Time in quantum gravity / Time in quantum gravity (contd.)

0:00 The role that time plays in Schrodinger quantum mechanics, it's a two separate role, at least this is the kind of view that we're taking. The parameter t that one sees in Schrodinger's equation is this type of order of time that I've been talking about, but the t itself, in the way it's usually used, also contains dynamical information. I don't know what t is except by looking at clocks typically. So, I'm going to restate, this is absolutely standard Schrodinger quantum mechanics, but I'm going to restate Schrodinger quantum mechanics by explicitly introducing a totally non-dynamical time that purely keeps order of events but doesn't have any numerical value, doesn't have any significance. Thank you. I kind of, without looking at the rest of the world, decide when I'm going to make a measurement. I then may look at a clock and I may look at the particles or anything else, but I decide upon the value of tau on my own without observing systems, and that's the sort of role that this plays, and then I'll look at clocks to know values of time. So the restatement of Schrodinger quantum mechanics... We have this tau in the framework of the theory to begin with. The Schrodinger equation, then, that applies for wave function and Schrodinger quantum mechanics is that there exists some function n of tau such that this psi, which depends on this tau and depends on dynamical variables, satisfies what would look... Likely, which if I calculated correctly would be the ordinary Schrodinger equation if I replaced the n of tau d tau with a t. The interpretation of this psi is simple, but at a fixed value of this non-dynamical ordering time, it gives amplitudes for observing these dynamical variables. And I put in two dynamical variables here. The one labeled capital T I have in mind in a moment using as a plot, but it could be any dynamical variable. And in fact, in following what I'm going to do in the next couple of new graphs, this tau is always the ordering time.

2:30 This equation doesn't tell us very much about what things are going to be like at tau equals one later, because we don't know what n of tau is. This is not known as specified in advance. The way we go about making predictions, typically, is to use clocks. Speakers would be suitable for using as a clock if over some interval of talk time, and again I mean typically you will not get clocks that work well over infinite ranges of time, formulating that idea without some notion of non-dynamical ordering time I would pose as a challenge to those of you who feel we don't need this sort of thing and just as supposed to rely on values of the dynamical system. Such an object would play a good role of a plot if it's Hamiltonian and the state vector of the system is such that the time variable decouples and the So with the wave function, I'll have to rush through this more quickly than I had intended to do, at least partly toward the end, but the wave function ends up being of a product form where the product factor involving t correlates very sharply with tau, and there's really no loss in generality in assuming that the sharp The values of t and the given value of tau occur at t equals the value of Schrodinger time at the interval of m v tau. The chi and the phi will satisfy then effectively separate Schrodinger equations. Well, when we have this situation of having a good clock, we can then pass to a wave function that depends only on dynamical variables. And to very quickly summarize what I've said here, what I did take is the wave function that depended on the other dynamical variables apart from t, that was a function of tau, as well, I substitute for tau using the shock peaking in t to get a wave function that just depends on the dynamical variables.

5:00 To the extent that t was a good clock then, this capital psi will satisfy the Schrodinger equation. I've gotten rid of the n of tau in it. And to the extent that t was a good clock variable, at fixed values of the clock variable, it will have an appropriate interpretation in terms of giving the values, amplitudes for values in x. The key point that I want to emphasize is that this object, this capital Psi, is in this case well defined and interpretable. Only to the extent that capital T is a good plot. But even if we don't have good plots, this psi of tau t and x is always well-defined and always has an exact interpretation. Well, when we had done this restatement of Schrodinger quantum mechanics, we were somewhat excited for about, I'd say, maybe a day or two, in terms of having a proposal, a possible proposal, an ad hoc proposal, admittedly. For a totally new viewpoint and equations really for that matter for wave function of the universe. Because in what we did in Schrodinger quantum mechanics, we started with an object that at least superficially looked like the kind of thing that you get in a parametrized theory, satisfying this sort of Schrodinger equation with an n of top in here. Though we Have no constraint equation imposed as opposed to what, I mean, superficially is a key point of view, and I guess I'd have to hurry up, so I can't try to convince you, lead you down the wrong path too successfully to try to unset you right, so let me try to be more straightforward. This object, at least superficially, one might have thought, looks like The object that we started with looked like the wave function in a parametrized theory, but we somehow ended up with an object that had the type of behavior of a wave function in the universe, and it satisfied the Schrodinger equation, which is the constrained equation of the parametrized theory if we had really started with the parametrized theory, and it's certainly analogous to the William DeWitt equation.

7:30 That was that in quantum gravity we're just going to assert the existence of some We can then pass from this to a wave function of the universe and the hope would be that wave function of the universe would satisfy the Newton-DeWitt equation. And would nevertheless reduce to, well, by satisfying the real or the real equation in the classical limit, it would reduce to something that would be a real general relativity. The problem, though, is that this simply doesn't seem to work. What we end up with is a Schrodinger-type equation. We end up with a d by dt, not the true Hamiltonian associated with this dynamical variable that plays the role of time, and there's no reason why they should be close to each other, so we don't appropriately get the Hamiltonian constraint, and therefore presumably do not get any relativistic-type dynamics or the dynamics that we were seeking to try to get. If I had time, I would have liked to make the side remark that the standard approach, it's not clear to us that the situation is any better. I don't know whether I should call it the standard approach, but... Roughly speaking, to say this in one sentence, in an Everett-type interpretation where you have imposed the Wheeler-DeWitt equation, it isn't obvious to us, in fact I'm asserting that it's not true, that observers in this universe perceive that the Wheeler-DeWitt equation is satisfying, granted that the whole object, including the whole superposition of states of the observer himself, satisfies the Wheeler-DeWitt equation. Observers have the annoying feature of perceiving themselves to just be in one eigenstate in this superposition of eigenstates, and it's far from obvious that that component, so to speak, of the wave function, which is all with the time remark, that in the standard approach, it's not clear to us that the situation is any better. I don't really know what you call the standard approach, but...

10:00 Roughly speaking, to say this in one sentence, in an Everett-type interpretation where you have imposed the Wheeler-DeWitt equation, it isn't obvious to us, in fact I'm asserting that it's not true, that observers in this universe perceive that the Wheeler-DeWitt equation is satisfied, granted that the whole object, including the whole superposition of states of the observer himself, There are a number of ways in which the Wheel of DeWitt equation satisfies the Wheel of DeWitt equation. Observers have the annoying feature of perceiving themselves to just be in one eigenstate in this superposition of eigenstates, and it's far from obvious that that component, so to speak, of the wave function, which is all that the observer really perceives, comes close to satisfying the Wheel of DeWitt equation, or comes any closer than R. So, the outlook, it is our feeling, is we really need some sort of time for an interpretation, I mean, I would have said a few more words about that if there were more time, we can attempt to use a dynamical variable, but I don't see anyone In sight, I argue we can't have one in a precise manner in ordinary Schrodinger quantum mechanics, and this is something people have been looking for for well over 20 years, we could attempt to introduce by hand, that's what Goh and I did, some sort of non-dynamical order in time creates a new theory that has that built in. Our proposal, I think, you know, really I could have made a case a little closer than it might have been. And I think the final option is to really do something that I consider to be more radical than anything that I've been talking about. What is possible to write a theory on mathematics? If you have a clock, an actual clock, in there, you can ask for the quantity of time you can do it.

12:30 I can't believe what I'm hearing here, because you're not satisfied with what the Bureau of Standards does, or anything in the future, no matter how accurate. Our time that we measure now is done by a physical system, not by some arbitrary thing. But you want something more. I can understand why you want something more. The best that can be done now, plus extrapolations from improved apparatus. I don't even know how to, I mean, so you want to say that our entire notion of time is this national view of science. It's, first of all, only for a limited value of time that that dial will typically, and this is an awfully hard thing, I think, to discuss because there's so much, you know, ingrained assumptions that people have. It has a lot to do with very accurate measurements, to understand time mechanics. I sure don't know how to measure time any better than that clock, and I want to use it, and that's why my capital T is there. I mean, if you want something better than the best. We should give the chance to people who didn't have a chance to speak in this session. Yes?

15:00 Now, I'm hoping you don't say that people talk about mathematics only with margin of error. You would have a problem with the order of things. There's a difference, but a... I don't see why this is a problem. Why would you not do this? Why would you not do this? Why would you not do this? Why would you not do this? Why would you not do this? I think because there we believe that the guy inside, if he stuck his ribs inside the box as well, he could make a measurement and yell out to his friend, and after that measurement would be made, we would know that the system is no longer an energy eigenstate. That's perfectly valid for me. Of course, but once he has made this measurement... Once he has made his measurement as Wigner's friend, or as Sidney's friend, then at that point he would know that, and you would know because you told me what his answer was, that the whole system is no longer an NPI system. There would be no problem with that. You don't care.

17:30 I didn't see you. Well, you will be talking about the topic that you only know about because you said you will not be talking about what you said you will be talking about. Why don't you announce the title? Well, you said I would be talking about it. All right. I didn't particularly prepare the title, so let me address it. I'm glad to talk. Can everyone see this blackboard as I use it? One thing that it wouldn't have been natural to talk about, and it's the thing that's in the Xerox illegible paper that we sent to some of you, was a proposal that I alluded to briefly this morning, that in natural time I'll just ask Bob and Bill. I hope that the sorts that they're seeking can be given by the space-time values to the past and the hyperspace, and that's partly in my mind as I give this talk, but it's not, I mean, since that's special to gravity, it requires much more preparation. It seems to me that there's not enough time to do it in such a short time, but in any case, it presupposes, the way I would like to interpret it, it presupposes a version of the sum over history interpretation, much as... There are a number of different types of algebra that can be used in mathematics and physics. There are a number of different types of algebra that can be used in mathematics and physics. There are a number of different types of algebra that can be used in mathematics and physics. There are a number of different types of algebra that can be used in mathematics and physics. There are a number of different types of algebra that can be used in mathematics and physics. There's a lot of flexibility to generalize these questions. We can run into trouble. The kind of trouble that we can run into is a violation of causality. Now why do we want, aside from genius, why do people like Jim and me, and maybe not too many other people, much fewer than I thought, favor some over-history interpretation?

20:00 I think that in gravity, there's a real crying need for... Four-dimensional observables, things which have a space-time region and lack a space-time region. In fact, there's a crying need for observables, and the reason this need is there, I think, is because of the, well, of a technical reason. Space-time is really active, and you can't get around that without taking time into account. You can't get around it in one moment. Let me just give two examples to keep in mind of the kind of questions that I think it's very hard to formulate without some space-time view. One is about a black hole. Suppose there's a black hole somewhere nearby, like in Cygnus X-1, and we want to know, what is the size of the horizon now? Well, even in the classical space-time, we'd have a little trouble saying what we meant by the horizon now, but we might give a prescription follows through the region of space-time locally defined. All of these terms can be used to stretch your parameters approximately equal to what you find and will intersect and arise at some point. We can call the radius of the area that we find here a measure of the radius of the horizon. Something like this. That's one aspect. The other aspect is a hyper-circuit. Where we are now, we follow it along somehow. We want the radius, the area of this intersection of the horizon. The other thing is, how do we even know where the horizon is? Well, in principle, we have to look into the arbitrarily distant future even to know where the horizon is. The horizon is a global conflict in relativity. Not having access to enough of the space-time to do this tracing and enough of the future to know where the horizon actually is, there's even no way of formulating the question about what the size of the horizon is now. That's the first kind of example. There are some people who have even proposed that one of the explanations why space is so close to being flat now is because the universe is very old in the sense that it's not a cycling universe and it's balanced at any time, and by some argument which I don't want to go into, in fact I'm not convinced. You can argue that after many balances it will evolve towards a space university.

22:30 The point is not whether that's true or not, it's just the kind of argument that has been made in which it's important for us to know how many times has the universe bounced, you know, reconstructed in a standard before the present era, the present cycle. That again has a thoroughly spaced time medium. I wouldn't know how to begin to formulate a question like that in a canonical framework or in terms of data now. Let me give you two examples. The moral, as I already said, is the example is supposed to be that observables have a space-time character and not just a space-time. And given that moral... The attraction of the sum over histories framework is that it seems to allow us to directly ask questions like that because we have access, our fundamental object is a history or a task, and with respect to that we can ask all these questions, find each one of the contributing tasks word-in-word and take some sort of expectations out of it, or whatever, or within each task we can ask how many times it's passed, the dispersion of, the probabilistic dispersion of tasks. However, the big problem that I want to try and get to in a very simple non-relativistic example is that it seems that at some moment it's allowing you too many questions. It seems that if you try and ask certain questions, the mere fact of asking them will have an influence on the test. It's much like the ETR experiment. Where if we find spin up here, of course we know it will be spin down here, but the mere fact that we decide to do the measurement here, or decide not to do the measurement here, has no influence, in that case, on the probability of spin up and spin down over here. What we seem to find by allowing ourselves to ask too many questions... And some of the history is that the mere fact whether we decide to do, to ask a question, to do an observation in the future will have some material effect on probabilities that we're already finishing by the time we ask the questions.

25:00 But to get, so the conclusion that I would drop in that whole thing is that three things, one of them, three closely related things. These terms have to be much better understood within the context of measurement theory that allows you to come up with a theory. Or put it another way, to get a good understanding of measurement theory, which really is the need of some of its use-lifes, we can be more confident in having enough of these requirements in some form of theory. I said they were related. I'm not going to argue too much in relation. Actually, we already had one. So let me, the other thing I should say is that although these are very important, it's very strange that causality or locality are present at all because quantum mechanics in some over-history formulations, particularly here, is somehow fundamentally non-local theory. That's something very subtle, and I think that deciding from the non-local framework and getting out local results is why it's hard to see a measurement theory from other institutes of view, which to me is more painful to the underlying structure of the theory. Certainly, it's related to some kind of unitary athlete. You could probably make it, and it's equally good accounting for Heisenberg's, but I think it would be no better than the account that you just gave. So the question is, when the canonical thing is innovated, as a basis for these properties, it's going to be... Probably we get this, yeah. And then the question is, do we need them? And I'm going to argue, well, it looks like we may need something like that, because otherwise we're doing some very bad violence.

27:30 Before getting to the problem, let me describe the framework in which the problem occurs. This is actually quite similar to what Jim described, besides the exact form that I'll write for the probability, which is quite different. Otherwise, it's quite similar. To understand this framework, we can go back to some article, which I've never been able to trace down, which I think I read an article by finding. In which he would describe them in several different ways, and he said, he imagined a particle moving, say, in one dimension along a line, and he thinks the region in space-time, some regions are, the particles either He says we ought to be able to design a principle, some kind of apparatus, that would do no more than determine one cat, not measure the position at all these time slices, many, many measurements, and figure out where it went, but just do one single determination, which is the question, is the particle of cancer that's based on reason or not? Those are very natural questions when we come up with these viewpoints, but if we could... It's just this sort that, if you think about it, that I'm just coinciding with the examples that I gave to the black hole and the black community, the problem is going to be, however, that this is also the sort of function that's going to give us a taboo of causality. Let me just take another example. Suppose that the particle begins at t equals zero, I'll just draw the t equals zero line, the x equals zero line at t equals zero, and here's the thing, the x equals one.

30:00 Does it have the character of the chat I'm drawing now, which is T equals 1, which is that sometimes before T equals 1, it's an excursion all the way to X equals 1, and then returns to R all the way before time 2 equals 1. Or another task of the system, which doesn't work, perhaps it gets a change in time, but after T equals 1. This is something I could question if you thought of this as an example. So, if you think about that in a more abstract way, what are we doing? We're taking the space of all paths, all possible paths, and we're dividing those... We divide again into k disjoint classes, according to whether they do or not satisfy certain criteria that we've set out, and then we ask about the relative probabilities, say, p1 through pk, of these k classes, and the function of the sum over histories, sum over histories, normally would be... The funny thing about quantum mechanics is that the probabilities of this can change if I subdivide it further into other classes, or even if I make correlations of other classes at the depth of that second thing between the two classes. We produce actually, the rule, which I'll write down in a little while, starts to produce absolute probabilities of quantum mechanics. Unfortunately, it produces only relative probabilities. The question I wanted to ask you was how you normalize the probability. With what other possibilities were you comparing the case where the test did go through? Don't answer.

32:30 There is a proposed answer. Oh, there is a proposed answer. Well, maybe you should answer. I'm afraid that you'll give the wrong answer and then my son will reply, not even to two people in the audience. There'll be nothing but a shooting gallery if there's no one except me to lose. So, thank you. Anyway, the point is that the answer is important, so I made it explicit here, the set of classes into which we divide the test. If you want to think of another example of what these classes may be, you may be doing a normal two-slit experiment or some major terms experiment, and then the classes would be determined by the points on the screen, the final screen, at which the project will arrive. That would be a very large number of experiments. A classical stochastic process. And why did I ask that question? I asked it because I think that philosophically, in the sum over history's interpretation, quantum mechanics is virtually identical to a serious stochastic process. The only difference is that the rule of force between the probabilities is a more complicated one involving matrices than it is graphically, which isn't interesting to me, but nevertheless, it makes it much closer to the truth, in fact, to the truth of science than many other difficulties. So let me say, the graphical rule is the probability of P sub i, which is what I call P sub i here, is just the sum over long terms, in fact, P sub i. Some keys of gamma. Well, here is some primitive solubility to associate with gamma. And a depository of real numbers, 3 and 1. All the key numbers add up to 1.

35:00 Now, let me say the quantum rule, which is considerably more complicated. It involves several steps. One is we pick a collapsing piece of sheet. Well, I call it a collapsing piece. Sometimes it's a piece, but it's not. It's a piece of a model. They consider paths which run forward in time to the time t and then go backwards, returning to the original state. And, furthermore, we satisfy the condition on the way down as well. There's a whole path, a double path, which is either the i integral or the lbc. And when you can use your backwards, you get the c backwards, you effectively get the complex conjugate of c. Which is the sum of all the episodes. So it's very similar. You might think it's complex, but we'll be able to get down to the real topic in a moment. But now it's only the relative sum. Two remarks I'd like to make. First is that when people refer to squeezing or decoherence or something like that, wasn't that... It's a few tabs, but all the contributions from the academic agenda are very, very similar on the way up to what I'm doing right now. And now, I'm almost kind of confused. It's complicated, but I think it's closer to me. It's a problem. All right. So there's no way I can actually give an example. Let me give an example which shows why we don't get trouble. In the case where we do, so what we imagine is, I've left out the whole region that you see over there, so you have the left in here, where we have the left section.

37:30 All of these points are left out of this picture. It's all below these points x and y. What I'm examining here is the effect of asking a further question at some time before the final collapse time whether the particle is found in region A or whether it's found in region A prime. So this is the two-fold possibility. And remember the rule is we take all, so now we have two possibilities even if all of them are in region A prime. What we want to know is, is the probability that the verbal stuff happened, plus it was in A, and also was in A, added to the probability that the earlier stuff happened, and it went through B, the two possibilities, equal to the commonality that the earlier stuff happened. That's what it would mean to say the later measurements had no effect. If the first thing is answered by pairs like this, they go through an A on the whole up and on the way down. The second thing corresponds to adding these pairs, which don't seem to take time, I suppose. What we know from unitarity, which is an input that I skipped over this one time, is that if we add up all the paths with fixed points x and y here, we'll get a delta function. That is, we'll pull these two points x and y together. We'll squeeze it at that time. The paths that are missing here are those which go up through A and down through A' Fortunately, those paths don't matter, because they are something like this, but by the unitarity applied to the second step here, they get to A' So in fact, the only paths that really contribute are either this one or this one, and therefore we have included all the paths. In other words, we can expect a perhaps time that's a little bit earlier, but this is also important because it means that the results are independent of our charge of time, which is a necessary consistency condition. So I'll just, here I've written it out symbolically because I was going to use the formula symbolism to discuss the thing that gives trouble. Since there's no time to do that, let me just state what is the case that gives trouble. I can leave you to work out for yourself if these probabilities do not add up, the probability of a certain outcome before t equals t1, let's say, is actually affected, is not equal to the sum of the probability of that same outcome plus possibility a at a later time plus the probability of that same outcome plus probability of not a at the later time.

40:00 The particular example is, let's do the later thing. Just take two position measurements of this sort and do the same thing. So here's our early test. So we do a position measurement here. And we ask whether it's here or here. And then we do some other position measurements. We ask whether it's here or here. And I'll call this A. The thing that doesn't work is that A and B All our time. So one question. Would you like to reply to the question which you were asked in the middle of the talk? Thank you for your attention. Including functions like Psi or Q, which give us commonalities in the usual way, of a position and a fixed time. When you're trying to use it to provide answers to more generally phrased questions, like did the particle pass through this column for a certain reason, you get this problem. Now, if you ask two questions, you can't know. Formulated at times, one of which is pervasive against the other. The mere fact that you set up your apparatus to measure this person has an effect on the probabilities of getting the answer to this question.

42:30 When I called it F, I called it E and F. E is the question, did it pass through this? F is the question, did it pass through this? What I'm saying is that the probability of E is not equal to the probability of E and F. But the probability of being and that's fine, we're after concomitant. That's right, so there's a variance. But physically there should not be a variance. Shouldn't there be? The answer to that question is probably a yes. Did I add the wrong thing? No, no, no. I didn't, I didn't, yeah. You couldn't say it was a yes, right? The answer to that question is probably a yes. Thank you for your attention. So you don't know whether you caught a bird there tomorrow or not. Well, that affects the probability that you know the bird was there tomorrow. Oh, I see. You're putting this at the end? This isn't the first one. No, but it's at the end. It's the last plane? Yes, that is the last plane. The last one that's trying to exhaust the air? Well, that's interesting. It should sound like the last one. It should sound like the last one. But it's not the first one. I don't think it's the first one. I think it's the last one. Oh, that would be the last one. Well, maybe we should give the chance to the last speaker, Paolo Romero, speaking on time and quantum reality. Well, that's the title. Introducing this workshop.