Arrows of time and generalized quantum theory / What is twistor-string theory?

0:00 Oh yes, at St. Peter, everybody who, and sports fans up here, your talk on the inside story of football at Penn State.

2:30 The football players said, oh, I don't want to talk on that. I've given that talk so many times. The Rotary's got it, various civic organizations. I'm just sick and tired of it. I want to talk about the Johnstown flood, the famous flood in the Midwest, which the man had also lived through. State football, the man insisted, and it went on and on. Finally, St. Peter said, well, all right, but I have to warn you, Noah will be in the audience. Chris Eichel, one of the people who has entertained the possibility, that usual quantum mechanics has to be generalized. We're going to apply it to quantum, because first, the usual formulations assume that the output of The theory is the probability of measurements made by observers and assumed, in one way or another, the usual quasi-classical world of everyday experience. But in the theory of the whole thing, there can't be any general division into observer and observed. Measurements can't be fundamental notions in a theory that seeks to describe the early universe where neither existed. And in a general quantum mechanical situation, there's no reason for there to be any set of variables that behave classically in all circumstances. So for that reason, quantum mechanics has to be generalized. That's the problem with closed systems. But also, quantum mechanics, the original laws, as they are stated, rely on incorporating the notion of fixed-actual geometry. There is no fixed geometry, but in quantum gravity, geometry is a quantum dynamical variable. It's fluctuating and without depth. It's not fixed at all. So for these two reasons, quantum mechanics has to be generalized, and the thesis of my talk today is that the generalization of quantum mechanics necessary for quantum cosmology and quantum gravity may shed some light on the arrows of time that are exhibited by our universe.

5:00 Time-asymmetric universes are clearly the errors of time, but governed by time-neutral dynamical laws. I think that's even surprising in a way. I think if you'd asked me, just coming off the screen without any knowledge of physics, it would suggest that a time-asymmetric universe is most naturally described by time-asymmetric dynamical laws. But it appears not to be the case. And I'll say a little bit more about why this is later on. In the face of the obvious time asymmetries of the universe and time-symmetric dynamical laws, the only source of explanation is asymmetries and their boundary conditions. And I'd just like to run through a few of the famous time asymmetries, or calls of time in the universe, and explain very briefly how these arise from time asymmetries between the initial and final boundary conditions. These ones I'll consider. First off, here are the second law of thermodynamics, the retardation of electromagnetic radiation, and the psychological distinction between past, present, and future, the psychological era of time, as it sometimes is. For the dynamic era, as Boltzmann said a long time ago, the second law of thermodynamics can be proved from the time-reversible mechanical theory that assumes that the present state of the universe evolved from an improbably special state. And Rodgers and Penrose, much the same thing, a little bit later. The dynamic over time is usually expressed in terms of increasing entropy. Of course, there are many different entropies. Entropy depends on coarse grain. So the particular entropy we have in mind here is the entropy associated with the coarse-graining defined by the quasi-classical realm of everyday experience, or more precisely, by coarse-graining by ranges of values of approximately conserved, intervals of approximately conserved densities, such as energy, momentum, species, in what were suitably chosen bodies.

7:30 The final boundary condition, the one of indifference, there was an early boundary condition in which that entropy was low, low compared to what it might have been, and therefore it has a tendency to increase since. And that asymmetric boundary condition is what gives rise to the term magnetic arrow of time. The electromagnetic arrow, similarly, can be seen to arise from asymmetric boundary conditions applied to time-symmetric laws, in this case Maxwell's equation. Any solution to Maxwell's equations with fixed sources, fixed currents, can be written in either one or two ways, either in terms of target means functions of the sources and free incoming fields or advanced fields of the forces on the sources and free outgoing fields. The asymmetric boundary condition that the in-field is approximately zero and the out-field is not zero This is what gives rise to the retardation, because then this is approximately zero in the entire field, and you can see the rise in the other part of the wave from the given sources. Of course, in cosmology it's not true that the free fields, at least if you start, say, at the time of decoupling, are approximately zero. All the energy in electromagnetic radiation and the cosmic background radiation, which indeed even today, at 3 degrees K, supplies the dominant source of electromagnetic energy in the universe. But that radiation has shifted to very low temperatures and very long wavelengths. Electromagnetic radiation. And radiation, for example, is visible today, then that will be approximately retarded because it's essentially zero again. The psychological era of time can be sort of similarly understood. Here's information gathering utilizing systems. Here's a little model of one. This is a model of a robot that operates on a discrete set of high steps.

10:00 Every time set, it shifts the contents, which is recorded and registered to the right here, and erases the final one, and it records a new piece of information from the external world. The first one, unless at any one time, it has a coarse-grained time history of what it sees, where this is the most recent information, and these are successively further away in the pack. It uses this information to compute processes of unconscious computation to compute a schema of the model of the universe, which we all do, I think, containing such things as the rules for success, cat food, yes, pea food, no, vanilla ice cream, taste good, and that's integrated with the present information, right, from the schema of process of computation to exhibit behavior. Well, the important point is that there are two arrows of time which contribute to this. The idea supports information, computes predictions, and exhibits behavior. The information that it gets, invisible wavelengths, for example, is from the past because electromagnetic radiation is retarded. So that's one of the reasons why... But more importantly, the recording of information is typically irreversible, not necessarily irreversible according to Landauer, but the erasure of information in this final step is necessarily irreversible. Therefore, the cyclontical arrow of time operates most simply if it's congruent with the radiation arrow of time, which is where it records information, and the trigonometric arrow of time, which governs the reversible processes. So that's an example of three arrows of time, rivers arising from asymmetric boundary conditions. There are other arrows. You have to be a little bit careful in this game to distinguish between arrows of time that are exhibited by the fundamental laws and apparent arrows of time. So an example of apparent time asymmetry, here are two of them. One is, or could be, so even if the states and dynamics and solar theories that are of limited circumstances can be time asymmetric.

12:30 And at least it's a possibility that the, for example, it's believed, well, it's incredibly difficult to get a straight answer out of a string theorist, right, the best, again, the theory is believed to be time-reversed and invariant, and therefore one possibility, for example, is the asymmetries and the weak interactions arise in, from symmetry-grade heat, and therefore the universe could exhibit on very large scales this be time-reversed and invariant, but overall time-symmetric. Another example is the expansion of the universe. We see the universe is expanding, of course, but in Stevens, no boundary at a wave function in the universe is real, and hence it's time-symmetric. Therefore, the ensemble of semi-classical trajectories is also time-symmetric, as it predicts, so with each trajectory expanding or contracting, but there are just as many trajectories, this radius of the universe versus time, that expand, as there are contracting here on the other side. However, because of the psychological arrow of time and the arrow of time associated with thermodynamics, the obvious meaning of information you have in your utilizing system, whatever one it's operating on, you'll see the direction toward the big bang as the path, the direction toward the metric. So the whole ensemble is symmetric, but apparent asymmetries arise from symmetry breaking. Time-symmetrical dynamical laws, which are governed by time-asymmetric boundary conditions, well, of course, I don't exactly know the reason why the underlying dynamical laws are time-symmetric, but it's true that contemporary fundamental dynamical theories are built on symmetry principles, one kind or another, and it's therefore natural that they incorporate symmetries that are related to time. A simple example is the CQT variant of quantum field theory, where Lorentz variance plus locality, then the quantum field theory of CQT, although you might object that CQT is not the same as D, but it's the host in some sense. So in the face of these dynamical models, there are now two possibilities to be observed in asymmetries.

15:00 And the first question we have to ask is whether all the asymmetries could be apparent. And the asymmetries that we see arise from just a special symmetry region in special circumstances, so that our little piece of the universe appears to be asymmetrical. A classic example of this is the idea, in the interest of the note, where we have an expanding and contracting universe in which it is time symmetric in the sense that it has time symmetrically related boundary conditions here and here, Enter the increases on one side, and then decrease is moving this way, it's increasing moving that way. It would be perfect time symmetry. We, of course, would live here in a regime where it's increasing over here, it's a pass in a sense. However, it appears that this idea, while it's not a firm conclusion, I think, is not looking too good. The key observation, I think, which was made by Davis and Twonley is, You would think if you made this time longer and longer, it would be harder and harder to distinguish this from one where the entropy just increased. In some sense, see the final singularity, because the optical depth, right, with realistic cross-sections from the point here to a corresponding point here remains finite no matter how long this time comes. And that's because it's true the time is getting longer, but the universe is expanding more, so the matter is getting more and more dilute. Therefore, the integrated contribution of the optical depth remains fine. So, I won't go into what it entailed, but there is a certain amount of evidence that, because we can see the time symmetric climate singularity, we don't have it. Davy Cray, right, says we're within a quarter of magnitude, showing from the brightness of the night sky, right, that we would... So we seem to be forced, and these ideas are true, to the idea that the time-symmetrical dynamical laws

17:30 They give rise to the asymmetries that we see by asymmetries between the initial and final condition, which is quite an obstacle to this point of view, which is the error of time in quantum mechanics. It's usually formative. The Schrodinger equation, for example, is transversal in the sense that you can run it through a forward and backward operation. But the second resolution, the idea that on a measurement, The wave function is reduced by the application of a projection operator as far as the outcome of the measurement, and then renormalized, which is equally essential to describing predicted probabilities of histories or sequences of things. There's not time for this sort of an effect. You can't learn this fact with some time. Take off the projections on an animal. You don't even know where the projections might have been put in, because the projections operate when the system was measured, and given time they're present, you generally don't know when it was measured in the past. Now, that's usually thought not to be such a bad thing, because he thought the measurement is an irreversible impact of amplification, as it's usually characterized, and therefore, the second law of evolution should operate in the same direction as the theorem in anatomy. But that's a little bit problematic, because it's after all, it shouldn't be possible to intelligent aliens, or in the future, They might get it into their heads to reverse the thermodynamic arrow of time in some large volume of space, so it runs the other way, through the prevailing thermodynamic arrow of time, which in particular they could do, and then, for observers in such a large volume, what arrow of quantum mechanical time do you apply to? Do you run it forward, like this, or do you run it backward because of the thermodynamic arrow of time?

20:00 Of course, if the existing quantum mechanical laws are merely effective theories, that is, which incorporate particular circumstances, then that won't be a problem, because the effective theories put in one part of the universe run one way, or in another part of the universe run the other, and that will be my basic thesis here. Because we argued in the beginning that quantum mechanics need to be generalized for quantum mechanics and quantum gravity, So that it doesn't incorporate an idea of measurement, it doesn't incorporate an idea of fixed space time, can it be also generalized so that it's time neutral, so that there's no quantum mechanical parallel time, and parallel quantum mechanical time that we observe arises from asymmetries in the boundary conditions in a completely time neutral framework. To do that, I need to describe to you a little bit about decoherent histories quantum mechanics and about Griss's generalized quantum mechanics, which is the framework in which this will be discussed. So just to begin with a little review of the quantum mechanics of closed systems, which is what answers the problem, the first key for generalization of any derivative in the notion of measurements, let me consider a little model of the universe that we commit to seeing in this audience of neglecting quantum gravity, which is actually a pretty good approximation any time after the first 10 to the minus 43 seconds of the universe. And thinking of the whole universe as being a large number of particles and fuels in a box, perhaps 10 to 20,000 megaparsecs on a sign, may be expanding. And then the usual apparatus of Hilbert space states, the Schrodinger equation, and so forth, can be applied, which will simplify my discussion. What's assumed for any closed system, and is given by theory, is the Hamiltonian, or whatever equivalent to it is as far as specifying the dynamics, the Hamiltonian event theory. The most general objective of any quantum mechanical theory of a closed system is the prediction of the probabilities of force-trivial target histories in that system. Thus, for example, we might be interested in the orbit of the Earth around the Sun. According to quantum mechanics, the Earth could take any possible orbit around the Sun, say, close to the Sun.

22:30 It's only a question of probabilities, which orbit it takes. We hope, of course, that in the situation that we have today, the probability is high that the Earth moves along the classical capillary orbit. The orbits are typically coarse-grained. For example, if we're describing the Earth in this capillary orbit, it's coarse-grained because we're only interested in the center of mass of the Earth and not in any physical degrees of freedom of the universe. It's coarse-grained because we don't consider the center of mass position in the earth to arbitrary accuracy, but rather we consider it to be some generous range of accuracy, so that's the difference we get. And it's coarse-grained also because we typically don't specify the center of mass at all possible times, but at a discrete series of times. But not every set of alternative histories that can be described can be assigned probabilities. Nowhere is that more clearly illustrated than the famous two-slit experiment, where we have a source of electrons that are passing through a screen with two holes to be detected at some point y in another screen over here. In a simple model, there are two possible coarse-grained histories, one in which the electron went through the upper slit and arrived at y, But it's not possible, in the usual story, to assign probabilities to those two alternatives. That's because the probability of y at point y will not be given by the sum of the probability of the probability of the opposite limit in y with y and the probability of the probability of the opposite limit in y with y. That's because the quantum mechanics probabilities are squares in amplitude, and the square of the sum is not equal to the sum of the squares because of quantum mechanical interference.

25:00 So some rule is needed to determine which sets of histories can be assigned probabilities and which cannot. I emphasize that it's not a matter of if we don't know which split the electron, that would be a case of probability for 50-50. We can't assign any of them probabilities at all. The usual Copenhagen rule is that you can assign probabilities to the histories that have been measured. So if we measured which split the electron went through, Then the interference is destroyed, the sum is obeyed, and everything is consistent. But that's exactly the kind of rule which you cannot have in cosmology, at least if you want to apply it in the real universe, because there is no notion of measure of splitting the system into two parts, one which measures the other. So the generalization, which is due to Griffiths, Annes, Gilman, and all, is just this. The initial, the closed system, as Hamiltonian, It has a state, and probabilities can be assigned to exactly those sets of alternative histories for which the interference between individual histories is negligible as a consequence of the state and the Hamiltonian. But now I'm sure you have formulated this precisely, or fairly precisely. Measures can be described in such a framework as quantum mechanics, but they don't play a fundamental role. The fine-grained histories, which are the most refined possible description of the system, the notion of coarse-graining, which is actually easy to get right away, the partition of the fine-grained histories into classes, and the measure of interference between the coarse-grained histories so that we can tell in which situation it vanishes. So here's the story in one language, the language of operators, the language of coarse-grained histories, which is to establish notation. Alternatives at one moment of time can be reduced to yes-no alternatives. Is a particle in this range yes or no? Those decimal periods are represented by projection operators, and the rotational u's, k, corresponds to the particular set.

27:30 Thus, when you're moving around the sun, you might divide space up into a lot of little boxes here. We can estimate the center of mass in this box or not in that box, or this box or not in this box. The set k corresponds to the whole set of boxes. Or we could ask about alternatives to momenta, which is a different set, and alpha lays the particular box, here's the earth in there, and t is the time in this Heisenberg lecture representation. A history is a sequence of alternatives of a series of times. So, the Europe and the Earth might be described by this box, that box, and this box. So it's a sequence of alphabets at a series of times, and it's represented by the corresponding chain of projection operators, like it's multiplied together. So that's how, in history, it's said to be fine-grained, if all the projection operators are one-dimensional, and coarse-grained if they're not all one-dimensional. Of course, people looking at the universe deal with enormously coarse-grained alternatives, right? It's specified in very few versions of the universe, so coarse-graining is really the general case. If you don't like that sort of operator picture, you can look at it in terms of sums of paths, where the fine-grained paths are just the finite paths. First-grading consists of bundling these paths into, if you're considering the set of alternatives for expanding the motion of the Earth, you might specify a range of positions set at three different times, and the first-grade distance consists of all the fine-grained paths of the center of mass that actually pass through those dimensions at that time. Then for each, each, each such history, each such class of paths, you can define, you can find them somewhere in the paths that would act in the initial state, that gives you an operator C acting on the side, which turns out to be exactly the same as the chain of rejections which I wrote down before, except for the fact that it could be the minus IHT, which is irrelevant to the probabilities, and this defines, well, that's how it works in some of our history pictures.

30:00 So we have, however, we have decoherence and probabilities. We have the set of histories which correspond to these class operators, the chains of projection. We can define branch state vectors for each S3 set by applying the operator to the initial state. We can define the decoherence functional by, which is a function, complex value function on pairs of histories with slightly overlap. The decoherence function was negligible for all alpha prime not equal to alpha, and the probabilities for histories within the diagonal elements of the decoherence function, or equivalently just the operator c applied to psi squared, and those, as a consequence of decoherence, those probabilities are consistent, that is, with the most general form of the probability symbols, which you remember from the two-slit experiment as the obstacle to assigning probabilities to history. That is, if you have a coarse-graining of a set of histories, that is, you take a given set and you group them into bigger sets, that's a coarse-graining, so when the operators are sums of the coarse-grained set are sums of the operators seen in the finite-grained set, then e-coherence implies that the probabilities of a coarse-grained set are the sum of the individual probabilities that are in that set, which is the most general. Well, that's a very speedy introduction. But the main point for this to argue is that this formulation of quantum mechanics still has a narrower time. It still has a narrower time because of the formula for the probabilities. In fact, it's much worse, right, than the quantum mechanics, because there's a psi on one end, the formula for the probabilities, and there's nothing on the other, so there's a definite parallel in your direction. It's much worse because the p's are no longer necessarily associated with measurements. Therefore, there's no question about them associated with any sort of irreversibility in the system. So this arrow of time would have to be fundamental.

32:30 It is helpful to have this arrow of time because it allows the definition of states at a moment in time. States which are sufficient for future predictions but not for the past ones. Let's suppose you have, for example, the probability of a set of histories. And if you'd like to calculate the conditional probability, you know a certain number of events have already happened. Say you're at the horse races, and so many horses have already come in, you want to calculate the probabilities for the next races. Well, that's a conditional probability for these events that have happened already, and these you're seeking to predict, constructed in the usual way. But that, it's easy to see, that that probability can be calculated by applying a chain of projections that have not yet happened. To a wave function, which is a function of the time, which you know the information, which is just given by the chain of projections that have already happened, and then normalize. That's the original story. The Seisenberg picture, the states are constant in between the action of these projections, and then add an alternative if they're reduced by the action of the projection. So it's evolved, reduced, evolved, reduced, but it's a particular function. That's okay for future predictions, but if you try and turn the thing around and calculate the conditional probabilities, given what we know now, what happened in the past, you can't do it using only a notion of state of the present moment. You need to look at the present information and the initial condition. The task now is generalizations of quantum mechanics that are time neutral, that don't have the power of time, but we're used to this one. So that introduced a little to the notion of generalized atomic theory, which is due to Eichel, Linden, Sorg, and Gell-Mann, and so forth. And, well, here they could only put on two transparencies. It's the same three elements that we've already seen in the usual formulation. You need a notion of fine-grained histories, which are the most possible, refined possible description of the system. Which, for example, Feynman has in the case of particles, four-dimensional field configurations, in the case of fields, as we'll see in general activity, geometries, you need a notion of coarse-graining, which are the partitions of these fine-grained histories in the classes, and you need a measure of interference, which is the decoherence function. The new ingredient here is that the decoherence function can be characterized axiomatic

35:00 And this is, in fact, this is by now a rather antique notation, compared to the one that Chris introduced, but I stick with it anyway, had ideas. So a decoherence function is any complex learning function that has the following properties. It's remission, it's positive on the diagonal element, it's normalized, and most importantly, it obeys the principle of superposition. Expressed in this bilinear form, if you have a coarse grain of a set, then the coherence function of the coarse grain set is a double sum over the coherence function of such a function. If you have over an information such a function, then you can calculate the final coherence. This is the off-diagonal elements vanishing, the probabilities for histories, which are the diagonal elements, and these probabilities are numbers between 0 and 1 as a consequence of the axioms, and they obey the most general probabilities. So the key point is that the crucial quantum mechanics, as I've described it before, is one way of satisfying those axioms, but it's not the only way. Therein lies the possibility of having a kind of quantum theory. And here it is. It's just a souped-up version of the work of Aramov, Bergman, Leibovitz many years ago. You just have a quantum, write down the clearance function, which has both initial and final conditions in it. There's a density matrix for the initial condition and a density matrix for the final condition. And this is time neutral because you can see from the second property of the trace, oscillating in a stack, you can interchange the initial and the final.

37:30 So the fine-grained histories are the same as before, the coarse-grained histories are the same as before, but the measure of interference incorporates not only an initial condition, an initial state, but a final initial condition as well. And that theory is time neutral. There's no quantum mechanical error over time. In such a time-neutral situation, the asymmetries of time are explained by differences between the initial and final condition. So, for example, the initial condition might not be a wave function, and the final condition might be a condition of complete ignorance. That seems to work pretty well for our universe, but in this theory it becomes a quantitative question. There is a price to be paid for this. As I illustrated, the notion of state in the quantum mechanical system is itself inherently an asymmetric notion. Therefore, in general, we won't be able to have a notion of state on evolving through space-like surfaces, In the original version, state was put away for future prediction. Now the theory is time neutral. It can't do either prediction in the future if it has. Now we turn to quantum gravity, or at least the cartoon version of quantum gravity. We talk about quantum space time. Quantum space time is already an issue about defining an arrow of time because there's no definite notion of time until it's quantum gravity. There's no fixed time to be talking about whether it's symmetric or not symmetric. But there are histories, and the histories are technically histories of geometry and matter, which have different boundary conditions on the different ends of the histories. So, for example, here's a cartoon version of the standing universe, this is a close-up of the universe, of the plans and the facts. A little bit, we could have a no boundary proposal in the beginning of the universe and then the condition of ignorance on the end.

40:00 That is not exactly time asymmetric, it's history asymmetric, but it will give rise to these different boundary conditions, time asymmetries, in the classical limit in which you get classical spacetime. Let me just illustrate how that might happen. Well, here's a cartoon version of the generalized quantum mechanics for geometry. We might take the idea that the fine-grained histories are four-dimensional geometries with matter fields on them, in relation, The coarse-grained histories or partitions of those in the classes, if amorphous in the very classes, is a parallel view to the question of whether the universe expanded with a big volume or not, you might pick a volume, and then partition all the histories in between by whether the geometries had a space-time surface that had a volume bigger than some of the crucial volumes, or that they did not. And you can calculate whether those alternatives do appear, calculate the probabilities, and you get the probability. It can be done in two planets, measured just as before in the usual quantum mechanics, by summing over histories, in principle, and taking the overlap, the sort of the overlap, and then equate functions so they're fine. That's a fully four-dimensional mechanics of space-time. And in general, you don't get classical predictions. But it can happen, right, that the sum over the action of four geometries could be dominated, for example, by one specific four-geometry. Therefore, this big interval over geometry here would be replaced by just evaluating this kind of classical geometry in a subsequent interval over fields. But that's field theory in Curtis-Stang's time. So that's the usual familiar story. The theory then is thus approximately equivalent to a field theory based toward geometry, with causal structure, time, and, as a consequence of this, time asymmetries, and asymmetries between them. So it seems that we can do it in the, at least, sketchy way.

42:30 The observed timing symmetries of our quantum universe can be seen as emerging phenomena in a classical approximation area of a classical space-time that arise from boundary conditions in a time-neutral theory of dynamics as well as a completely time-neutral formulation of quantum mechanics with no built-in arrows of time. In case you missed it, that was my conclusion. In this discussion, I didn't live up to the imperial standards of Rinder when I was a pioneer in the Imperial College, so I'll just do my real-time reading of this. Thank you, Chris, and have a good evening. Course grading is a fact of life. As I look at you and think, first of all, what do I need to describe you? There are all sorts, there's a position of every single molecule inside you that you're trying to ignore, and the key densities, and then also they've ignored all the other variables of the entire 10 to the 8th particles and 80th particles in the universe. So coarse-graining is the way we deal with it. If we did not coarse-grain, in general, then there would be no decoherence, because physically, the decoherence is the disappearance of phases between histories, which is the founders of quantum mechanics said. That comes about because they move from one to some degrees of freedom in products. So that's expressed precisely by the notion of course training. So course training is both what we do and it's essential to the story. So it's essential to get good at preparation and registration? Well, I'm talking about cosmology here. So the preparation and registration would be in the projection algorithm.

45:00 I don't have to believe that you have to do cosmology if you want to do that. Well, it would be the same thing, I think, right? I mean, you'd have then some initial statement and some final statement. I could do all that for your statement. Sorry? I can do it. I wouldn't have stakes. I would just have a preparation and registration, and I could use an ensemble of these things. Well, I think if you deal with any realistic system, you would have pure strength. One. And two, what I'm talking about is the era of times that are exhibited by the universe, not the era of times in some laboratory experience. I think the answer is yes. But let's discuss this afterwards. That's it. I explained the whole universe and I don't have enough time for that question. So, problem. Just a clarification. You talked about history in this context, one of your alphas, alpha 1. You introduced them as sequences, time-ordered sequences. In your general life, do you think that they don't need to be such? Well, it might have been a little swift in here. I mean, for most of the talk I discussed the quantum mechanics of the system in a box with a well-defined time in a particular red screen that we have. I'm asking what happens when you actually go to the general picture to describe it. Oh, in the gravitational case. Yes, so in the gravitational case there isn't any time. So alpha, a given alpha can be something which is not, it's not possible to divide it in a sequence of... In a four-dimensional sense. So, for example, you might ask the probability that the universe has three space-like surfaces with given value. But in a general quantum mechanical theory, you'll get histories. So classically, maybe the volumes would be ordered in time, the bigger volumes later. But quantum mechanically, the analog is defined in paths, where there will be no such order. But in general, there would be no more.

47:30 Something that's always slightly bothered me about this is how the density matrix begins. Why should I want every moment of time in between? That's equally a generalized problem. Can you repeat the question? Well, Chris noted that I had an initial and final density matrix, and I could have had another decoherence functional satisfying axioms if I put two more density matrices in. And that's perfectly true. It does satisfy the axiom. Well, what can I tell you? We need a theory that's a very weak structure of generalized quantum mechanics. Those would be, for example, if we had divine interventions or something like that, those would be represented by such and such matrices. I'm trying to get the minimal theory that's consistent with the present that exhibits time and time again. Yes, so some formulations, which are, for example, I guess a natural problem, not formulations, but for example, I don't think there's any inherent error, for example, in dynamical triangulations or regi-calculus, those sort of discrete formulations. No, you have to, I agree with you, you have to augment it a little bit, I'm not sure exactly how to do it, but certainly the everyday entropy in the arena of atom physics and chemistry books is the end of the associator.

50:00 For example, in the Navier-Stokes equation, that deals with these hydrodynamic variables and dissipations. You missed the draw back there. Luckily, it's about you. I have to ask, this is aimed at quantum gravity theory, as I presume. It's a quantum kind of code for quantum. What would your view be on the singularity issue, which is the problem with quantum gravity, which after all is related to what you feed in at the beginning and the end? So if you have an asymmetry that's sort of plugged in there, do you expect to find that place in... Well, I'm not a card-carrying, no-valuable person, so we don't have any singularities at the beginning of things, but it's certainly consistent with the general point of view that there are the cases where fluctuations are very small, and so we would expect with this picture, in the classical domain, we would start from a regular... All of these, everything that I've said presumes that we specify as a part of the theory the asymmetric initial and final conditions. I'm not expecting you to produce those. It would be nice if I could deduce them, but nothing I've said allows me to deduce them. Because you could have time-neutral theories, as I indicated, with time-symmetric initial and final conditions. You could have time-neutral theories with time-symmetric initial and final conditions. It's just that the evidence and the observations seem to be contested against those. Well, I'll describe that for you. Oh, I see you meet your objective.

52:30 I take it back. Indifference. Now you say, who's indifferent? Don't leave things out. Maximum missing information. What? You like it better? Well, information is the whole thing. Well, let's not go there. Any more questions? This is a comment. This is like Prime Minister's Question Time. It breaks everything in terms of the questions. I thought that there was a subtle difference, maybe important, between the way the symmetry and boundary conditions in the quantum case came about both in the gravitational case, which was that in the quantum case, the initial condition is really on the wave function of amplitudes, whereas in the gravitational case, where you don't have an external time, the condition was on the histories themselves, which is what they look at one end, what they're like at one end, and what they're like at the other end. You were going to disagree with that? No, no, I think it's... Is there no more questions for the next lecture?

1:02:30 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, agreement of the sphere. 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 you transform from one to the other where 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 thing 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 sub-manifold of this complex space. The developments that have been happening recently, which I want to give some indication of, are developments in string theory, this is work primarily done by Ed Witten, based on some other work that we did earlier, and in which, roughly speaking, the target space for your strings is not space-time anymore, or not some space-time to which you attach extra dimensions, Projected twistor space is what it has. I should say just a comment which those of you who are very quick at calculating or counting, I should say, will realize that the number of dimensions of the space of light rays is actually five, and for a complex manifold you need to have a linear number of real dimensions. You actually get another dimension by considering this thing to have a helicity, notice the spiral that's going on here, so this light ray is not really... And now, in space-time, something with a bit more structure, which comes about from its specificity, and also the fact that the energy is involved, so that gives you another parameter, so you do 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.

1:05:00 It does have certain implications later on, say you're trying to do quantum gravity, and you might say, in a more conventional picture, you'll metric some quantum operator or something which means that your life cone, or at least the bias of your life cone, is fuzzy in some sense. Subject to quantum uncertainties, but the picture that one has of a twistor theory is something more like this, where the light rays being the primary objects, you don't lose them, but nevertheless whether two of these intersect or not becomes something which is subject to quantum uncertainties, become an operator, so that you then have a bunch of points. So the idea is that it gives you a bit of 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 in the geometry you start from. Now, one of the troubles that I've always had with string theory as it has developed is this problem about the extra dimensions. There are a number of 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 can make people try to excite them. If you're thinking of non-linear instabilities or something like that, with geometry as a whole, that amount of energy is trivial on a cosmological scale, so it's very hard to see that you can be immune from exciting different dimensions.

1:07:30 The argument is much more involved than just this, but this is part of it, and in any case you can appeal to singularity theorems which tell you the classical theory. This is one of the reasons I have difficulties for looking at higher dimensions, so that although I have nothing against string theory as such, in fact I rather like the ideas when I first saw them 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, to regard this space here as the target space for your strings, or in other words, your Riemann surfaces, you see you already have Riemann surfaces sitting there, and you might be concerned with more general ones, but not just these Riemann spheres that represent points, but other kinds of Riemann surfaces, which represent structures over here, which would be rather hard to understand directly. You can see much more easily what they mean over here than over here. So that is very exciting to me. People have commented. I hope that people who have heard me talk on this before don't mind me telling the same joke over again. People have commented to me, isn't it very refreshing and exciting and thrilling and all this to see the string theory ideas now in twistor theory. It's giving the whole subject to a boost. I'd 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 in a safari, maybe in Africa, and you look through the binoculars and you see these water buffaloes, you know, rampaging all over the place, trampling over everything, and then you look over there and you suddenly see they're going straight towards you. Yes, there is something that worries me about this. It's just a case by case. It's flattened by the recognition of consciousness. But still, you know, you just have to be able to... It gives a new meaning to target space. Sorry? It gives a new meaning to target space. Target space. Well, yes, I know that. I think you put what's needed in there. The original. Let me be a little bit more specific about what's going on. This is more or less the picture I showed you before.

1:10:00 Let's get a little bit better on this. I'm not going to assume that everybody here knows about Christopher. I've got a subject that's been around long enough. This is meant to be the Minkowski space, with these coordinates, the orange ones, r0 up to r3, this is the time coordinate. Over here, we have the Minsky space. The basic relationship which relates these two spaces is its incidence relation here, which takes the first two components of the Zs, takes the matrix of these things here, its emission matrix, and this... The relationship is called the incidence relationship and you can read this either in here or here. If you fix the blue ones, the acoustic coordinates, and ask for which points correspond to that, which I'll explain in a moment, you get a light ray over here, if you do it the other way around and fix a point here, that can fix the orange ones and let the blue ones run around. Then we find the locus over here, which represents the point, and that's this projected line in this free space, which is the ring sphere. Sometimes I draw it with a line, sometimes with a sphere, but this is a complex line, so it's got two dimensions. It's indeed a ring sphere. Where does this relationship come from? This comes from the fact that you want your points in spacetime to be real, and you look at your incidence relationship and see what that gives you. And that's just what I've done here, the relative conditions, distance relations there, and this over here, pre-magnified by the complex conjugates under the transpose of that vector, which you could imagine because of the I.O. and the I.O. distribution. So this is the twistor reality, or the reality condition of your life, or something, which, like this, by defining a twistor as a subspace object here, the complex conjugate of it,

1:12:30 It involves taking complex conjugates of all these numbers and interchanging the first two with the second two, which is that thing, and it's a dual object, say, a product between the twistor and the conjugate twister, and this condition here is the situation I wrote down on the previous transparency at the bottom, which if you say, roughly speaking, is a reality condition, real space-time here, if you drop this condition here, you get complex and complex space-time over here. So that's the basic correspondence, a little bit more about some of these things, but first of all we go to spin the notation and write this thing which matrix is extended with one prime index and the incidence relation which is this thing in the corner here, omega equals i r pi, so it's going to be this, a minute ago, equation over here, only that's the omega, this is the i r and that's this equation in the corner. Now, the first two components of this, upstairs and prime spinner, one of the slight irritations of this thing, not exactly like being trapped to 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 you use different letters and all sorts of things. And then I sort of... I've used that to improve what I call improved notation ever since. The trouble is that Witten always refers to the original paper. It's the only one I wrote where the conventions are considered to be all the wrong way around. This is only the beginning of certain problems I have which I'll mention in a minute. Let me just say a bit more. Interpretation of these objects, these two spinors, they only get to the tie, not just where the indices are, but the letters also have to do with what they mean. One has momentum and angular momentum to represent, in terms of these spinors, the momentum, which is called a null momentum, it's a massless particle we're talking about,

1:15:00 and then we translate that into two spinors, factorise it into a product. So this p is the momentum of that product pi over here. You represent that in terms of the electric product of the momentum of pi. It is proportional to the momentum of pi. So it's a very neat way of representing massive particles, which is otherwise a little bit complicated, because you not only have the condition that this should be a point in an elevator, which is automatic when you have it from here, but you also have the relation between n and p, The S here, this is the factor of proportionality between the injector, is simply the square root of the half of the spin of the product, and it enables you to interpret the twistor space, not just the lightweight space, which is the spin in the middle, but also the top and bottom halves in terms of spinning photons, although they're not specifically photons, they're mass of particles, it's a description of mass of particles, and the top half corresponds to the right hand and the bottom half doesn't. So this is the basic physical interpretation of twistors. Just a few more points here. Geometrically, if one wants to interpret a dual twister, that would be a plane in this projective space. So the point is represented by the root of the twister, and the conjugate argument which has to be a dual twister is represented by a plane. In those particular cases where the point lies on the plane, that's the space Pn that represents the light waves that land right before it, after the moon.

1:17:30 If you don't have this condition of equation Pn, then it's the light wave space, which is zero. Or I should say, let's put it another way, let's say we don't have the condition of the arcs being real. One way of translating from twisted space to space-time, it's fully complex structures, we 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. So you get this equation here, and now you say, instead of having light rays, you now have the so-called alpha planes. These are complex planes, which you find that for any arbitrary twistor, you have a plane which The points on it satisfy the incidence of the appeal of what's called the beta plane. I didn't invent this terminology, it's actually very classical. In fact, it all comes from classical geometry, which is what's referred to as the Klein correspondence. The thing is that what Klein was interested in was how you represent lines in free space. He thought of it as a projected free space. Five projected dimensions, that's called Klein's representation of lines in three sets. If this is a real space, then this will be a real quadric, which is of some relevance to what Witten's been doing. But in general, it's a complex space, in which there's a complex quadric. And the points over here are represented by alpha planes, and the planes are equal planes. So that's the geometry. This is its relation. This is also due to Witten's line in H.I.O. and all that here. But that's the classical geometry which underlies twistor theory. It's just that in twistor theory you read the correspondence in the opposite way. So Klein was thinking this is a nice way of representing lines in space, points in this manifold, and what I'm doing is going the other way around and thinking this space here is space-time now, it's four dimensions of space, and the complex case is the four complex dimensions of points here representing lines in this twistor space over here.

1:20:00 So you have this decline in correspondence and reverse. OK, that's geometry. The thing at the top here has just to do with the shift of the origin. Now, remember that this is a momentum thing, so it's natural that it's not affected by the shift of the origin. Whereas this thing is an angular momentum thing. It is affected by the shift of the origin. We go from the point O to the point Q. The O gets transformed very simply. But here there's something new on this transparency, which is when you go to the first complex, twistor theory, where here I'm imposing canonical connotation between the twistor, z, and its conjugative bar. So it's very neat because one just has two things, the primary, what is it like, and its conjugative, and also the canonical conjugative. If the twistor can use it itself and canonically conjugate to its conjugate, then all these set of points unchanged, except that you have to be a little careful about the multiplicity, it turns out to be just polarizing, not ZZ by Z by Z, and then you would say, what's the analog of a wave function, a single particle, in twistor theory, well, in order to read... In physics, you say you can use either the position representation or the momentum representation, and to say your position representation, you say that your momentum variables are represented as operators of the wave functions and doesn't depend on the coordinates or the other way around. The momentum coordinates and it doesn't depend on the position coordinates. But here we're saying in the z representation, you want your f to be independent of z bar. What does that mean? Well, that means the f by z bar is normal. In other words, f is homomorphic of z.

1:22:30 Complex analytic objects. This is one of the important underlying principles of twistor theory is that one is going to holomorphic or complex analytic structures and there's a lot of mathematical power in such structures trying to harness such power. Z representation. Z bar is the operator of Z. S dot is its operator. And this thing is its decoy. The homogeneity operator is mostly shifted. This is minus 2. And the helicity is basically the homogeneity of the function. So if you want to represent the pure helicity state, it's not just homomorphic in Z, it's homogeneous in Z as well. And the degree of homogeneity corresponds to the helicity. The very basic mass of fields. This is the wave equation for spin zero. You can have the mass of field equations with unprimed indices, with the negative velocity and the primed indices with the positive velocity. Here, the scalar wave equation. This is the field equation for mass of spin zero, and the Dirac-Weil equation. The neutrino, assuming mass is here, maps the photons. The left-handed path is the anti-cell fuel, and that's the red one. And you have homogeneity 0 and 4 for the left-handed particles 0, right-handed particles minus 4, and linearized gravitons minus 6 for the respectively left-handed and right-handed parts of the gravitons. So you have this very neat way of describing mass as particles in a pure helicity state.

1:25:00 In terms of homogeneous functions, then you want to see how to get backwards and forwards from that to spacetime, and basically the way you do that is to draw a contour interval, I've done it here for the spin zero case, so this function is homogeneous to the degree minus two, spin zero, and what you do is you basically think of this as a function of omega and pi, you substitute the incidence equation here, so now it's a function of pi, and then you integrate out the pi dependence. And then you have the function of x. And doing that, you automatically get the field equations, in this case the wave equations. For the other opportunities, you automatically get the fields that I just described from the last transparency, which is all the different elicits coming out. Versions of this formula were known to Whittaker and Bateman many years ago, but this gives you the general expressions of arbitrary. There's a point I'm trying to illustrate here, which is that if you're going to do a contour integral, the function you're integrating has to have some singularities that you integrate round, and the typical case, this is the simplest case that you're going to write down, this is the homogeneity of degree minus two, this case, the wave of origin case, so you have a product of two linear factors. And these will go singular on you in two planes, that's the a-plane and the b-plane, and if you arrange that these intersect in the bottom half of the twistor space, then you find that in the top half, these singularity sets are always separated, so wherever your line is, they will always intersect these separated singularities, and you take a contour which goes between them. In particular, because that lies in Pen, it's a real field, and it gives you 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. In the twisted space that I've been talking about, it's divided into two halves by this space which represents the light rays. And that was a very important early motivation for the twistor theory, because it's like the way in which the Riemann sphere can be applied to the top and bottom half, and the idea is that you have a function which is defined on the equator of a sphere, of a Riemann sphere, and those functions which extend into one hemisphere give you positive frequency fields, those that extend into the other hemisphere negative frequency, and what this does is just that, but globally for the whole space-time.

1:27:30 It comes from this fact here, but if I consider complex points, as I said before, a complex point in space-time is represented by a line somewhere in this space that may cut across Pn or be somewhere in relation to Pn. But let's look at the imaginary part of the complex position vector of this complex point. That imaginary part could be zero, it could be space-like, it could be... The different possibilities for the imaginary part of that complex position vector are expressed over here. Page 1 is where the imaginary part points into the past inside the light curve, which lies in the top part of the crystal space, and if the imaginary part is past the forward tube, And what one may be interested in quantum field theory is field amplitudes which extend into the forward tube, that's a positive frequency condition, and what that means in twistor theory is that we're talking about something which is defined in the top half of the twistor space, here, as the lines line up here. An important ingredient to twistor theory, we have a nice representation of this positive frequency condition, which is basic to quantum theory, and going back to the transparency which I was showing before, when this line applies in the top half, divided, these planes only intersect in the bottom half, you can see that the singularities are separated, wherever this line is, in the top half, and so you have a non-singular field in the top half. And then, of course, positive frequency. This is a nice way of generating positive frequency fields. In fact, it's completely general. If you've read something that's appropriate, there's not going to be any details of that.

1:30:00 But it's still a little awkward. You say, what about having singularities just 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 non-singular in the top half. That gives you positive frequencies. It took a long time to appreciate what this was all about, and the answer, re-education from my old field, which was very helpful, this had to do with sheep 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, a simple case that we're looking at here. So covering would be, say, the top part where you leave out one batch of singularities, And the other patch of things we've got, the other thing we've got, the other thing we have, is the union of the two cultures at all, not just the space, but the union of the covering of the space with open space, I won't talk about this in detail, because if you know about it, that's fine, if you don't, it would take me too long to explain what it's all about, but I think what's rather 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 building sets, and you want to somehow to deform something which was initially set 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 the top and build up a curved manifold in this way. And in fact this is exactly a sort of what you call a nonlinear version of cohomology. If you just think of this as an infinity catamount, then you have the first term on the element. That's exactly what I've been describing before. If you imagine, before making this one final enough, Some sort of nonlinear cohomology. 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 that 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 and shifting actual factors across each other in an active way.

1:32:30 Okay, well that's basic twistor theory. I think that's pretty well all I wanted to say, the basic twistor theory. 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. I should say these are from the Jack Mills fields. And one thinks of these as gluons, represent your gluons in terms of illicity states, and some of them are possibly represented by arrows going in, 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 could qualify it, say you get zero again. They come in, then we get interesting amplitudes which can be worked at explicitly, and this is what these people did here. I think there was some conjecture with Hawking and Taylor and these people here, and Neyar did in fact notice that this was part of some of the twistor theory. I'm not totally sure about the relationship between all these things, but he certainly did point out the connection to twistor theory. And it's a very attractive thing in many respects to the likes of me, in the first place, when it's talking about honest interactions which are 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, it may not be so nice for others to do it, but let me start by showing you something that I did, if you think of a function of your momenta, or say a single momentum, and you can write that as a function of the pi and pi bar, and then you can do a Fourier transform, not a normal one where you can do a Fourier transform with respect to p and the function of x, but you transform with respect to one of these, it's very lopsided, it's asymmetrical,

1:35:00 Say we're expecting a pi bar, and remember that pi bar are canonical conjugates, so it's not unreasonable that you go from one to the other by a Fourier transform, 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. This very much underlies what people have been doing on this subject recently, but in a certain sense it's not quite right. The trouble is that when you go from, I mean, what you could imagine, suppose you did it not just on one variable, but you then did it on the other one, so you could Fourier transform, sorry, let me say it the other way around, if you have a function, a twistor function, and then you Fourier transform on the other one, that gives you one of these functions, which is now a 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 Fourier transform on this one, and then you get a function of a dual twistor. So you could say, this is the way of getting from your function of a twistor to a dual twistor, by doing two Fourier transforms. But we know that's not right. When you Fourier transform, you do what's called a twistor transform, when you go from a twistor to a dual twistor, and it's something which is not correctly represented by Fourier transform. If you like, it's morally a Fourier transform, but in detail you find it's not really that. And one somehow is throwing it overboard by using this transformation. And this does begin to show up in certain places and stuff. So I'm afraid, well, I don't understand what's being done here. Every now and again I try to read it, so little things come into my mind. I say, well, that isn't quite right, but I can see that you get wonderful things out of it, so I like to understand what's involved. So, I think there's great potential here for trying to, what I would say, there's a lot of sort of string baggage, which makes use of a lot of ideas from string theory, and that seems to be very much part of what's done here, but the motivation for a lot of these things comes from thinking that you're starting in a space-time theory, but you're not really doing that, and so... I'm not quite sure I can follow it. I'll perhaps explain what I mean a little bit more in a moment.

1:37:30 But let me make another comment here, which has to do with signatures. You tend to use plus plus minus minus signatures. The twistor theory which I've been prescribing to you is really about the bottom here. Here you have this twistor space divided into two, the top half, the bottom half, and the middle case. So this is physically what you really do want. This is certainly a level of what one directly experiences, the way in which twistor theory should be used. But pure mathematicians, if they're not so interested in physics, they like positive definitions. Then there's of course all the reasons which have to do with the fact that if you want a spacetime in which the curvature, the vial curvature, is anti-self-dual or self-dual, then you find it comes out complex if you start with one of these signatures, whereas if you start with a positive-definitive signature, then you can have perfectly real space which is self-dual or anti-self-dual. And so the twistor correspondence actually turns out to be something But it's certainly something you can much more readily realize in geometry if you adopt really positive space-time signature. And then you have a twistor complex conjugation, which is not the kind that I referred to where the conjugate of a twistor is a dual twister, but where the conjugate of a twister is another twister. And this case is one where you don't have any real questions, that is if whoever has z equals z bar, it would imply z equals 0, and you have certain structures which are related to that, and it's also related to quaternions, in the quaternion case, and it's much studied in differential geometry. The case that Witten has been interested in is the plus-minus-minus signature, where, I did describe this briefly, it's if you're considering a real. I think that's the kind of thing that we've been working with. I think for various reasons that it's just sort of easier to see what's going on in a real projected tree space.

1:40:00 But the view I take is rather like what would have been the case in working with positive dependency until we've got some. Signature, which you admit is wrong, and then you say, well, we're going to go with all this art, and then we're going to get something analytic, which we then extend into the physically appropriate situation. Now, it seems to me this is, in a certain sense, going the wrong way, because this is where you want to end up, so why not work with this? And it has kind of structures like this, between the positive and negative frequency parts. See, that's the sort of thing you want in quantum field theory. So, to me, it seems to me one ought to elude all this again and try to do it in the right signature case directly, rather than hoping to do some kind of pseudo-Witt quotation at the end, which is going to give you what you want. So these are kind of minor quibbles, if you like, but they're points of some relevance which might begin to be more and more important as the field develops. And what he uses is within an out-space are basically things which look like momentum states, and then you do this Fourier transform and he comes up with things which are delta functions supported on alpha planes. This works fine if you're in this sort of real case that I was talking about, the signature plus plus minus minus, this case here, so you get real alpha planes. But in the appropriate signature, there's something a bit funny about these things because they can't be supported on alpha planes. I think these have to be singularities of some kind which extend into the complex in some way. Anyway, let's not go into that, but I think that one needs to study this a bit more. Now, the kind of string theory that Witten seems to be talking about, although here, as far as my understanding, is stretched to a considerable degree, but one has this Chern-Simons action which is used in twistor space. Well, first of all, I have a problem here, because what are our action score and what is our logarithmic score of all that thinking? In, uh, all of your physics, you use the logarithmic, the quantilaterals, and field equations, and how your data propagates in time, and so on. But there's no time here. If you like, the time is all built into the twist and geometry. So it's something that doesn't matter. But, with this particular type of action, we can have this...

1:42:30 Well, essentially a topological quantum field theory, and actually in this context, what you might call a cosmomorphic quantum field theory, where it only depends on the complex structure and doesn't depend on the particular connections that are involved in here. So it has this very remarkable property, the kind of reaction you're using, but it doesn't give you any meaning. I mean, usually they're going to give you field equations, and that's what you want, and with a topological quantum field theory you might say that's very interesting mathematically, but why do we want that in physics? Because you want field equations in physics. 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 where I was suggesting that one might actually We have a version of topological quantum field theory, which is really a holomorphic quantum field theory, and I call this thing holomorphic linking, and it's concerned with the Gauss formula, which is the integral, the linking number for those curves in R3, and you have this wonderful formula, using Gauss, which tells you the linking number in terms of the double integral. But then if you make this thing complex, even in a complex, it still makes sense. And then, fiddling around with it, you'll learn the things related to twistor diagram theory, and this is work which Andrew Hodges has carried forward to his students, and it's a work writing quantum theory theory in terms of twistor theory, so it's very nice to see this stuff relate to twistor diagram theory. Let me just mention one thing which is... For me it's personally interesting, because if one is trying to represent these curves, curves which maybe now are not straight lines, but are hierarchical, it actually goes back to work that I did when I was trying to do algebraic geometry as a research student way back, and the problem I was concerned with had to do with how you represent spaces of non-maximal dimensions, but not hypersurface as you said. The first case you notice is this curve in tree space. So how do you represent the curve in tree space by an equation? You probably paid it for. This is your curve.

1:45:00 And instead of thinking that it's a quite low curve, you think of it as a condition of line. And that is an equation of line. So it seems to me this may well be the right way to represent curves in twistor space. Now, one thing I haven't yet mentioned is that... I mentioned behind it dimensions, but there's also supersymmetry. There's no evidence as yet, at least from observation, that supersymmetry really is the future of nature, whereas string theory, as it's developed, has very much been dependent on symmetry ideas, which is slightly cheating when I describe what Woodland and Anthony were doing. They were really talking about not just C-4, but symmetry generators. Now, this is all related to the question Anomaly cancellation, which has originally required force, strings, linear parameterization, and variance to be regarded as a power-providing force in string theory, carrying his other ideas to the twistor ideas. Now, do you really want? What you find, actually, is that, if you wanted to pull out the Yaw space, obviously, this anomaly cancellation would work, whereas here, what the... The company at Ban is that if you regard CP3 with four supersymmetric generators, then it is actually a supercalarly outspace, and this means you get the polynomial cancellation and so on. But to me, I would say, well, I'm not sure I want to use supersymmetric 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-string theory because you really want it to be independent from parameter theory. In twistor theory, you see you can have a version of twistor theory which is completely conforming invariant, or you can have a version of twistor theory where you actually know where infinity is, which is represented by this line here.

1:47:30 And this breaks the formula of variance. Now, is it the case that if you throw away your three parameterization variance, as you would be doing in this case here, if you have this projection down here, well then you know where this curve, it's got an active parameter all the way along it, which has to do with the planes which pass through this line across this infinity, and it has to do with this. Canonical 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 supersymmetric parameters. We'll know about this is already part of the question. You'll tell me I don't understand what I'm talking about, which I will accept fully. Thank you very much. Well, the general comment, I think, that has to do with how you represent things which are not analytic. I mean, there are things called hyperfunctions. Which I believe are the right way to talk about non-analytic problems in this sort of subject. See, one normally thinks of distributions, which are the dual of C-infinities functions, and you can talk about del functions, not just distributions, but del 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 hyperfunctions are defined fundamentally in terms of holomorphic functions. But it's basically the jump. If you take a holomorphic function which runs up to a boundary somewhere and you extend it with another one, the jump between one and the next is a hyperfunction. So, hyperfunctions are, in fact, basically described in terms of normal functions.

1:50:00 So it seems to me that you come full circle, that you generalise and generalise away from the whole of the content and then come back again. 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 are junk and so on. And yes, I think you're absolutely right, we want to be able to handle such things, and it wants, somehow, the theory of hyperconnections to be brought in to the subject, which is not something which has yet been done properly, but it is. They sort of marginally played a role in things that we've done, but I don't think that's too far. I have a couple of questions. Like, first is, what is the physical problem that one is trying to solve here? Is there really anything that people have not been able to solve? 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 felicity-violating amplitudes, which is where you have two, one felicity and the other one's all another felicity. Now, as I understand it, although I don't understand it, I don't understand it, I shouldn't say 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-stealth dual solution and then you turn away from that to get the stealth dual components. Now I don't know to what extent exactly done by it and there seem to be some anomalies in what they talk about 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 from my point of view. Well, I would say two things here. One is something to me which isn't surprising. Namely, okay, you're looking at conformally invariant helicity, amplitude, helicity states, and twistor theory is the right framework for making that look neat. So, that's just a mathematical point, and there may not be any physics in it. Now, it's not clear that this is a... To what kind of 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 and what they call almost space-filling D-brains, sort of boulder patterns, but the strings, in one approach, they have homomorphic curves, which are supposed to have nice relations to these attitudes and points along 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.

1:52:30 Well, that makes no sense to me until you've got a real theory which 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's done things before which have had certainly mathematical implications which are often true and surprising, 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 does it 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, I think, is this form of supergravity. I don't quite see, you know, how should I interpret all that. That's not gravity I know well. So, is that the right thing to do? I haven't done it. But it's just a leap. That's all I see. This is a leap into... I mean, twistor theory for a long time has in some sense been stuck. But you've been able to translate that into twisted diagram theory, you've been able to take the answers that you know from finding theory and translate them into twisted theory, and you get some nice diagrams, and Andrew Hodges has 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 a central principle which will supply what's been missing from twisted theory for a long time. But I can only say this in a conjectural way. I don't see it there. The basic formula that you wrote down, which is I and all these things, not all had to do with just zero-resonance fields. It had nothing to do with us. So, is there any connection between them? Well, if you call it the deep, where you actually solve the nonlinear problem.

1:55:00 Right, yes. No, I agree. That's one point. I don't see any gravity connections yet. All I see is a claim, and here you have to confess that I'm not very good at extracting these papers myself, I stare at them for a long time, but I have a colleague, Alan 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 end-of-self dual solution, the Yang-Mills solution, and then you perturb away from that. So it's not representing the four Yang-Mills things in a twisted expression, but you're sort of half using this. 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 it. Well the trouble is that 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 TPP. That's supposed to be at the stage later on when you can do some pseudo-Wick 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. So they don't talk about that at all, which to me is, I think one should take that picture and see, translate what they've done and see how much it can be translated. I don't know. I mean, it's basically the same philosophy as one you have with rotations.

1:57:30 I mean, in some respects, it's obviously good in path integral or something where you might have some convergence, better chance of convergence being possible to see each other. Now, 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 would quote, pseudo-quote, as to say you can change one of your pluses into a minus in whichever way you want, by having the continuation. And this could make sense. I mean, maybe it should be a right positive frequency or a finite number of keys coming up using that. 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 you're going to get what you want. And you don't have the advantage that there is of the positive-definite signature, the ordinary sort of linear approach, where at least you have a chance at certain things that converge with that signature. I'm a little bit amused by what the advantages really are about using that. I think it's mainly that when you talk about real projected free space, somehow you can picture what's going on, and you can talk about these alpha planes, and you can see, in some sense. It seems to be somewhat limited and that no view is not a gentleman. That's my impression. When you end up by having explicit expressions, which are in terms of his 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 double function which expresses momentum conservation, you take that away and you've got left with some polymorphic amplitude.

2:00:00 But that's very conventional. 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, well, 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 would get out of this sort of... Assignments approach, I think, is probably more related to debacle, which is equivalent to that, but it's equivalent to the things I've said, but in a rather, 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 delta function, which expresses momentum conservation, you take that out, and then what's left is sort of 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 amplitudes you get out All of these 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 implicitly you have to have some way of representing that in terms of hyperfunctions and so on, so you don't actually do it that way. But, I mean, the answer to your question is he hasn't actually used that. Well, the alarm tells me that somewhere in the depths of one of these papers it is actually used, but I haven't seen it myself.

2:02:30 Announcements. For those of you who have not registered or paid or collected your dinner tickets, please do so. We really need the dinner tickets. Also, there was a small number of empty dinner tickets left, if anyone could still go and get some dinner. These dinner tickets may, by the way, include the infected copies, so get there quick. Coffee is outside level 2. It's going to be very crowded, so can you perhaps move out onto the patio after you've got your coffee?

2:05:00 He always does. Which is one of his great strengths.