the other thing I should come back to, especially in non-uniqueness, because even in zero-th order, I wrote down this first diagram as the zero-th order in-to-out-state amplitude. this one, or I could have written down this, or any length of the chip and chain there. So they'll still have the reproducing kernel aspect of these things here, so they're not just like prime diagram elements which are all singular functions. So they should bring all of these things as like representatives of some more abstract object. I can't say we really know what that is. So that's something which is unclear at this stage. And I'll say more about that in the moment to these first-order ones, I've already noticed that for the four fields scattering in the cross-plus-minus-minus case here, I've got two alternative forms which are the same, they give the same answer, and in fact you can find others, for instance these, this one, these are all the same, they all give the same functional fields, they're all representative of the same abstract object, and these are made form chains, you could extend these chains this chain, as far as you want it to the right, or extend this one downwards, will give rise to the same object. So that's an example of the non-uniqueness, and how they really must be referring to some more abstractly defined object. And I better deal now with the other, the plus-minus, plus-minus case is equally simple, in this case it's a loop with all boundary lines running around it, and for that, here's another alternative representation for it, but in both cases, the polygon or loop that you get in the middle has the vertices in the correct order that we need for the trace calculation. So, and another feature I'll point out is that if you compare, if you compare the one

30:00 I had, if you compare the two different channels, if I compare this one with this one, they're actually the same form that's being integrated. The only thing that's different is where the boundaries are. You have to switch around the things and turn that one upside down. You see it's actually the same form that's being integrated. the choice of the ordering of the vertices on the outside only affects where the boundaries are and not the thing that you're integrating. So those are other powerful things and that gives you some half a sense of where these things might be more abstract. These might be concrete versions of something that's really more abstract. We'd really like to think of these as representatives of some more abstract object and an example would be of really describing all these fields as deformations of twist of space, and then describing how these deformations interact with each other in some highly non-linear way, but some potentially complex analytic manifold way, then these things could arise as describing that deformation, that compound deformation, which is an intrinsic thing, but actually expanding it in powers of the coupling constants, and so you get these are the calculational ways of getting those coefficients, and the different ways of doing it, of giving rise to these different integrals. So that's the kind of thing that I have in mind. Of course, people who come from a string theory background have lots of other suggestions to how you can think of these things more abstractly, and I think that's part of our work is to try and put all these different insights together. Right, so let me just put all those things together. I've got a thing which I think of rather, I'm going to write it down to an abstract version of a twister diagram here, in which I bring out the fact that it has this essential polygon which relates the four vertices, irrespective of which representation we choose, it's always got the same connection of the vertices here. And so there is an identity between these object which is not by any means obvious so it is in fact just the spinner identity in one of its many guises i think i believe we've seen that for the sputum identity we call it the spinner identity in the motility but it's obviously well known it gives rise to this fact which does

32:30 support the idea of picturing this thing naively as something like a two surface with edges and then And then the complete amplitude, of course, is given by the, by taking these colour-stripped amplitudes and also finding them by the appropriate traces, so we have an expression like that which has been written down by many people now, and, but I'd just like to, just for a moment, I'll just drop the diagram notation and write the thing as an integral. What I'm saying is the complete amplitude A, and just to assess how simple it is, because The only two things in the integrand are actually those two quadruple poles, which are there. Everything else is in the boundary, and in the integration over the fourth twister variables and the Ws, which is really just pulling these things together on so that the Zs are collinear. And the external one functions as well. So everything's in that. Well, really, everything's in the contour and the choice of boundaries. So the contour consists of, it's these v's appropriate for each ordering, but those are relative homology classes, relative to the allowed boundary subspaces, where they're given coefficients, which are the gauge theoretic traces. So, again, it's, once things are these, there's a connection between these spaces, V, which is the, again, the solution that has to do with the spin of identity, it has a very, it is actually a very simple expression, so that's the kind of thing that makes one feel that there's not an air, so something is going on, we don't know quite what, but there's something in that. It's ridiculously simple, and it remains ridiculously simple for the next bit, which which is the question of the five and six field amplitudes. Alright, so, the five fields, well, I'm going to tell the same old story again, basically, to reassure you that the same features persist. That wasn't just a fluke for four fields. There are general features here which persist. So, I'm just thinking about MHP three amplitudes, though, at the moment. Alright, now again, if I had the five-field problem I could go back to my original thing using just the one twist of representation and mastermind altogether, so on, that's perfectly true, and that's the thing which gives rise to all sorts of powerful calculations, LVPR

35:00 but we're taking this different approach, and from this approach, the point of view is that we need a different representation of the delta function, and it turns out that this is such a representation of the delta function that is just the scalars here and a bit of homogeneous to minus 2 on the outside so that's the thing which follows, again from the algebra of the two-twister function space that's really what's involved here, that's what's coming in here why it works, because it's the internal structure of the two-twister representation rather than just one twister and if I then go through the whole procedure I went through integrated by parts. So I have to stick on all these poles on the outside and some numerators. Lo and behold, they all cancel by integrated by parts at the cost of introducing for all these boundary lines. So the whole thing comes down again to, this is the plus, plus, plus, minus, minus, minus case, plus, plus, plus, minus, minus case to, oh, I seem to have got So this diagram is five external fields, and again, very similar form. Everything is conformally invariant, except the suppressed conformal breaking boundaries. So that's an encouraging result. And for the other type, again, actually yes, in that case there are several non-unit, it's not unique, there are other things I could have done simply because of different choices you can make about how to put the five fields around the itch of that diagram. it's an asymmetry there, it's only one way that works to give a neat answer however for the other type of five field, MH3 process and that gives this diagram, which is again similar and again has this feature it's the same integrand, just a different selection of boundaries and so a complete amplitude takes a sum of the form of 12 terms, that's 24 divided by 2 6 of, the first side of 6 of the other with his thirdly deranged in all possible permutation or orders. And again, Dr. Van described the Feynman textbook approach. I hope there'll be some new textbooks soon now. So it's a rather slur on textbook, perhaps, I think, but I know what you mean. So you get these 25 pages of close type describing the Feynman diagram.

37:30 They boil down to doing this, because the integrand is the same for all these different pieces of the amplitude. Everything's going into the choice of the concert with which you integrate this. And that's not too bad for a disgusting mess. And again, if you feel that something is going on here, you don't know quite what it is. There's one feature which is new because we're talking about five fields. And then I can look at the soft photon limit. I can see what happens if I drop one of them. That's easiest to see if I go back to the preceding one. zero-homogeneity function stuck on here, this plus here. So if I just replace that by one, which I can, well, lo and behold, that just degenerates to the, to one of the versions that I had for the, the four-field amplitude. That's as it should be, that doesn't look formal, that doesn't really make any mathematical sense, as I'm doing things to the concert there, but that's encouraging, and you've made the same argument that all the other soft photon limits it should be. The six fields. Of course, we've got more work to do. We've got 24 of the type where all the minuses are sitting next to each other. 24 is in one apart, 12 is in two apart. So we've got to look at all of those. And I won't go through any details in here just to assure you that they do. And the same features occur. You get the diagram of exactly the same type made up They're longer, though, the sun. They've got three boxes sitting together, but still, the overall polygon here has the vertices in the correct order in each case. And in each case also, that's the third one, in each case also what you integrate here, though it may not look it, you can redraw them a bit, what you integrate is exactly the same in each case, only the choices of boundary are different. Then you can investigate what happens if I drop one of the pluses, one of the plus fields, and it degenerates to the things I just did before in fields in a consistent way. Now, I haven't really pursued this, but basically the pattern's obvious.

40:00 It's what happens if you do 7 or 8 or 9 for mHV. You've got a longer chain here, and basically the two minuses have to sit next to each other on the diagram for it to be simple. And so this looks very like the BCF representation as well yesterday. I don't know if there's any form in the real connection here, but it looks very encouraging as a viewpoint on what the MHV are like. So that's the basis of the feeling that there's something to this which might contribute to an understanding of what's going on here. using more interest of theory, which is to do with its higher forms of representation, which are not just the zero resonance field one at a time, but actually looking at it in combination. I should mention also I skipped straight to looking at the gauge fields without looking at any of the other massless field theories, but historically we looked an awful lot at 5 to the 4th, and Roger Sands would always have 5 to the 4th, and the master's QED, and then we've... So that's with Merler and Carlton Scattering, and then we've got some generalizations of that. So the point is, they all work just as well. I think that's the overall claim, is that you find double lines coming for the scalars moving through a diagram, single and triple lines for spin half moving through a diagram. So we're looking at the special case here, where it's just spin one object moving through, Essentially all the massless field theories have the same features as I've discussed here. From my point of view, it's a very important special case which brings out these amazing features of the MHV amplitude which we see in the Gagefield case. Now, I want to discuss what will happen if we look at the non-MHV processes. and of course one of the most important threads seems to me is what we've been coming at is how we join sub-processes together to make an art of a process that seems to be a problem it seems to be a work far better than when you had a right to expect again, but it seems to me there are many approaches but they all give wonderful, powerful answers and if you'd like to do something like this in this picture how to join things together so this is still now a conjectural stage

42:30 or of the conjectured result, whatever. And it all goes back to an idea that Roger had over 30 years ago for even how to get the first-order apertures, things I discussed earlier on. And it's a thought, really, about deriving these, which is where we can correct, you see, the first-order apertures, but how do we get them in a guessing way rather than an analytical way? Well, you might just look at the zero-order ones, to fields just going through unchanged, without any interaction, and think what could be mediating between them to represent the interaction by another gauge field. So I think Roger's first thought was, well, he just draw a line like this. I'm sure he could correct my details on this, but this is a first thought. But that thought is not quite right, because if it's just a single line like this, it corresponds to something which has definite validity. part of an on-shell zero-res mass field. And then his guess was that you need two of these, and we now see that more clearly. Really, what we've got here is a two-twister, these internal lines technically thought of as giving a two-twister representation of the final propagator. Whereas this is the on-shell's propagator, the delta plus rather than the delta x. And one is the period of the other. One is the discontinuity or the cut that you get from the other. But I should mention that everything to do with these boundary lines seems to be intimately connected with all these questions about cuts, discontinuities, coefficients of logarithms, and these very, very, very close connections between them, as well as the infrared divergences, the process divergences which arise, are really what's going on in here. I hope that in many ways what people are discussing here turns out to be the same thing, only addressed in apparently different languages, and that by bringing these together will have extremely powerful results. Anyway, that was the first idea, and I simply began to apply that idea to not to one-to-one join to one-to-one, or I'm trying to join a two-to-two process to another one-to-one. So, schematically, I've got diagrams which have this form, where there's an essential polygon which seems to describe the guts of what's going on here.

45:00 So I've got a simple two-two here and a one-one here, and I want to make a bridge between them and so get a thing which treats the six fields in the correct way. So it's not absolutely immediate how to do that, but, let's see, yeah, yeah, so if I do, and that's what I've got, I'll let you talk about the plus minus, plus minus, plus minus case, so I want to join this one to this one, that's my, that's my job, and I can't do that, doing it directly doesn't seem very helpful, but I have to do something a little bit more indirect. I choose another representative for the two, I actually draw this out into a chain of three boxes in the way that I described, and then I draw out one of those lines into a chain of three lines. And that, if I then join this process to that resulting figure, I get something which has a threefold symmetry, and so it expresses this idea, but it also expresses that I'd have got the same thing if I'd chosen to make a cusp here, or here, I'd have got the same thing. And this does seem to me very reminiscent of the important results that we've been hearing about about breaking compound processes down into elementary vertices, and the results should be independent of how you do joining up. And I think this is, although this is not proved, I think this is a powerful candidate as a requiring investigation. And much more generally, what on earth the connection is between these wonderful techniques with the reference momentum for joining sub-processes together, and what's going on here, where we're really appealing to the idea of two twists of functions. I think these things must be intimately connected, but I don't think of any theory of how this can be. I think the freedom in the irrelevance of the ESA, the reference momentum, be connected with the freedom of representation in the two-thrister function set-up. So that's one thing that comes out of this function. I don't know what the answer is, but I think it must be sitting in there. And there's an important symmetry embodied in this. If you see this whole thing in one, it's got a threefold symmetry, which you don't see when thinking of it as two bits added together.

47:30 I can do actually, and again, there's a non-unitence in this, because you can see I could have paired that these plus and minus pairs could have been paired in the other way, so there's one that's the equivalent of this, but it's retroflective, so again, these things are only, we know they're only representative in their nature. I could do the same for the other two, three to three apertures, so there's nothing special about this case, it's just the simplest to, what is the illustrative one. And, again, one can investigate some basic constraints on these things. Well, there's duality, and that's particularly easy here. I mean, you can just see if you're basically turning this upside down, it has the effects of complex congregating it, and then you see you go back where you started, and that's basically what you want. I look more carefully, but I've really got at this through an infrared divergence argument, and that's really what this joining together amounts to, is looking at the residues of the infrared divergence and then guessing what the thought thing is from that so I can look more careful at the infrared divergence for the whole amplitude and that definitely seems to be consistent too and then I can look at the diagrams arrived at by the software where you drop down, drop one of the external ones and get back to the ones that are MHB that we had before so that's an important part of the argument And there's another important part, which actually is more, and this is something which is proved, and it's quite, I think, quite significant, in that I'll just turn for a moment to another master's field, maybe master's fight for the fourth. Now, one of the earliest ideas that was dropped, you know, in the realm of the early parts of Richard's theory, if you look at the first non-trivial science of the fourth process here, and that's the Feynman diagram. It's the first Feynman diagram that anyone's written down at this conference, and I think there really won too. It's a shame, I think we should really honour the memory of Richard Feynman and this absolutely incredible thing that he started off. But anyway, here's one. And it was conjectured early on that maybe this would correspond to what you get from a twister diagram in which you just join together this one and this one. And it really is the first example of sticking two MHB processes together.

50:00 It's a scale here. It's not quite the same thing. That's essentially the same idea. And by just identifying that twister. So that was an early idea. But that simply isn't wrong because that connecting line there is on the shell and it would just represent the period of the correct climate diagram where this is actually the delta plus rather than the final propagator. So it doesn't represent the internal structure correctly. In fact, the right answer, or a right answer, is actually this. Actually, perhaps I could have put them on top of each other, yes. You've got it like that. So, where it was just, you need to have some more structure here. And the structure that you need, with these coded fields here, is actually, consists of drawing more boundary lines. Now, I can think a bit about what this means. So, this is a remarkable problem. This is not at all obvious. This is a very non-trivial thing. It's not at all obvious that this has got anything to do with this analytically with that Feynman diagram because it obviously loses all the symmetry of the three states here, which in the Feynman diagram aren't exactly the same footing, but here look completely different, and down here as well. But the reason for that is that really what's going on here is that we're using a three-twister, not the two-twister, but now the three-twister representation of a general field the product of the three fields that are coming in and going out, and we're using a three-twister representation, and this, across the middle here, is a representation of the Feynman provocator in the three-twister representation. And again, the freedom in that representation gives you this arbitrariness in how a lap of symmetry and how the thing appears. Now, my guess is that the same is true of my conjectured non-MHV, my next to MHV conjecture here and that this structure across here is actually the appropriate corresponding to a three-twister representation of the final propagator involved here so that really the internal structure of the three-twister group is what's necessary to understand this three-to-three process that's my guess coming out of these investigations And I think this is given as an example of gluing things together, but my impression would be that if you can glue these together, you can glue anything together.

52:30 I mean, the whole concept, if this is understood, how this works, and its connection with the other very powerful methods which are already exploited with such amazing effects, that would actually, this would be not just, again, just a glue that works for this process, but in much more general circumstances. It also describes, of course, that it made you think I could be, which is absolutely crazy. I mean, what I've just done is given you a scalar diagram. Everyone knew what the answer to it was. Everyone knew the answer was just this. And it's the simplest thing you can imagine. It's just 1 over a momentum square times a delta function. And I gave you an incredibly complicated looking structure to describe it. And this might seem completely wrong way of doing things if I'm trying to find the simple expressions But I think under the surface, there is something important to do with the structures of these things, the singularity structure and the algebraic structure, which will be illuminated by studying this, as it were, completely twisted viewpoint, rather than just partially twisted viewpoint. So as part of my conclusion, then, I think this approach, using twister diagrams, which have been very striking features about what you're integrating in the contours, the boundaries and so forth, that actually, first of all, should connect this idea of breaking down into MHP sub-processes, but I think actually what it's got to do is to bring in more advanced twister theory, not just the twister theory of one particle, but of one zero-massive field at a time that are those in combination and the algebra associated with them if that's this kind of joining together works it should give a new way of looking at that loop diagram which also i've been got that kind of process of drawing together the divergence regularization here is different from other schemes and therefore might give a new handle on the problem which arise in loops that's just the hope because i've only looked at an infrared thing there's no we're going to know that it's going to be the same for an answer to our questions. Another factor that's coming into this is that the conformal invariance is now and the conformal breaking are absolutely explicit. And a big thing like this, everything here is completely conformally invariant. The conformal breaking is supposed to take place only on these extra unmarked boundaries which correspond to the boundaries of real McCarthy space.

55:00 And that's the only place, that's the central dogma, that's the only place where the form of breaking takes place. So that's a new concept you only see in this kind of representation where you're keeping everything to a story, and maybe it's another way to doing that. The other things I mentioned, I've done it with pure gauge field, but this last example with Scalus shows how there should be ways of relating massless field theories one to another with spin-raising operations, and so making perhaps new connections between and that sort of field of theories, and I was very, very struck by Dr. Nair's results for Gravitons, which suggests there was application to these ideas to spin too, which really, again, looks simpler than one. Dead hope, so that looks very promising. But really, fundamentally, at the end, the object of all this is not just to provide a calculational technique, in fact, it may not be very good as a calculational technique, It may be that there are all sorts of other techniques which are much better for getting the mental space answers, but the idea is that this will actually suggest things which are on a more fundamental level, which will cohere with the other aspects of the program, the description of gravity and the description of the elementary particles, which is something we've had very little exposure here, because it's not really the subject of our confidence, which is an important part of the whole battle under motivation. That will conclude my remarks, and thank you very much for your interest in this area. It's a very nice opportunity. Thank you. I have the impression that for MHV amplitudes, you argued by showing these singularities in cell channels. I'm evaluating it, yes. They all arise from, no, they're not, those aren't, those aren't, they arise from a particular representation of delta functions, which is this. So you have to justify this first, and that's, I mean, it's non-trivial, but that's where it comes from, and after that, it's just been raising operations and integrated by parts and so forth. To evaluate that. Well, I mean, that's the justification of this thing. It follows

57:30 from this. So that's where so it's not so I'm not quite sure of the question. My question is how you learned of the first section for five, use the part failure. Oh, I see. I'll just put it in by hand. All right, so like I did with the four-field case. In the four-field case, I just wrote down the... So what I've done, I've missed out the intermediate stages here. Starting with this representation of the delta function, there'll be... For scalars, then I would represent the delta function. I put in the X, the Park Taylor numerator and denominators. Those are just either numerators or poles which connect with the vertices here. They, those are two component spinners, but the twisters are for them. Well, yes, but then the factors of the form... Oh, I think I've got to write something down. Yeah, I'm sorry, I trust I should have emphasized this. Two component spinners, yes, but that's where the infinity twister comes into it. Right, so you've got a thing like this, That's a typical example on this. The other line. I, alpha, theta. That's what that, this is the object, it's the infinity twister, which turns the four-component object and picks out the two momentum parts of it. So that's how it turns the twister into one of the spinners that appear in the formula. Does that answer your question? That was meant to be... Well, let's proceed. I can't remember if you see, sorry. No, it's out there. Yeah, thank you. But thanks for bringing it out. I didn't make that clear. Well, the six-floor effort, how would you hope to show that the diver reproduces the same thing? Well, the three-to-three? Well, I think that's got to come through an understanding and translation into this twisted geometry of these methods with the, with the joining together by probably both reference momentum and so on. And you see, there is another approach which is to start with a scalar one, which is established, and then use spin racing operations on that. But my efforts with that certainly haven't got very far, and I think it's better to go straight for the transcription of the methods which have been successfully employed

1:00:00 for in the CSIW approach, and try and translate that into the two twister. The spin racing would be multiplying by Z, or do you have to change the sense? Well, I mean, the point is the homogeneities, isn't it? It's just a question really of homogeneities being right. If you look at the original thing that I wrote down for the, yeah, if you just look down, it's no different from what I did in the four-field case. your 1, 2 to the 4th here well, 1, 2 is cubed and then the 3 denominators here that simply translates into that numerator factor it's just that, if that's x and that's z then that dash 9 there, the 3 vector just means the cube of that thing treats as a numerator whereas this hole here it just represents exactly, it's just the 2, 3 here here is the 3-4 and this one is the 4-1. So, you do that, and then delta is everything in the inside, and often that's just integration by file. I mean, these automatically force the right consciousness for the functions on the outside. I mean, that's just the... I mean, that's essentially what went into this formula in the first place. So, it's not doing anything new. So, it is just transcription. There's no jump in concept at all. It's just that is just a delta function in a twisted way where all these simple poles are pulling the the external the four twisted variables onto a common nine those three poles there are just those three similarities there and that numeric factor is that it's not certainly different well back to bringing it up so i really skipped over this so as to get a feeling for the more advanced uh points but that really um of course it's not retrieval at all is that it's quite And I think it's also true throughout this conference that differences in notation between different people here can cause a lot of problems, even when we're really talking about the same thing. We're just in different notation for spinners, and they're not bad enough. But thank you for bringing that up. We're just supposed to have a comment and then a question. The comment is, I think you do yourself a gross injustice in attribution to me. But the question had more to do with the K. I'm interested to see if it's the same as you get out from dimensional regularisation. Or is there a difference?

1:02:30 Well, I'd say the gamma arises. The way the artist constants arises, the reason why it's a natural value is the same reason that you get all the artist constants coming out from dimensional regularisation. The question I've really got to raise, can you see a way, at least in principle, that you might pick up the value of that and look certain? Well, I'd have thought at the higher loop level it could, simply because the regularising massive things also play a role, and it seems like the lamb shift actually starts to appear informally. So I'd have thought there was a... But there's another factor which I might mention to people, which, again, I'm slightly oversimplified. I said K has a natural value, e to the minus gamma, but actually it's no more natural, it couldn't be minus e to the minus gamma, it's really no more or no less natural, there's no way to sign it, but they, they know it wouldn't affect the results really, but they affect the, what the geometry would mean, because it would mean that we're fully, essentially we're, it has an experience of it that you can do this. It's a very remarkable fact, when you think it would just perturb everything or other place, making this kind of change, and it's a very remarkable fact of these homogeneous integrands that if you can make this, to make this homogeneous shift, it doesn't make any difference, it just brings new concepts that do come into existence that didn't turn analogue in the projective portal now. But the thing is, what we're really doing now, instead of pulling these things into coincidence, you're pulling them into non-coincidence. And it's extremely odd, but pulling things into non-coincidence shows exactly the same effect as pulling them into coincidence. It's very normal. But anyway, this essentially corresponds to, this is the quantized version of enforcing that. But then it could be out that, or it could be minus that. And so, really, especially whether you're working in the top half of just the space or the bottom half, there's actually a time asymmetry involved in that, a very subtle kind. It doesn't come out in the results of the calculations, or might do only at a much more advanced point. So there is a genuine point of programmatic interest, that one. In my point of view, that would be ethnicity, wouldn't it? But this isn't a... Yes, I know, but it's very hard to interpret directly. So this is why I deliberately have a very cagey view of what it means.

1:05:00 It works for this, and its simplicity seems to just have got some significance. But I think it would only really make sense when combined with something which hasn't got an air in here, which is very, I mean, Roger's views of how you can deform, twist the space to deal with the other half of gravity, actually seem to bring in similar ideas, but in more fundamental ways. This is really like a linearised fix for something. The idea is to kind of linearise everything. But really, these things are all non-linear, and so it really should flow from something more fundamental. Questions? We'll take a half-hour break. Thank you.