Arrows of time and generalized quantum theory

0:00 I hope you've enjoyed your lunch and also the beautiful weather that Jonathan has arranged for you. I'd like to say that he is the miracle conference organizer. So although there are other people listed on the organizing committee, really we have absolutely nothing to do with it. Jonathan has done everything. And our first speaker this afternoon is Jim Hartwell and he'll be talking about arrows of time and generalised quantum theory. That's a little bit daunting to talk about in hours of time when Robert Penrose might be the audience. I'm pleased to see that he is not here. I have a story of a famous Penn State football player who was very good at his trade and he died and went to heaven. And when he got to St. Peter, St. Peter asked him what the detail was. And what the title of his talk was. And the famous football players in the talk, I didn't know what it was. Oh yes, at St. Peter, everybody who gets in here has to give a job talk. And we're, there are many sports fans up here. We're all interested in hearing your talk on the inside story of football at Penn State. And the football players said, oh, I don't want to talk about that. I've given that talk so many times. I'm just sick and tired of it. But I want to talk about the Johnstown flood, the famous flood in the U.S. The one the man had also lived through. And St. Peter said, well, we do really would like to hear about the Penn State football, and Ancestor, and on and on. Finally, St. Peter said, well, all right, but I have to warn you, no will be in the audience! It's certainly one of the people who has entertained the possibility that usual quantum mechanics has to be generalized before we can apply it to quantum cosmology and quantum gravity. That's because there are two reasons. First, the usual formulations assume that the output of the theory is the probability of measurements.

2:30 Made by observers and assumed in one way or another the usual quasi-classical world of everyday experience. But in a theory of the whole thing, there can't be any general division into observer and observer. 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 and the ritual laws, as they are stated, rely on an incorporated notion of fixed-factor geometry. For example, the Schrodinger equation from the 19th century is a metrical notion, which is applied by the nature of time. Fixed geometry, but in quantum gravity, geometry is a quantum dynamical variable. It's fluctuating and without . 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. Now, it's a remarkable fact, I think, that we live in a time-asymmetric universe, where we're floating in arrows of time, 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 street 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 tiny symmetries of the universe and kind of symmetrical dynamical laws, the only source of explanation is asymmetries and their boundary conditions. I'd just like to run through a few. Of the famous time-based symmetries, we'll talk about the time in the universe and explain very briefly how these arise from time-based symmetries between the initial and final boundaries of the universe.

5:00 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 parallel time, as it's sometimes called. As Paulson said a long time ago, the second law of thermodynamics can be true in the converse of a mechanical theory if one assumes that the present state of the universe evolved from an improbable by a special state. And Rogers and Penrose said the same thing a little bit later. It's usually expressed in terms of increasing entropy. Of course, there are many different entropies. Entropy depends on coarse-graining. So the particular entropy we have in mind here is the entropy associated with the course training defined by the quasi-classical realm of everyday experience, or more precisely, by course trainings by ranges of values of approximately conserved, intervals of approximately conserved densities, such as energy, momentum, and species, in what were soon to be chosen models. The fact is that we live in a universe, apparently, where if there's no final boundary condition, it won't make a difference. 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 decrease since. And that asymmetric boundary condition is what gives rise to the thermodynamic error. The electromagnetic error, similarly, can be seen to arise from asymmetric... Boundary conditions apply to Einstein's laws, in this case Maxwell's equations. Any solution to Maxwell's equations with fixed sources, fixed currents, J-U, can be written in either one of two ways, either in terms of the target green assumption of the sources and free incoming fields or advanced fields in the force on the point of sources and free upcoming fields. The asymmetric boundary condition that the infield is approximately zero and the outfield is not zero is what gives rise to the retardation because then this is approximately zero and the entire field can be seen as arising and retarding whether it gives the sources.

7:30 Of course, in cosmology it's not true that the free fields, at least at the start, say, of the time that we happen, are approximately zero. All the energy in electromagnetic radiation and 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 low wavelengths. Electromagnetic radiation, and radiation, for example, is visible today, then that will be approximately retarded because it's essentially zero to infinity. The psychological error of time can be sort of similarly understood. Here's information gathering utilizing systems. Here's a little bottle of one. Every time set, it shifts the contents, which is recorded in these registers, to the right here and erases the final one, and it records a new piece of information from the external world on the first one, and thus 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 recorded in the past. It uses this information to compute from a process of unconscious computation to compute a schema of the model of the universe, which we all do, I think, contain such things as the rules for success, get food yes, be food no, vanilla ice cream tastes good, and that's integrated with the present information. Well, the important point is that there are two arrows of time which contribute to this. The idea is that the core is information that gives variations of exhibit's behavior. Information that it gets in visible wavelengths, for example, is from the past because electromagnetic radiation is retarded. So that's one of the reasons why it was passed, but more importantly, the recording of information is typically irreversible, not necessarily irreversible according to Mandela, but the erasure of information in this final set is necessarily irreversible.

10:00 Therefore, the cycle of functional parallel time operates most simply if it's comparable to the radiation parallel time, which is regarding information, and the thermodynamic parallel time, which regards the reversible processes. So that's an example of three hours of time in universal rising from asymmetric boundary conditions. If 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-based symmetry, here are two of them. One is, or could be, the time-based symmetry of unique interactions. So even if the states and dynamics of, for example, the string theory are very effective theories that are applicable in limited circumstances, they can be kind of asymmetrical. And at least it's a possibility that the, for example, it's believed, although it's incredibly difficult to get a straight answer out of the string theorist, right, the best I can get is that the theory is believed to be time-reversed and invariant, and therefore one possibility, for example, is the asymmetries and weak interactions arise in, or asymmetry breaking, and therefore the universe could exhibit on very large scales the time-reversed and invariant theory. Hours of time running one way or the other in the movement of action, but overall time-symmetric. Another example is the expansion of the universe. We see the universe as expanding, of course, but in Stevens, no boundary wavefunction of the universe is real, and hence it's time-symmetric. Therefore, the ensemble of semi-classical trajectories is also time-symmetric, as he predicts. So, with each trajectory expanding, we're contracting, but there are just as many trajectories This is radius of the universe versus time that expands and is sort of contracting there on the other side. However, because of the psychological arrow of time, and the arrow of time associated with thermodynamics, the aegis, and the information gathering utilizing system, whatever one is operating on will see the direction for the big bang as the past, as the direction for the country.

12:30 So the whole ensemble is symmetric, but apparent asymmetries arise from symmetry breaking. Now, what's the reason that we have time-symmetrical dynamical laws that are governed by time-asymmetric boundary conditions? Well, of course, exactly 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 CPP theorem in quantum field theory, where Lorentz invariance plus locality, I mean, in quantum field theory, is applied to CPP, but you might object that CPP is not the same as T, but it's post in some sense. So in the face of these dynamical problems, there are now two possibilities for the observed asymmetries. And the first question we have to ask is whether all the asymmetries could be apparent, that is, basically, the whole picture of the universe could be found symmetric, and the asymmetries that we see arise from just a special symmetry breaking in special circumstances, so that hardly a piece of the universe appears to be asymmetrical. And a classic example of this is the idea, in the interest of you all who first thought this, where we have an expanding and protracting universe, In which it is time-symmetric in the sense that it has time-symmetrically related dominant conditions here and here, and the entropy increases on one side, and then decreases over this way, it's increasing over that way. So it would be perfect time-symmetry, but of course the idea is that we would live here in a regime where it's increasing, and then over here we can see it pass in the sense that it's increasing in that direction. However, it appears that this idea, well, it's not a firm conclusion, I think, is not looking too good. The key observation, I think, which was made by Davis and Plummer, is you would think that if you made this time longer and longer, it would be harder and harder to distinguish this from one of the geometric distinguishes.

15:00 The point here, the corresponding point here, remains finite no matter how long this time becomes, and that's because it's true that time is getting longer, but the universe is expanding more, so the matter is getting more and more dilute. Therefore, the contribution, the integrated contribution of the optical depth remains finite. So, I don't want to go into it in detail, but there is a certain amount of evidence that because we can see the time symmetric, we don't have it. David Cray says there will be orders of magnitude showing from the brightness of the night sky that we wouldn't have a time-symmetric universe. Raymond Buffon says that if the universe were time-symmetry, it would necessarily be irregular at both ends of the Big Bang, and that they crunch, and that's something we don't see because we see it highly irregular in the Big Bang. So we seem to be forced, and these ideas are true, to the idea that the time-symmetrical laws give rise to the asymmetries that we see by asymmetries between the initial and final condition. But there's one obstacle to this point of view, which is the arrow of time in quantum mechanics. Quantum mechanics has built into it an arrow of time. This is usually formulated. The Schrodinger equation, for example, is kind of cursive in the sense that you can go right up to the forward and back of time, where the operations can be applied to, but the second law of evolution, right, the idea that on the measurement The wave function is reduced by the application of a projection operator which starts off to the outbound direction and then we normalize, which is equally essential to describing, predicting probabilities of histories or sequences of things. It's not time-reversible, is it? You can't run it back at the same time, take off the projections somehow, you don't even know where the projections might have been put in because the projections operate when the system was measured at a given time or present. You generally don't know when it was measured even then.

17:30 Now that's usually thought not to be such a bad thing because it's thought that measurement is an irreversible act of amplification, as it's usually characterized, and therefore the second law of evolution should operate in the same direction as the firmament and amygdala. But that's a little bit problematic because it's after all shouldn't be possible to intelligent aliens or in the future And then, for observers in such a large volume, what power of quantum mechanical time do you apply to it? Do you run it forward, like this, or do you run it backward because of the quantum mechanical time? Of course, if the existing quantum mechanical laws are merely effective theories, that is, they should cooperate in particular circumstances, then that won't be a problem because the effective theories put in one part of the universe one way or in another part of the universe one way or the other. And that would be my basic thesis here. We know, as we argued in the beginning, that quantum mechanics need to be generalized with 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 be also generalized so that it's time neutral, so that there's no quantum mechanical parallel time, and the parallel quantum mechanical time that we observe arises from the asymmetries and the boundary conditions in a completely time neutral framework. I argue that it can.

20:00 Decoherent histories of quantum mechanics versus generalized quantum mechanics, which is the framework in which this will be discussed. So if you begin with a little review of the quantum mechanics of closed systems, which is what answers the problem of the first need for generalization in the remnant of emotional measurements, let me consider a little molecule universe, let me admit the same in this audience, of neglected quantum gravity, which is a pretty good approximation when you find out the first 10 to the minus 43 seconds in the universe. And thinking of the whole universe as being a large number of particles and fields in a box, perhaps 10 to 20,000 megaparsecs of the sun, maybe expanding, then the usual apparatus of Hilbert space states of short interperation and so forth can be applied, which we'll simplify in our discussion. What's assumed for any closed system, and is given by theory, is the Hamiltonian, or whatever the present theory is, as far as that's concerned, the dynamics of the Hamiltonian event theory. The most general objective of any quantum mechanical theory of a closed system is the prediction of probabilities that of course bring about part of the histories of that system. Thus, for example, we might be interested in the orbit of the Earth around the Sun. 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, that we're in 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 Earth. It's coarse-grained because we don't consider the center of mass position of the Earth to arbitrary accuracy, but rather we consider it due to some generous range of accuracy, the less we could possibly do. And it's coarse-grained also because we typically won't specify the center of mass at all possible times, but at a series of times corresponding to classical observation.

22:30 So in that sense, we're dealing most generally with coarse-grained alternative histories of the system, but not every set of alternative histories that can be described can be assigned probabilities. Now where is that more clearly illustrated than the famous two-slit experiment? Here we have source electrons, which if are passing through a screen with two holes, can be detected at some point y to another screen over here. In a simple model, there are two possible coarse-grained histories, one in which the electron went to the upper slit and arrived at y, and one in which the electron went to the lower 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 arrive at point y will not be given by the sum of the probability to go to the upper slit and arrive at y, and the probability to go to the lower slit and arrive at y. That's because the quantum mechanics probabilities are squares analogies, and the square of the sum is not equal to the sum of the squares because of quantum mechanical interference. So some rule is needed to determine which sets of estimates can be assigned probabilities and which cannot. I emphasize, it's not a matter that we don't know which list of electrons went through. That would be a case when the probabilities are 50-50. It's that we can't assign any of the probabilities at all. The usual rule is that you can assign probabilities to the histories that have been measured. So if you measured which split the electron went through, then the interference is destroyed, the sum will lose a wave, 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 to the Earth universe, because there is no notion of measure. You're splitting the system between the measures of the Earth. Which is due to the preface on this Gilman and all. It's just this. The initial, the closed system has a Hamiltonian, it has a state, and probabilities can be assigned to exactly those sets of alternative histories.

25:00 Of course, the interference between the individual histories is negligible as a consequence of that state. Measurements can be described in such a framework as quantum mechanics, but they don't play a fundamental role. So, for quantum mechanics and closed histories, we need three reasons. The fine-grained histories, which are the most refined possible description of the system. The notion of coarse-graining is actually easy to get right away. In addition, the fine-grained histories make classes. And the measure of interference between the coarse-grained histories so that we can tell in which situation it vanishes. Here is the story in one language, the language of operators defining coarse-grained histories. Alternatives in one moment of time can be reduced to yes-no alternatives. Is the particle in this range yes or no? Those yes-no alternatives are represented by projection operators, and the notation on use, k, corresponds to the particular set. Thus, when you're moving around the sun, you might divide space up into a lot of little boxes here. We can ask for the center of mass to be this box or not in that box, or this box or not in this box. A history is a sequence of alternatives to a series of times, so the orbit of the earth There could be this box, that box, that box, that box, that box, that box, this box, this box, this box, this box, they're all in many different parts of the history. So it's a sequence of elements at a series of times, and it's represented by the corresponding chain of projection operators that gets 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. There are a number of alternatives, specifying only a very few variables. So, of course, Grady is good with the general case.

27:30 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 sort of fine-grained paths. Of course, Grady consists of bundling these paths into little groups. So, for example, if you're considering the set of alternatives Of course, not even the motion of the Earth, you might specify a range of positions, say, three different times, and, of course, brain history is consistent with all the fine-grained paths at the center of mass that actually pass through those regions at that time. Then, for each set of history, each set of paths, you can define one of the several paths that are active in the initial state. The objective is to get an operator C acting on the side, which turns out to be exactly the same as the chain of projections which I looked at before, at least up to the factor of n minus i hg, which is relative to the probabilities, and this defines, well, that's how it works in some other histories. So again, how do we get the coherence of probabilities? We get the set of histories which correspond to these class operators, the chains of projections. We can define branch state vectors for each S3 set by applying the operator to the initial state. We can define the decoherence function, which is a complex value function on pairs of S3 despite the overlap. The set of S3 is decoherent if the decoherence function is negligible for all alpha prime values of alpha. And the probabilities of histories are then the diagonal elements of the decoherence functional, the equivalent of the adjusted platen, just the operator you see, the platen size 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-slip experiment as the obstacle to assigning probabilities to history. That is, if you have a coarse grainy set of histories, that is, if you take a given set and you group them into bigger sets, it's a coarse grainy, so the operators are sums of the coarse grainy set are sums of the operators sitting in the finite grainy set, then the equilibrium supplies the probabilities of the coarse grainy set are the sum of the individual probabilities that are in that set, which is the most generous. Well, that's a very speedy introduction.

30:00 But the main point for this talk is this formulation of quantum mechanics still has an arrow of time. It still has an arrow of time because of the formula for the probabilities. In fact, it's much worse than the usual formulation of quantum mechanics, because there's a psi on one end, and there's nothing on the other. So there's a definite arrow here for 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 periodicity or versatility of the system. So this arrow of time would have to be fundamental. It is helpful to have this arrow of time because it allows the definition of states in a moment of time. States which are sufficient for future predictions but not for the past. Let's suppose you have, for example, a probability of a set of histories. And I also want to do that again, given that I've come over here, and we'd like to calculate the conditional probability of a certain number of events that have already happened. Say we're at the horse races, and so many horses have already come in, we want to calculate the probabilities for the next race. Well, that's the conditional probability for these events that are happening already, and these we're seeking to predict, constructed in the usual way. But that, it's easy to see, that that probability can be important. All of this is calculated by applying the chain of projections that have not yet happened to a wave function, which is a function of the time, which is just given by the chain of projections that have already happened, and then normalize. That's the usual story. In the Seisenberg picture, the states are constant in between the action of these projections, and then add an alternative to reduce by the action of the projection.

32:30 That's okay for future predictions, but if you try and turn the thing around and calculate the prediction probabilities given what we know now, what happened in the past, you can't do it using only a notion of state at the present moment. You build the present information and initial prediction. What I want to discuss now is generalizations of quantum mechanics that are time neutral, that don't have a model of time, and the use of this one. So with that I introduce you to a little bit of the ocean generalized quantum theory, which is due to Eichel, Linden, Zorc, and Gell-Mann, and so forth, and, well, here you can only put on a few transparencies. It's the same three elements that we've already seen in the usual formulation of quantum mechanics. You need a notion of fine-grained histories, which are the most possible defined by the description of the system, which can be, for example, fine-grained mass in the case of particles, or in the case of field configurations, in the case of fields, fields will see general relativity, geometries. You need a notion of coarse-grained, which are the partitions of these fine-grained histories and their classes, and you need a measure of interference, which is the decoherence function. The new ingredient here is that the decoherence function is going to be characterized axiomatically, and this is, in fact, this is by now a rather antique notation compared to the one that Christy used, but I can stick to it anyway. So a decoherence function is any complex linear function that has the following property. It's normalized, and most importantly, it obeys the principle of superposition, expressed in this bilinear form, where you have a coarse range of a set, and then the decoherence function for the coarse range set is the double sum over the decoherence function of the finite range set.

35:00 If you have all that information, such a function, then you can calculate the finite decoherence. The off-diagonal element is vanishing. We can define probabilities, trajectories. So the key point is that the usual 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 ending the kind of quantum theory. And here it is. It's really just a souped-up version of the work of Akronov, Ervin, and Nikovits many years ago. You just have to write down the generic 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, if you keep oscillating this back, you can interchange the initial and the final conditions. 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 condition as well, and that theory is time-neutral. There's no quantum mechanical matter at all. In such a time-neutral situation, the asymmetries of time are explained by differences between the initial and final condition. The initial condition might be a little-longly-weighed function, and the final condition might be a condition of complete ignorance, and that seems to work pretty well for our universe, but in this theory it becomes a quantitative question, a question for prediction, a question as I discussed at the time-symmetric university before, what is the best final condition to put in here, or how well do we know what it is? But there's a price to be paid for this. As I illustrated, the notion of state in the quantum mechanical system is itself an inherently kind of asymmetric notion, and therefore, in general, we won't be able to have a notion of state on revolving through space-like surfaces because the original version of state was good only for future prediction, and now the theory is time-neutral. It can't do either predictions in the future or the past.

37:30 Now we turn to quantum gravity, or at least this cartoon version of quantum gravity, talking about quantum space time. Quantum space time is already an issue about defining the number of times, because there's no definite notion of time after all. It's quantum gravity. There's no fixed time to be talking about whether it's symmetric or not symmetric. Well, that's true, right? But there are histories, and histories are technically histories of geometry and matter, So, for example, if there's a cartoon version of 10 years, this is supposed to be four years in the universe, and we can have the no boundary proposal in the beginning of the universe and then the condition of ignorance on the end, 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 we get classical spacetime. Let me just illustrate how that would happen. Well, here's a cartoon version of the generalized quantum mechanics for geometry. We might take the idea that defined brain histories are four-dimensional geometries with matter fields on them. That's the math of generalization. But coarse-grained histories are partitions of those in the classes. If you morph them into classes, it's parallel to the two. So, for example, if you're interested in the question of whether a universe expanded with a big volume or not, you might pick a volume and then partition all of the histories in between by whether the geometries had a space-like surface that had a volume bigger than some crucial volume, or whether they did not. And you can calculate whether those will turn into equal here, and calculate the probabilities, and you get the probabilities. And by taking the overlaps of the overlaps and the wave functions that we find, that's a fully four-dimensional quantum mechanics of space-time.

40:00 In general, you don't get classical predictions, but it can happen that the sum over the action of the four geometries could be dominated, for example, by one specific four-geometry. That's the sort of, for example, the stationary phase-dot approximation. That's a concept of geometry. Therefore, this big integral over geometry here kind of fights by just evaluating this kind of classical geometry and subsequently the integral over fields. But that's field theory in current space time. So that's the usual familiar story. The theory then is thus approximately equivalent to the field theory that fights for geometry with causal structure, time, and, as a consequence of the structure, time asymmetries. So it seems that we can do it, in the least sketchy way, that the observed time asymmetries of the universe, our planet universe, can be seen as an urgent phenomenon in a classical approximation here of a classical space-time that arise from boundary conditions in a time-neutral theory of dynamics, as well as a completely time-neutral. In case you missed it, that was my conclusion. But you can hardly fail to observe that in this discussion I didn't lead up with the imperial standards of rigor, because I'm here in Imperial College, so I'll just do my real conclusion.

42:30 What do you gain by the coarse-graining? As you pointed out, it's essentially what ABL did, and actually earlier, Bernoull was on the same screen. So what do you gain by the coarse-graining? Coarse-graining is the fact of life. As I look at you, right, and think, first of all, what do I need to describe you? I've also experienced the position of every single molecule in the society that I've ignored. I have a few densities. And then I also have ignored all the other variables of the entire 10th and 80th particles of the universe. So coarse-graining is the way we deal with these. 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 some degrees of freedom into others. So that's expressed precisely by the notion of coarse-graining. So coarse-graining is both what we do and it's essential to the... So you can eliminate the preparation and the registration. Well, I'm talking about cosmology here, so the preparation and the registration would be in the projection area. I don't believe that you have to do cosmology to do quantum gravity, so for me it's not necessary. Well, it would be the same thing, I think, right? I mean, you'd have to have some initial state and then some final state. But I can do all that for his data. I could do it, I wouldn't have stakes. I would just have a preparation and registration and I could use an ensemble in between, a la Brunewald or ABL, if you're questioning me. Well, I think if you're dealing with any realistic system, there's one, and two, what I'm talking about is the aerotons that are exhibited by the universe, not the aerotons that are exhibited by the universe. If one doesn't want to do cosmology, does one need perspective? I think the answer is yes, but let's discuss this afterwards. That's it. I explained it all in your ears and now you're going to have to do it again. That goes without saying. Perfect. Just a clarification. When you talk about history in this context, one of your alphas, alpha 1, you introduce them as sequences, time-ordered sequences of... In your generalized way of thinking, they don't need to be such, is that correct? So they don't need to be sequences of vectors or sequences of... Well, it might have been a little swift in here. I mean, for most of the talk I discussed the quantum mechanics of a system in a box,

45:00 with a well-defined time, in particular, when it's granted to one. I'm going to ask you what happened with it. Actually, go to the general... In the gravitational case, there isn't any topology. So, a given alpha can be something which is not possible to divide into a sequence of properties that are different. No, you have to deal with it in a four-dimensional sense. But in a general quantum mechanical theory, you'll get histories. So classically, maybe the volumes would be ordered in time with the bigger volumes like with some expanding regions. But quantum mechanically, the analog is defining paths where there will be no such ordering. Therefore, the ordering would be an aversion property only in the classical limit. But in general, there would be no order. Something that's always slightly bothered me about these time-symmetric formulations, you have a density matrix at the beginning of time, the end of time, but why shouldn't I have one every moment of time in between, we logically can't see quite reasonably, with the lack of a technical field. You can, it's equally a generalized quantum theory. Yeah, but isn't that a bit too much? Can you repeat the question? Well, Chris noted that I have an initial and final density matrix, and I could have had another Decoherence functional satisfying as axioms if I put a few more density matrices in various other times. 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 density matrices. I'm trying to get the minimum theory that's consistent with the present that exhibits time and time. If you didn't deal with the zeros, because the finite system wouldn't have to be glided, it would just be intervention. Yeah. Do you think it would influence the issue of the result time if space-time was as great as some approaches to quantum gravity?

47:30 Where's Raphael's work? He would say yes. So, some formulations, which, for example, is a natural growth, that also defines an arrow in some sense, but there are lots of discrete formulations, for example, in, I don't think there's any inherent arrow, for example, in dynamical triangulations or regi-calculus, those sort of discrete formulations. When you talked about the course grading appropriated by the entropy, you talked about the hydrodynamic variables, but of course, you mentioned it in connection with Robert's point, which was heavily relying on the entropy of black holes. I agree with you. You have to augment it a little bit. I'm not sure exactly how to do that. But certainly the everyday entropy that you read about in physics and chemistry books is the entropy that I described, because of the high dynamic variables. Otherwise there would be no particular reason for it to be associated with it. For example, the Navier-Stokes equation is an equation of the entropy. That deals with these hydrodynamic variables and dissipations associated with that increase in entropy. Ah, NOAA! You missed the joke, I told you. I missed the joke. Lucky me. It's about you. I heard one of them. Now, I was going to ask if this is aimed at quantum gravity theories, I presume. I mean, it's a quantum mechanics which should cope with quantum gravity. What would your view be on the singularity issue, which has always been one of the big problems in quantum gravity, which after all is related to what you feed in at the beginning and the end. So if you have an asymmetry just sort of plugged in there, do you expect to find that based in some quantum gravity theory at some stage, or how do you view that? Well, I'm not a hard-carrying, dual-boundary person. Singularities at the beginning of these, right? But it's certainly consistent with the general point of view that they're the case, the fluctuations are very small in the beginning, and the matter degrees are going to grow, so we would expect your picture in the classical domain would start from a regular...

50:00 And the initial condition, which is supposed to be singular, can evolve to time asymmetric. Could you just plug that in? I mean, you're not expecting, out of your quantum gravity theory, to come something which tells you what you're supposed to plug in. 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 to deduce those. Well, I mean, 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 conditions on the finite conditions, it's just that the evidence of the observations seems to be infinitesimally against those. Well, I described that for noise and media, you objected that they were infinitesimal. You described the finite condition on the interlimit states with the words totally ignorant. Who is ignorant? Oh, I take it back. Indifference. Now you say, who's indifferent? Totally mixed, I would say. Maximal missing information. Well, like I said. Well, information is just like that. Well, that's math. Let's not go there. Any more questions? This is a comment that you made. It's a pretty good question. I disagree with it. The question for the comment is, This is like a prime minister's question time. I thought that there was a subtle difference in the importance of the way you... In the quantum case, the initial condition is really on the wave function, so it's some asymmetric attribution of amplitudes to the histories, whereas in the gravitational case, where you don't have an external time, the condition was on the histories themselves, and there should be some difference between them.

52:30 What they look at one end, what they're like at one end, and what they're like at the other end. That's correct. That's what he said. You were going to disagree with that or something? No, no, I think it's... I like the observation. Maybe showing one thing is quantum gravity as such, you have the same answer. Thank you.