String theory without space time (contd.) / subsequent discussion

0:00 The metric is somehow physically complex. From theory to analysis, for the fields which you know so much, as we can integrate on a space time like that, can you construct solutions which are based on the fields without using actually the metric? You did it for your ghost variables. You make a similar procedure for... Some fields without theory put in in the next section. Wouldn't that give semantics something like a theory which isn't propagating? Okay, if I understand you, are you asking the question whether I can find solutions which in some sense have a background electromagnetic field but no space-time, no metric? Okay, that's actually exactly the question that Bryce asked me right after my talk. Okay, my answer to him was I think that's a great question to look at and I plan to look at it. I never thought of thinking of it before. Okay, so I don't know whether there are solutions like that, but I don't think it will be that hard to sort of analyze. If there are, what would one say? Well, I think that would be yet another, you know, example in this wide class of classical solutions that one would have to, you know, argue that in the real solution, that quantum solution or whatever, just doesn't exist. Obviously, we need, in sort of the real ground state, a solution which has a space-time metric and a field and all of that. So... I think that's something which should be looked at. May I ask another question? I am returning back to the question of time. Suppose that you talk about a ground state in ordinary finite dimension of quantum mechanics. You mean a state with the lowest energy, which supposes that you have a time independent and a dominant, and the energy is generally a conjugate across the time. Now, suppose you talk about the wave function of the universe. We know that there is no time independent plane, an infinite time independent plane.

2:30 Now, originally that wave function was somehow loosely called Gram State, but I think that it's in fact a wooden story, Gram State, nowadays. No, you can't tell them anything. Unpredictable. But even that concept then is far away from what the Gram State was originally. Now you are coming to a theory which doesn't have any time, doesn't have any space-time, and you are talking about gramscale. What these grand states mean operationally? Is it not a far-fetched extrapolation of the formalism which started firmly grounded in experience and then taking the former structure without actually the other structures being present which gave the grand state its primary meaning and then Well, I completely agree this is a far extrapolation from fields we're used to, theories we're used to. I'm using ground state only because I don't know other words. I don't know the sort of appropriate language to describe the physics in this... I think it would be very desirable, distinguished solution. Distinguished solution, preferred... Actually, it's much more likely to be something like least action, energy. And in any case, what you really mean is that, in general, it's a tremendous problem. The classical solution is proliferating, and we need to know whether there's some way of picking it up. Some people then say, if we look non-perturbatively, and to all orders, most of these may go away. The possibility is that there is some principle of selectiveness that you can call ground state just as a metaphor. Another possibility, less pleasant, is that... The choices for quantum mechanics will come of course with choice, and for this universe, certain things are this way, and for other universes, they would be the other way. That's the least pleasant. They're all being discussed.

5:00 What do you expect the most highly symmetric solutions to be? Not the most highly symmetric, I don't know. Broken down dimensions. Is that satisfying conditions only in four dimensions? No. In fact, I wrote down some cues which work in any dimension. You want a cue which would work only in four dimensions. I don't know of any. I don't sort of know a reason why there don't exist such cues. So sure, one can hope. So that would be sort of a, would that have a physical interpretation? Right, so the question would be something like what's the spectrum of space if you then linearize, perturb about that solution, look at linearized fields and see what kind of space you get. I don't think we would sort of figure that out. But you could better the solution, right? Yeah. So there's no background space in there, there's just a string of fields. That's right, that would be one without. What is the demand for the master's degree? Would you still have a third degree? I would guess you would. The discussion's got extremely good. Possibly we should wind up this particular topic fairly soon. There's just one more. I'd like to come back to a more technical issue about what the taxonomy really means. Because you have this stupid taxonomy with no kinetics in it. Now, usually when I write down a path integral, I think of it as a transition amplitude between some initial configuration and some final configuration.

7:30 Now, since you don't have your integral that doesn't have that structure, you're clearly just summing configurations, right? You're just adding up all these different phi's. But a phi is a configuration strong phrase, at an instant, okay? And so... I mean, it's just a configuration, right? It's just one string. No, no, no. Field five would correspond to an ordinary field of space-time. No, it's space-time. It's definitely space-time. Five is the field over the whole evolution of space-time. Right. At an instant. Okay. I mean, it's just a configuration, right? It's just one string. Yeah, phi, no. Field phi would correspond to an ordinary field on space-time. Yeah, so on space-time. It's definitely space-time. Phi is the field over the whole evolution of space-time. Right. So I would integrate over all phi. It would be like doing the covariant function. I would be integrating over all phi subject to feasible asymptotic conditions such as phi goes to zero at infinity. Okay, that's good. See, what I'm trying to get to is an understanding of, if I'm going to try to treat this integral outside of the classical limit, so I'm going to do some semi-classical evaluation, you showed me that I could find certain q's which were a function of the solution to the equations of motion, and I could perturb around them, and they gave me this thing. Now, when I do a path and I go, I have this, I'm generally going to want to sum over all stationary functions of the axis. Presumably I'm getting some interference between the different backgrounds. I'm going to accept what you said, that this thing is mathematical independence. So I get a sum over different backgrounds that are contributing in this task integral. I'm just curious, does it induce what you suggested, that we get a superposition of possible universes? I mean, doesn't that pose a problem by your statement that it's background independence? You're asking sort of in the semi-classical approximation. Right. I want to evaluate the capital in some sense. It's not quite graphic, but I'm going to do some of it. So I'm going to do backgrounds.

10:00 Did I just sort of confuse you? Well, I would just maybe be a little bit more optimistic. I mean, you're saying... It was a very open-ended question. I apologize for the last question. I think perhaps we should stop at this point. Let me thank Gary again for just giving me the opportunity. Now, the cut-off time is half past twelve. We have three short contributions of ten minutes each. So, and he said we'd have discussion on various things including time before that. So I think perhaps we could just have half an hour at most, or three, four, and then I'll take the three ten-minute formal class to fill out the time nicely. So, who would like to... Thank you very much for your time, and I hope to see you again in the next lecture. I don't know if they would agree with that, but maybe we can have the question.

12:30 I suppose what I was saying yesterday is that one might try to look for a framework in which the asymmetry that is in general, that the asymmetry would emerge as a consequence of particular initial conditions occurred times in the universe. So that there would be one possibility. For example, if you have a simple physical system, which actually have a clock, which have a preferred direction, then you do single out one sort of direction. And the arrow of time in quantum mechanics is encoded in those initial conditions. Now those are not the kind which is no-boundary proposal. Rather there we see that rather something more like a superposition of things which are symmetric in time. In that case we have to understand the semi-classical regime and how such different classical solutions equal here. Now the prediction of the no-boundary proposal is roughly as follows. Don and Steven and others have used it semi-classically. It corresponds to an ensemble of classical trajectories with one end is simple, the other end is easy.

15:00 But if you like, there's just as many ones which run... the ensemble is symmetric, right? I think roughly, because it's kind of... So I would find that in some sense we're implementing Roger's proposal. Does anybody know the answer here to the question of variations? It is conjectured in what Jim said that maybe the Hartle-Hawking proposal has a mixture and therefore has both boundaries. But there's another boundary condition, for example, that offered by Vignenchi and anticipated perhaps by Salabrini. Does that one have to see in the IMF? I think that was the main motivation. No, he said he didn't know. Well, the claim is that it's true in an outgoing way. Yeah, but that's enough. The question is whether that means the same thing. Well, I asked him, and he said he hadn't thought of it. Maybe subsequently he had. I think I would guess so, actually, so that that would be an illustration of the difference, a gradually emerging... Anyway, what I was trying to do yesterday It can be a separate area, but I would like to emphasize that I think it would be different.

17:30 I'm not quite sure what you mean by the mind science. Can you give an example of what you mean by the mind science? It isn't going to the canon. That's just the point. It's taken literally and exactly. It's a generalization of some ill-understood kind where the composition law doesn't hold. And yet that may be what Heinrich Hawking said. Eh, we ought to understand it better. B, we ought to examine other suggestions, like that of the Yankees and these other people. C, if they have, if it's truly easy to react. I mean, you do the path integral, and you put it for minus I-s, and then you add the root of the I-s, and then you also add the I-s, and then you put the root of the minus I-s, and you sum up the whole problem. Well, that's a conjecture. I don't really know. I don't think anybody's studied it. How does anybody's I-study it? I don't think that's what they do. There's no sense in getting these things through all these different forms of science. That's correct, but there may be some connection. I was just asking. I don't actually know. I don't reply myself because Mary's asked me this question a lot of times. I've given various incoherent answers. If you decide to say something and you just want to say a comment, then you go to a class. I think, let's see, I'd forgotten what my response was. Let me make some general response. Well, take that up and exhibit these various possibilities for Raphael and also the various possibilities and causal anomalies coming about from not having, you know, conserved or conserved time.

20:00 Right. I think if these ideas fail, right, that's the most likely way to fail you in making some statement. However... These causal anomalies are, the universe is extraordinarily classical as far as its space time is concerned, to sort of change the plot given to us by space time, non-classical behavior, we have to get this whole system, this whole system, to learn backwards, to be able to hope that the violations of such anomalies will be extraordinarily small. However, a report can be produced which is, which is live, contrary to experience. Now, Lathiel's anomaly is of a different sort. It's not an anomaly, so to speak, with the formalism, but with the definition of observable, because, recall, one has to extend the definition of preferred time just in the choice of observables, even if the framework of probability seems to matter. That is, you could have a space-time probability, but then if you went back and said, okay, but it only works for observables which are confined to one hyperspace, it would be reintroduced into time. He exhibited these more or less, but he would have exhibited these more or less if he had enough time. Maybe you should do that this time. There's a big untouched question in this whole area, which is called measurability. That is, it's just easy to go right to quantum mechanics to say that every Hermitian operator is an observable, like x cubed p p x cubed. Now, as Mary was describing to you on Monday, the class of things we can observe is, of course, much smaller than the truth back in the world, and why he's proposing something and why that's true.

22:30 They're bounded in particular by, I think, the second law, otherwise we'd get that. What has to be investigated there is whether what you've heard naturally in the theory is limited, sort of statistically, like the ability to actually detract mathematics. I think that's essential, right? It's no good standing up and saying, well, this is an observable, right, if you can't describe it as an observable. But that's, I would argue that one is not in a much different situation than the linear quantum mechanics. Observable, but... Can I respond to that? I think that this is an often missed point, that with the difference between doing quantum mechanics when I can bring in an external... We all know that if I have an apparatus and I want to actually calculate the point of evolution of the system in the apparatus, I have to evolve them both in the system. And we have a presumption based on what we were taught in class and what Glenn Norman felt was that it's meaningful to calculate expectation values of observable rules in the absence of having a measuring apparatus that's ever even discussed. And the hope is that those expectation values will be extracted by our measurement apparatus when we actually couple the pieces together. Now we know also that if the evolution is such that there's more than one classical path, or that there's interference, or that there's more than one outcome in the sense that we have a superposition, that our expectation values are getting at averages over possible outcomes and not specific coupled outcomes. The average correlation between subsystems would say that if my measuring detector sees this, then I see this other thing, but my expectation value wouldn't suggest that I see that at all.

25:00 So if you're really careful, you have to evolve the joint system, which is what you do in the quantum cosmology, because you're part of the whole thing. So you have to be careful when you say that ordinary quantum mechanics doesn't really have these problems, because we wanted to do something different. Now, just to respond to that, I think that the attitude of those of us who are skeptical about these things in quantum cosmology is that in ordinary quantum mechanics, it is exactly the presence of an inner product and Hermitian Hamiltonian, which guarantees for us that the probabilities of probability expectations values are meaningful, and that if you have a mutually exclusive set of possibilities, the sums of the probabilities of the theory predicts that the different things add up to one. I think if you try to do, to get predictions out of quantum mechanics which are probabilistic without an inner product, there are many suggestions, and for example there's a suggestion of extending the definition of the relative frequency operator that Damien, David, Graham and other people... However, if one looks into their discussions, then one can see clearly that in the context of a situation where there is an inner product, the notion of the relative frequency operator depends on that inner product. It depends not only on the state that you're discussing the frequency to see, but on what's not that state, which demands a notion of orthogonality. All of these are examples of showing that you can have a sensible notion of probability interpretation without a notion of an inner product, the unitary evolution. That burden is on the people who told me to do that, and it may be that one can do that, but I think it remains to be seen. When there is no time, there will be no cause or effect. But there still should be a notion of probability. The interesting question I think is not that. It's whether, when we go to the limit where these notions are approximately defined, do we enlarge or not enlarge? And that's what Ten was suggesting. I would like to return to the question of what is an observable, either classical run or quantum run in parametrized theory.

27:30 And they would like to strongly object to the standpoint which says that they are invariably the quantity which commutes with all the constraints. Now there is a basic difference... I can sit down at this point. There is a basic difference between H-tier... Could you write on the blackboard three or four quantities that commute with all the constraints? Which commute with... With all the constraints. The operator 1, the operator 2, the operator 3. Sorry, I wasn't very careful at the beginning. I said in parametrized theory I easily write the quantities in simple parametrized theories because they're properties. You would probably not give me any quantities which commute say with a constraint in any parametrized theory which is sufficiently complicated like a nonlinear sigma model in a completely curved diagram, so let me go to my original point to make it. If I have a constraint which is a gauge constraint and which is typically linear in the momentum, that constraint generates on the constraint surface the trajectory, the orbit, in which the points are merely different descriptions of the same physical state. If I take two points on such an orbit, I cannot physically distinguish between these two points. And then the standpoint that observables, those quantities which commute with against them, or classically have vanishing points on them, is totally justified. On the other hand, if I had a parametrized theory, Then the commutation of a quantity with a constraint means that the quantity is a constant of motion and in fact the trajectory which is generated by the constraint is a dynamical trajectory and I would say that the state

30:00 Say of the people in this room, now and five minutes from now is not to be identified. These are not merely two different descriptions of the same state. They are physically distinguishable situations. Now, I am expounding a standpoint. You can then answer to what I am trying to say. Now, there is a scheme which Carlo Rovelli proposes, in which to every quantity at a given time one can construct an observable, the observable being the quantity which starts with the quantity of the initial value and in fact reconstructs the initial value from the data given at another instant of time. And I agree that this is possible. However, the problem is, if we have, and this then comes in touch with a matrix in half a mate, we know how to measure certain operators, but we do not know how to measure highly complicated operators constructed out of the fields and the geese. I now say how to take few at a time key and add that time to measure that quantity by the apparatuses which are at my disposal. But if I wait another five minutes, take Karos observable. That observable is a highly complicated function of the present coordinates and the momenta. I mean present after five minutes. And I do not have any apparatus in Panama which would measure that quantity and make the inference about what he was at the time. Now in some special cases this is possible. In the quantum non-demolition experiments this is what people actually propose to do, but for systems with very simple dynamics like harmonic oscillators with linear dynamics. But if I have nonlinear dynamics, I think that this proposal is highly unrealistic, and of course there is also the question, if I have nonlinear dynamics, how do I take such an observable and factor order it into a quantum mechanical operator? It's a highly complicated function of use and peace.

32:30 And in fact the factor ordering is usually handled by factor ordering the Hamiltonian and then the Hamiltonian does the job for us in quantum mechanics. So somehow that proposal of eliminating dynamics at the classical level and then quantizing... Just stumbles upon the problem of sector ordering of highly complicated expressions and there is no guiding principle like taking a sub-algebra of those variables and taking it into quantum theory without any change, like the dear way of taking the algebra of piece and piece and mapping it isomorphically. I think this is not a question, this is a sequence of questions. So let me take one at a time and try to give an answer, at least what I think. The most crucial one, the central one, I think, and also the one you started from, is what is observable in a parameterized system. So let me try to start from the complicated example instead than from the simple example. So from general relativity instead than from a simple model of general relativity. And I think that general relativity, pure general relativity, from the observational point of view is an extremely complicated system. From the point of view of an experimentalist is an extremely complicated system. There is a very easy way to change it by transforming it in a system that we can study much more easily by adding just a little bit of matter to it and studying the coupled system of general activity plus some matter.

35:00 This system is still for diffeomorphic variance, so for what concerns this discussion it is absolutely equivalent. So I went there to those people. They asked, but what do you measure? What are you measuring? And I said, suppose that I... And the answer after a long discussion was that first they would change the system and add something to the system. And they would launch around the sun. And this rocket is dynamically coupled to generativity, in the sense that its trajectory is not determined by putting it there as we do it, but it follows the trajectory in the space ship. On this rocket there should be a clock that is also coupled with time inside the gravitational field. And at this point one can start to make experiments in this coupled generativity class and ask, for instance, what is the scalar curvature in the rocket. When the clock has a certain value. And one can do experiments there and measure. Now, this quantity, which is, so, generativity, the point of an object is, so I'm measuring things like that, which are exactly, I'm absolutely sure if I design, if they design an apparatus like that, that whatever they are measuring will be something that produces a score.

37:30 This is what I know for sure. Next point, what I was stressing, that I don't think that if in classical theory we measure only things that commute with a constraint, I don't think that we can measure in quantum theory things that do not commute with a constraint. This is the content of what I'm saying. So this is, what I'm saying is simply I want to, that what is true in classical theory is also true in mathematics. I can translate these into the simple, and the simple explanation is this Hermitage series, and I can design an apparatus that may have something like that, which is down-dependent, and this apparatus is weighted up to a certain value of x naught, and when x naught is equal to p, measure x1, and I can call this x1 at the value p, and now I know what is this expression of a phase phase. I agree with you, this is a complicated expression. Now, the second question, your second question was, well, this, I don't know how to order this. Now, my answer is the same. I don't know, in general, how to order this operator, I don't know how to write this operator. I know that in simple case, I don't have to quantize x1, x2, x3, and then order it, because I have alternative ways. I can directly look...

40:00 The main argument is not that there is a unique way to achieve, but the main argument is that I just had a few remarks that are somewhat related to this this this same sort of picture about time and I'll talk about basically timers as conditional The dependence of conditional expectation values and so on. Okay, I'm going to assume that we can formulate things in terms of density matrix. Now, Jim Harlow has sort of posed a big challenge to this approach to say maybe you should form it in terms of path integrals. And admittedly, just hearing the beautiful superstring talk this morning led me to say I don't know how one would do this or what one would put in as arguments of the density matrix. But I mean, I'm supposing here that the density matrix should be written in terms of the full string fields or whatever it is. And then I'm saying, pardon? So I'm saying that in ordinary quantum mechanics you take the expectation value of an observable by multiplying it by the density matrix and taking the trace, or if the observable is a projection operator then you get the probability, but now if we're within the universe it makes no sense to talk about absolute probabilities. There's no way to test absolute probabilities. You could say what's the probability that this conference exists?

42:30 All we can do is calculate conditional probabilities, because an observer within units requires some condition, and so we can calculate conditional expectation values by projecting the density matrix onto the condition, and then putting in the observable, and then dividing out by this normalization factor. Or you can calculate conditional probabilities if that's that. I mean, then you project under the condition and project under your result, and you get that. Now you notice, of course, that this is the same as applying this onto this condensed density matrix, where the conditional density matrix is this. That's what people normally talk about the collapsed wave function. Here, of course, I'm not saying that there's anything ontological going on. There's no physical thing that collapses that. It's just that for each question you ask, you could answer, it may be convenient, more convenient to calculate this rather than this at each time. This is the same thing that Gelman was saying in his first talk, except I replaced the condition by one single condition, and I have in mind that I'm going to test things by doing things on the present. For example, take the density matrix, and I want to look at the observables that represent me, so it's in some local region. I don't need to have projection operators in all possible paths. All I need is to apply projection operators onto whatever system that has the records of what it thinks is in the past. Okay, now from that, let me just jump to, in the Wheeler-DeWitt equation, if one instead of a density matrix had a pure state, for example, then all right, you get the same sort of thing. You could do it, you could apply it to wave functions. And I'm proposing what seems to be rather unpopular view here of using mod psi squared for the measure in super space. And this does turn out to... I mean it's something at least you could say is positive. It's not going to be normalizable over the whole super space, but if you calculate conditional probabilities, it can still make sense because even though the full density matrix is not normalizable, when you put sufficient conditions on it, then you can get a normalizable thing. And then it turns out in the classical limit, when you have semi-classical wave functions and sum of WKB parts with S being the Hamilton-Jacobi equation, then if this term's a negligible, of course then that solves the Thank you for watching.

45:00 There's sort of a pencil of these trajectories that correspond to a wave packet or a tube that's moving through the super space. Now if we postulate that the fundamental probability is calculated by taking mod psi squared, then in this WKB approximation one can reduce this down to find that it corresponds to the flux across this region multiplied by the proper time that the system spins here. And I do have to use, if I do the canonical treatment, I have to make the lapse function. All of these are independent of the spatial metric. So the lapse function can be a function of coordinates, but not that and so on. So you can get out, what I'm saying is you can get out a classical interpretation even from using mod psi squared. And in fact, one other thing I think, which is, well, a point that Bob Wald raised yesterday, and actually I had this transparent, I used this in Moscow last spring, so I didn't write it after Wattletop, and it's exactly the same point that Carol is... We want the full state to obey the constraint equation, but what we're going to start tackling about these conditional density matrix, it may not obey the constraint equation. I think that's, well, it's almost what you were saying. You were talking about observables not competing with the constraint. Here I'm saying it's convenient when you're calculating conditional probabilities, it may be convenient as an intermediate step to calculate this thing, but it's not to be interpreted as physical. It doesn't obey the constraint equations. Well, for example, if this was... This isn't, by the way, what I will say. Yeah, okay, okay. Maybe I should talk later about what exactly... Alright, but, yeah, you're not using the probabilities in quite the same way that I am. But suppose I did do this... This thing, it's just an artificial thing that I'm calculating, but I'm saying this artificial thing I'm calculating in general would not abate a constraint equation because, for example, if we projected it onto some conditions such as our existence and the existence of this conference, well, this conference doesn't last for all time, whereas the full solution of the Wheeler-DeWitt equation is going to, in a... In a certain sense, last for, well, all time where the semi-classical approximation holds. So therefore, if we project on any finite range of some variable that really should range over a very large range, then we're going to violate the constraint equation.

47:30 So in a technical sense, you could say that our existence is unphysical. By that I don't mean it's unreal, it's just I'm really criticizing the use of physical in the sense when people talk about physical density matrices or physical states of being this constraint or a physical thing that you put the BRST operator there or whatever. No, all I mean is when people talk about physical states, when they say that something has to be annihilated by the constraints, by the Hamilton-DeWitt operator or the BRST charge or whatever one uses, then I'm just saying that the conditional density matrix, it's an intermediate stage in calculations, may not obey that. And what I'm saying is that you might say that we should use only these things, and some people have argued to me that one should only stick to these, but then it seems that you can't write down all conditional probabilities, such as, you know, does our existence, you know, what are the values of things given our existence, and so on. Let's see, I don't know whether I should, maybe I don't have time for this one, but... This was a slight alternative to Jim Hartle's thing with clocks, which you didn't present here, but I'm just saying in non-relativistic quantum mechanics, you also, in practice, actually use, you can use density matrices that don't depend on time, and I like to view it in non-relativistic quantum mechanics, saying that time, the t, represents an inaccessible observable, for example, an ideal clock off an infinity, or an ideal clock here, but that I don't have, I have no access to, and then you can say because it's an inaccessible observable, To calculate the probabilities for things that we can observe, then we should sum over all possibilities of that, so what I should say is we should trace over T, I think of T as some argument in the density matrix, I trace over it and then I normalize it by tracing over both T and X, and that basically just gives me the time average of this, which of course you'll recognize as, just gives me the time independent row, or in other words, in terms of energy eigenstates, the density matrix now will have no off-diagonal terms connecting states of different energy. And this is, of course, exactly what you'd expect from the energy super selection rule, that you should not be able to, there should be no observable effects from states of different energy in the density matrix. So even in non-relativistic quantum mechanics, one should stick to a, to a, to this sort of thing, and then one can, one can indeed get time out in terms of evolution, correlations between one part and another. If you have a system of a clock plus the rest, and you define clock states, then you can ask, given that the clock reads a certain value, what's the other thing doing?

50:00 Bill Wooters and I had examples of precessing particles, and then you can get a stationary state and find a correlation between one particle and the next. The correlations, you take the conditional expectation values of observable given that a clock reads a certain value and then you ask how do those expectation values depend upon the value of the clock and the idea is that that dependence that represents what we think of as a time evolution even though the state has no dependence on the Schrodinger T or we just don't have that in our formalism. And if you have a very idealized case where the clock and the other thing are uncoupled then Then you'd find that it obeys the Schrodinger equation or the von Neumann equation. In a more general case, it won't obey that, but probabilities will still add up to one, and it remains a calculable problem to what extent the approximate Schrodinger equation will be sufficiently different to violate observation. And I think unitarity has to become a physical, has to become an experimental problem, and one should test to what extent is it experimental. You can always reformulate it this way to make probabilities add up to one, Do the expectation values follow a simple enough equation? Yeah, this may work for a system with a finite number of degrees of freedom, but... I'd like to remind people that we're in an infinite dimensional system, and even at the linearized level, the Hamiltonian constraint freezes out unphysical modes for example the longitudinal modes, and if one has an inner product on states which are not solutions to the constraints, then at the linearized level one is essentially putting probabilities on unphysical longitudinal modes, and I'd be very skeptical about whether... Any inner product on the unphysical state space in the full quantum field theory could be worked out even with your unphysical projection operators in a way that didn't have lots of probability leaking into the unphysical modes. I think it's something to be shown and given the nature of field theory seems very unlikely.

52:30 It's not so clear to me that the answer is small. It's not clear to me that the answer is small. Because, for example, the universe consists of lots of different bubbles. Many worlds are sort of realized concretely, based by separations. So I think it's a meaningful probability. Well, I've got losses to which probability you're calculating. The probability is that there's another constant. Given the initial conditions, there's another constant. Oh, oh, oh, oh. I just don't know. The second remark is, it's of course clear in quantum mechanics that there is a well-defined description for dealing with clocks that divide a series. The big difference, though, with what I'm proposing is that in essence, the corresponding variable is the amplitude. Yeah, so I think this is all fair for non-algebra physics. Yours works much better, of course, when you get to quantum gravity. One more very, very quick question. Who hasn't? Anyone who hasn't spoken yet? One of these things that Don and Bob Wald and I were doing last night was trying to understand what would happen in one of these cases where we're trying to put the system... The best analogy is to cook them, because what you're really doing is turning them into a thermal equilibrium state. So if you take your observer or send him a friend or something, put him into a big box and try and put him in his energy eigenstate, it's really equivalent to putting him in thermal equilibrium, i.e. cooking the guy, at which point asking him any questions is going to be somewhat difficult, naturally. So I think that the quantum mechanics of these systems where you try and put things into energy eigenstates... The predictions are going to be extremely much different from what we might expect them to be, and it's not clear, but certainly in conventional quantum mechanics I don't think it works.

55:00 Perhaps in quantum gravity one might be able to make it work because of the structure of the constraints and so forth. But the fact that in classical gravity you get nonsensical results makes one worried. But I think in that sense, if you ask the conditional probability that the guy is there, when you look at it later, most of the probability he's not there, but if he is there and can answer the question, then you could ask that, and you might have a history of remembering what he saw after putting in the box. How do we know the universe isn't there? Yeah, that's what I was basically arguing. I mean, it was this thing with Reuters came out that... We were asking, you know, how can it be consistent with the energy super selection rule, and, you know, if it isn't in a big box and asymptotically flat, then all our observations should be consistent with assuming that the density matrix is diagonal in the energy representation, or even that it's just one term in that, and yet everything is reproduced. About time, I mentioned to you last night, I don't want to mention it more. Have you included it in your list? If I haven't, I don't know if it's okay. It's okay, I don't know if it's okay. Then John, and then Ted Dixon. I hope it's not going to be long. Well, the reason is basically because fortunately Gary did most of my job. Certainly in conventional quantum mechanics, I don't think it works. Perhaps in... Quantum gravity one might be able to make it work because of the structure of the constraints and so forth. But the fact that in classical gravity you get nonsensical results makes one worried. But I think in that sense if you ask the conditional probability that the guy is there, when you look at it later, most of the probability he's not there, but if he is there and can answer the question, then you could ask that and he might have a history of remembering what he saw after putting in blocks. How do we know the universe isn't in there? Yeah, that's what I was basically arguing. I mean, it was this thing with Reuters came out that we were asking, you know, how can it be consistent with the energy super selection rule and, you know, if it isn't in a big box and asymptotically flat, then all our observations should be consistent with assuming that the density matrix is diagonal in the energy representation. Or even that it's just one term and yet everything is completely reproduced.

57:30 I've said this already. I hope it's not going to be long. Well, the reason is basically because, fortunately, Gary did most of my job. Ironically, I was in my sleep. Some of the points sort of came up in informal conversations during these few days, talking to Chris and talking to Carol and Gemara. So this canonical quantization idea and why one is doing it. So I have to backtrack a little bit and sort of remind ourselves that in fact we have been reasonably successful in thinking relativistically in the sense that for example when it comes to black holes and so on we really have a space-time point of view or even special relativistically. Many of our S2 physicists' friends, for example, don't do that. I mean, when they sort of want to know about a black hole, they sort of really pretend space-time is really in the zone and there really is this thing, and all that black hole does is really give you some boundary conditions. Not all of them, but some. They just give us boundary conditions, that's all. So in other words, they still think in terms of the older concepts. I would like to sort of say that that's exactly what we are doing in quantum gravity, particularly. Also in quantum mechanics, but much more in quantum gravity. In other words, we're not really being... We have made almost no progress at all in extracting ourselves from some space-time which is looming in the background. Most of our questions, most of our concerns are formulated in that way. Gary already pointed this out. For example, many of the things that Roger has said, including the time reversal invariance or things about how gravitational field itself may reduce the wave function and so on.

1:00:00 Meaningful you want to formulate if one has a background in space-time. Similarly, Rafael yesterday reminded us of some questions that one would like to ask, and these questions actually had the nature of what is the horizon size of a given black hole? Again, one has the idea that in some sense there is a bias. Space-time is well defined when we are asking such a question. What I would like to suggest is that, in fact, One would gain, I mean, one needs enormous amount of insight about, so to say, the other half of the problem that Gary, again, reminded us this morning about, namely, well, what are the new concepts that we need and what are the issues that arise, really, when we try to get rid of this space-time concept as much as possible. Unfortunately, when one tries to do this, of course, I mean, one has no handlers, I mean, it's difficult to forget everything, and one does have to depend on some formalism. I would like to suggest that one has to hang on to formalism, I mean, has to, in sort of a time-established good custom, to hang on to formalism without, with some degree of blindness, without understanding its It's meaning at each time, each step in the argument, if at all you want to make a conceptual jump. Again, I can remind you about what happened in the beginning of quantum mechanics. We can talk about points or brackets going to commutators and supposing when we didn't really quite understand it, I mean quantum mechanics, everyone could say well these points or brackets is all a formalism, you're hugging on to that, you're saying that it has to go to commutators, because this has such a different conceptual structure in classical mechanics of course, And what I would like to suggest is that we should use mathematics as a framework to guess where we want to go.