Time in quantum gravity / Time in quantum gravity

0:00 I have detailed arguments. I also have my briefs. They're contained in three papers, two of which are in final form, and one of which is in the draft stage, which I put some copies over here back on the table. If you were sent these by Lee Smolin, it's only paper two, which is the final form of the paper two, which you might want to take a look at. So in the 20 minutes I have, of course, I can't hope to present detailed arguments for this case, but I shall at least try to state a view of what the problem is, first of all, and what the proposed resolution is. I'm afraid that means that the arguments will be very schematic, but I hope at least some major points will emerge. Now, as Marie described to you on Monday, the fundamental formula of standard quantum mechanics is the formula by which Which gives the joint probability for a series of must-know questions, a series of yes-answers to yes-no questions, which are represented by projection operators in the Hilbert space, alphabets as the question is asked, and the time in which it's asked. Operators in the same questions at different times are connected by the Hamiltonian. This is written down here in the Heisenberg picture. These two formulas contain within them early familiar elements of the Schrodinger picture story, the small evolution by the Schrodinger equation, and the reduction of the rank factor, if you will, in this formula for the probabilities involving projections. The important point to note is that the projection operators occur ordered in the variable t. This is the expression of causality in familiar quantum mechanics. The special role which is played by time, I think, is completely clear in this formula. You can see it certainly in the, most particularly in the time modeling of the operators, but

2:30 also in the way it's written. We're allowed to choose any alpha here, that is any sort of subspace or Hilbert space from observable. Every single observation for which the probabilities are predicted are characterized by one particular observable, the time t. Now, when spacetime is classical, the time in this formula is the familiar time of classical spacetime, and we know how, for example, because of the communication of operators in space-like separations, this is consistent with the relativistic causality. The question for quantum gravity is, roughly, what is the status of this fundamental formula when spacetime becomes a canonical variable? And I shall outline here one proposal in the following, but first I'd like to discuss... Two ideas, two points of view, I think, which will not work. I'll be brief about this because Lee Smolin mentioned some of these arguments this morning. The first point of view that I'd like to describe is the idea that a fundamental formula is needed. This is the view that if clocks are included in the system, then predictions involving time are predictions for correlations between the clock indicators and other variables on a single space-like surface, the surface map. This is the view that history, when we write for the universe, is entirely the study of correlations between present records. This is the view that all interesting probabilities can be computed from one single wavefunction of the universe on a single space-like surface. Or, to put it differently, this is the view that in the fundamental formula, one need only consider commuting insensitive to order projections at a single value of count. It's perhaps most honest to say that we have direct access to time only through present readings of the plaques and the history only through present volume of records. The probabilities, however, of these particular correlations in the present are not the only probabilities which we need to compute in order to consider these properties. A plaque, for example, is a mechanical system whose indicator is correlated with the location of successive space-like surfaces with some high probability. A record, right, is a mark whose present value is correlated with past events that also apply probability. So just to define what we mean by good clock or record, some sort of notion of multi-time probability is needed in the theory. Otherwise, we have no way of guaranteeing, we have no way of understanding, I think, what we're talking about. In a relativistic language, we have to consider probabilities which involve observations, if you will, which involve different numbers of space-like surfaces.

5:00 The second view I'd like to talk about is the view that we only need to define time classically. This is the view of acro. We're dealing with our observations in cosmology, but anything else occur when we're talking about classical epochs of the universe, now in the late universe. And that therefore, since the spacetime is approximately classical, it's that classical spacetime. Which supplies the notion of time, which supplies the ordering and the fundamental formula, and so forth. However, the semi-classical spacetime is not an exact statement, it's only an approximation. We have a system that's said to behave classically when certain properties on different space-like surfaces are correlated according to classical laws with high probability. That means just to define what you mean by classical behavior, or by extension, semi-classical space-time, you need to have these multi-time probabilities, such as the rise and fall of quantum mechanics. So I don't think we're going to get away with doing everything either in the classical regime or the regime in which we work just on one space like this. If we can agree on that, that we need to have multi-time probabilities in quantum mechanics, then I think we're faced with two options. The first option is to keep the fundamental formula. In that case, we must identify the preferred variable t. Is it trace k, with some hidden value at a time, or something else? I won't say anything more about this approach because it was discussed at length this morning. The question is whether there's any choice of t which is consistent with the covariance of spacetime q. The second option, which I was to speak about today, is to generalize the fundamental formula. In this case, we have to find an expression for joint probabilities, which occur on different hyper surfaces of the space time, which reduces the fundamental formula approximately when space time is classical.

7:30 We want some expression for the joint probabilities, such that when space time is classical in the present domain, it reduces the fundamental formula of standard quantum mechanics, not exactly. So there's an approximation appropriate to that classical behavior of the quantum system, which is space time. Now, this certainly can't be a general statement in more of a classical space analysis. It can only be an approximation which is appropriate to specific initial conditions. Now, sum over history's quantum mechanics gives us a natural route, right, for looking for such a formula. We can, following Feynman, use the sum over history's formulation of quantum mechanics as the starting point for defining joint probabilities. Let me just remind you of how that goes in non-relativistic quantum mechanics. Suppose we want to start at, we ask for the probability of starting a non-relativistic particle at a position x0, and we detect it in successive intervals, delta1 v delta n, or times v1 v vn, and that joint probabilities can be constructed as follows. We first take the sum of e to the is over all paths which start at x0. All the regions where the particle is supposed to be detected will wind up at some point xn in the final region. They square, and then it winds to square over the final region. That's the joint probability. Of course, we could also compute that by evaluating this trace over the operators. We start with the initial conditions, if you will, that the particle is in a definite state of zero. We calculate the trace of the projections down on these regions. The point is, it's the same, right? As Feynman, Hayes, and I learned this morning, John Statrell showed, right? It's just a mathematical identity, but this, the same as this equality, could be regarded as an interesting mathematical identity, or we could start, which we start from here, another identity, or we could regard this as a starting point for quantum mechanics. Let's see how that would work in the quantum mechanics of choice cosmology.

10:00 In the quantum mechanics of closed cosmology, the histories are, of course, four geometries and a suitable gauge. I can think of the histories as curves. In super space, the super space is three metrics in space by surface, and matter fields, all the rest of the three fields, upon that surface. Now let's consider some multi-surface probabilities. It would be how we calculate them according to quantum mechanics. There are many probabilities, but the one which is most analogous to the multi-time probabilities of standard quantum mechanics is the joint probability to ask whether a spacetime has a set of surfaces, in this case I'm only going to consider three, with geometries in confused configurations of 1, 2, 1, 5, 1, 2, 3, 5, 2, 2, 3, and 5, 3, which line regions of super space, which I'll call O1, A3, and A3. So the question is, given certain initial conditions... What is the probability, if you like, that the history passes through these regions O1 and O2 and O3 of superspace? Given certain initial conditions, what is the probability that the space-time has those surfaces with those geometries and field configurations upon it? Well, the proposed prescription is the following. The sum of the histories. You first of all consider all histories which satisfy the initial conditions T as through the particular three geometries and the successive surfaces in... And wind up, let us pick for a moment, at a definite point in O3. So you calculate the sum over all paths which satisfy those criteria, passing through O1 and O2 in any order. You then square that and integrate that over this final region. You then repeat, because we have no way of distinguishing whether this was the final region and that was the final region, or the other thing was the final region, and sum up the answers. And that's the proposed expression. Now, we could ask, right, is this equivalent to familiar quantum mechanics? So, to answer that question, let me first address the question of how, starting with some over history's quantum mechanics, we get back the ordinary Schrodinger-Heisenberg formulation in the case of a simple example of non-reliable quantum mechanics.

12:30 There is an easy, let's for example consider the amplitude we get from A to B, the overall paths which connect A to B, and the paths in non-relativistic quantum mechanics move forward in time, and therefore this sum can be factored in the following way. If we consider the sum of all paths which get to A at point X on some intermediate surface and connects to B, And finally, we then sum over all x. That sum, right, reproduces the sum over all paths which connect A to B, because the path had to land somewhere on this intermediate surface, and sum over all those things, and it only landed at one point. If we write the sum from A to x, right, as psi A of x to tau, and from B to x, e to the psi B of x to tau, we recover, through this composition law, the inner product on the Hilbert space. I'm going to do the sum over history's quantum mechanics. The recovery of the Schrodinger-Heisenberg picture of quantum mechanics is possible because there's a particular property in the non-relativistic histories, right, that there are surfaces which the histories cross once and only once. Now what do we expect in quantum cosmology? In quantum cosmology, there are no surfaces in super space which the histories cross once and only once. If we, for example, construct a surface which corresponds to, say, some particular volume of the universe, there's no reason to expect that a general history, and I'm distinguishing here between a general history and a classical one, should have surfaces with that volume in only one part. Put differently, there may be good time variables for classical histories, as Jimmy York described to you this morning, for example, trace k, but there's no reason to believe that for a non-classical history, we couldn't have surfaces with arbitrarily many surfaces having the same value of trace k. They go up and down. There's no geometrical variable which makes it good. And therefore, we do not expect to recover a Schrodinger-Heisenberg formulation of quantum mechanics and post-cosmologies with any choice of the time variable in the case of...

15:00 There are certain circumstances in which we would expect to be able to cover it approximately. What is claimed, right, is if one takes this expression for the probabilities, the natural generalization of the sum over histories framework, that if the initial conditions C are such that there's a region of super space where the space-time behaves classically, then the probabilities of suitably crude observations will be given approximately by the fundamental If we have a series of three regions here, and superspace and space-time behaves approximately and classically, then approximately it will be true that the particular construction, as I illustrated before, will be given approximately by the fundamental formula, where the p's are ordered by the time of the class of the geometry. It's not difficult to see this, because if you recall the construction, it involved summing either the IS over various paths which cross the regions. And if it's the case, for example, that there is a single path, a classical path, which dominates, as far as the geometry is concerned, which dominates that sum over history, then we will recover that particular ordering among the surfaces. And a familiar notion of time will then re-enter quantum mechanics as the... As the time associated, not exactly in the formalism, but as an approximation, appropriate to specific initial conditions which give rise in a certain region of the universe to approximately the classical spacetime. We should come back to that. The status of time in the sum over histories formulation of quantum mechanics is the following. The distinguished time and associated causality of our familiar quantum truth is not so much found in the formalism as exact features. Rather, there are approximations in the light universe made appropriate by specific initial conditions. It's becoming clear, I think, that many of the features of our world find their explanation not in the form of dynamical laws, but rather in the form of specific initial conditions.

17:30 The second law of thermodynamics is certainly one such feature. Murray has explained to you on Monday the existence of variables which behave classically. There may be another. In quantum gravity, for example, the existence of currently classical spacetime, which we see all about us, can only find its explanation in the properties of some specific quantum state, which is the general feature of all states that behave classically. Could it not also be that the familiar formulation of quantum mechanics involving a distinguished notion of time, which is so closely connected With the notion of classical space-time also finds its origin in the same very special initial conditions. Thank you. It's wonderful you ended two minutes before your time. Congratulations. Let me ask you a very fast question. In your problem with path integrals in Newtonian physics, you integrated over x-rays at a given time at your gates. However, in your problem in geometrodynamics, you integrated over all of the three geometries, which... There are several that time enters not only in the ordering of the, as we call it, the fundamental point of the lens again, but in this diabolical projector, which has a fan, which blows from transparencies. I don't know why I heard speakers manage to get around this, but anyway. The preferred time enters not only in the ordering of the operators, but in the fact that every operator for which you make a prediction is characterized by this special number. It's a remarkable property when you think about it. Time is just as observable as any other operator.

20:00 Why should every prediction that's made have that particular observable as a label? Therefore, in order to generalize the formula, we not only have to generalize the formula with respect to the ordering, if you like, we also have to generalize the class of observables, because if we said, right, that we could go back and say that, well, okay, the formula is perfectly symmetric and doesn't involve any particular ordering, but we only are allowed to discuss operators which are observables which are confined to one special class of space-like surfaces, we once again, in fact, introduce the... And therefore, it's necessary to generalize the notion of observables, and this is a proposed one way of doing this in the Newtonian framework, and I think Raphael is going to explain some difficulties with that later on with respect to causality. But it seems to me that if we're going to avoid this problem, we need to do both. It's characteristic of path integral formulations that they prefer a particular configuration space if you're going to regard them as a starting point for setting up a theory of quantum mechanics. That is, in ordinary quantum mechanics we deal with paths which are an x. You then have to analyze experiments, right, to show how you can, for example, discuss measurements of momentum. That is, there's not an immediate transformation to it. But it can be derived, as Feynman showed. Of course, it's also true, I think, of the traditional theory that while it's very good at predicting probabilities for operators like x cubed p plus px cubed, it's just about as bad as explaining exactly how you observe it. So, what a great tragedy to have this specialization. Here, what I tried to do is to single out one particular, we have a, it goes over geometries. It's the space-time, if you like, that supplies the basic observables, and we construct other ones, like these regions, from that notion of space-time. How far you can go in recovering a transformation theory, I think, is at this moment unclear, but it's also not very clear how necessary it is.

22:30 No, I just wanted to ask if you have any more elaboration as to why we can't do it on the present hyperspace. I mean, this all looks very beautiful, and maybe it is more fundamental, but in practice, whenever we test things, it would seem like we're really just comparing our records of one kind of clock and our records of one history is recorded by one means and history is recorded by another means. And if we get agreement on that, it seems good enough. But how do you know it's a clock? What if I say, this is a clock? Well, I mean, are there a lot of other things that agree with it, that would give the same value? I guess I'm slightly paranoid. If I found 100 cups and it said 12 o'clock, I still wouldn't believe it's a clock. I would do the calculation to show that it actually behaves as a clock. Let me put it this way. I'm not... I guess if it should turn out that this point of view involving the internal states of our brain is a possible way of, is complete in some sense, and I'm expressing some skepticism about this, that we can reduce all our observations, marvelous moment, and never do another calculation, I'm agreeing that that's what's accessible to us directly. It still seems strange, for example, if you're going to say it's an internal setting. States of our brain, for example, do we believe this surface defined by our brain is defined to 10 to the minus 43 seconds? But it's extraordinary. I mean, yeah, but see that's, that's, that's, I'm a little bit reluctant to inject these psychological notions into the fundamentals. I just don't think the brain is sort of defining time intervals. I don't think there are fine intervals to be accurately, arbitrarily precise. So I... Let me say the following. Should it turn out to be that we can formulate physics in this way, nothing would be changed here. It would just be slightly superfluous. People who would believe in that would still have the obligation of supplying the measure, if you like, on the inner product.

25:00 And this is one proposal that that measure is that induced in these particular probabilities by the sum over history. We have time for one more question. Why don't you recognize who you want to answer? No, no, I think you... No way. I can't. Your choice, James. Well, who was first? Why don't we take our distinguished organizer? Well, I just wanted to take a few more comments. Neither I nor any of the people I know really... I think talk to the information is going to be the service of my body, so... This point of view is consistent with your point of view, because these regions... We don't have access, certainly, to information over a whole space like Cyprus, so that just means these regions, though, are extraordinarily big, and there are whole lots of geometries that could fit within. Well, let's see, I wasn't supposed to take another question. The trouble is that Ted Jacobson would like to make a comment at this moment, and I think that we should give him a couple of comments. I hope you don't have transparency, so watch out, don't put them down here. Well, I just want to make a couple of cautionary remarks about attempts to extract probabilities from quantum theory when they don't have unitarity at some time underlying the dynamics or evolution of the system, and I'm saying this just because it's very tempting when you have a passenger role sitting in front of you to try to say, well, of course I can just... Specify some final conditions and square numbers and add them up and normalize some of them and get probabilities, but I think there might be some problems with that.

27:30 So, first I want to make an argument. If you have a background in cosmic structure, which is of course not what we have in quantum gravity, but if you do, then basically the requirement of locality implies that evolution must be unitary. And then I want to follow that up by a discussion of, well, if there's no background in public discussion, does that mean this is irrelevant and we can give up the entirety? And I want to suggest that the answer is no. These are very qualitative remarks and they're in the language of quantum mechanics rather than quantum field theory or something. But I think they could be generalized probably. Just a very simple observation. So I want to argue that if you have a background causal structure and physics is local in that background causal structure, local and causal, then evolution must be unitary. And the argument is simply that, well, okay, what do I mean by unitarity? Unitarity means that every vector of Hilbert space, I mean, the evolution goes to a vector of the same norm. But if I don't want to talk about every vector, I only want to talk about a basis. And in addition to saying they go to something of the same norm, I have to say that orthogonal basis vectors remain orthogonal. So that's what I'll take to be the definition of unitarity. Now let's think about choosing position eigenstates as a basis, that is states that are localized in space, and consider a state, psi, which is a superposition of psi 1 and psi 2, which have support in space-like separated regions. I could define the relative probability of finding the particle or the system in region 1 or region 2 by the ratio of P1 to P2, where that's defined to be the norm of psi 1 squared divided by the norm of psi 2 squared. Now suppose the system evolves a little bit, and I'm calling this evolution E rather than U or something because I don't want to presume that it's unitary. So it evolves by E to these new states. And here's where the locality assumption comes in. I just say, well, there's some light coming forward of this region and some of that, and so if I go for a little bit of time, the evolute or whatever of this piece of the wave function is still going to be space-like separated from that one.

30:00 So I now compute the relative probability that the system is here or here, and that's given again by the ratio of the norms of E psi 1 and E psi 2. Or at least if E doesn't scale all norms by the same factor, then the relative probability that the system is here and here will have changed, even though they're space-like separated and there's no connection between them. Now, that's obviously very fishy, but you might say, well, as long as I can't transmit information with that, I don't care. We're just assuming some non-unitary evolution. And thereby, by will, I can affect how much the norm, how much the relative probability that the system is All of these states localized in space-like separated regions must scale by the same factor. The other part of unitarity is that orthogonality is preserved. A similar type of argument using causality would imply that. If, after evolving a certain time, these states were no longer orthogonal, then that would simply mean that what I observed in one region would depend on what the state was in another region when I compute the probability. And so I think it's fair to say that this also implies that the evolutions of these two states are orthogonal. In this picture where I take smaller and smaller regions and I fill up space with those regions, I seem to conclude that, at least in a core experience sense, I have to have unitarity, which I argued in the position basis for this system.

32:30 I said these are qualitative remarks. I don't know what it would take or even if it could be made absolutely rigorous, but the basic idea is clear that if you just give up unitarity, you're almost certainly going to violate locality. Okay, so now the response to this might be, well, we're interested in giving up unitarity in quantum gravity theory. Certainly in the semi-classical limit, things will be almost unitary and therefore maybe almost local, and what do we have to worry about? Because when we have no light cones that are well defined, then we don't care about non-locality. And I think that's not even, well, then in that case I would say that you have the following problem. Suppose we try to just forge ahead and define relative probabilities of two alternatives by projecting onto vectors that specify those alternatives in Hilbert space, E1 and E2. I'm just trying to follow what Jim said we should do. I don't want to give up the answer. I don't. I think it's really dangerous. Jim's suggestion and other ones that want to hear from time to time involve giving up unitarity. I mean, I'm saying that... That's not something that everyone likes. No, I mean, I should say it. I should have made it clear. My opinion is that this is really a problem, and when we figure out how to do quantum gravity, we'll probably have to do it in a way that would preserve unitarity, otherwise... Now when we project on these to define the ratios, it's important that the norm of E1 and the norm of E2 are the same. Otherwise, some completely insignificant things such as the relative norms of E1 and E2 will affect the relative probabilities that we derive, and that's certainly not what we want.

35:00 Furthermore, experimentally, I mean, it's hard for me to imagine how I would set up a measurement where I compare the probability of a spin being up in the z direction and a spin being right in the x direction. That is, I normally have to set up orthogonal possibilities that I measure, so I also want these two vectors to be orthogonal. Although, to be fair, I could compute this if they weren't. and in a sense it would still be meaningful to talk about those relative probabilities. Now suppose the evolution is not unitary on this particular subspace spanned by E1 and E2. Then these conditions on the vectors are in general not preserved. So if I later try to compute, and this doesn't have to be later in time, it could be later in whatever parameter I'm ordering things in terms of whatever variable I'm thinking of as a clock or even as just a label. These conditions are no longer satisfied, so that if I want to compute some relative probabilities at a later time, I'd better not use the same unit vectors to project on, or rather I'd better not use their evolutions. I'd better go back to the drawing board, find two that are orthogonal and have the same norm. Now you might say, well, here's my final comment then. You might say, this is all in the framework of ordinary quantum mechanics. I don't care about, I mean, I have a special basis. This seems to be what sometimes people say, I have a special basis selected out or to project on, like certain coordinates that are selected out into some over-distributed formulation, for example. And you just say, I'll always use the basis defined by those coordinates. And the problem then is simply just that you see from this argument that the answers you get really depend crucially on the fact that you use that basis and not another one, and if you change variables in a functional integral, let's say, and then define your relative probabilities in this other basis, there's absolutely no reason why they should have anything to do with these probabilities. So somehow you're putting into the theory some special status for the variables that you happen to use in a functional integral. Now there's no, you know there's transformation theory in the 75th volume that says that's wrong, and I can't believe it all of a sudden becomes right in the very quantum regime.

37:30 So, I think it's very dangerous to just extract probabilities by normalizing them in the absence of neutrality. I feel we are in the best situation in which we should postpone any extended discussion for tomorrow, so the next speaker is Bob Walsh, and he is talking about the role of time in the interpretation of the wave function of the universe. My belief of this, I should say, is all joint work with Bill Unruh and written up in some other version by Bill, which is there too, so everything I say should be interpreted as joint work with the possible exception of those points in which Bill gets up and screams. Our feeling is that time is extremely important in the interpretation of quantum mechanics and The lack of time appearing in the formalism really is a serious problem, and I think there are in this room a lot of people who feel that way and a number of people who don't, so I guess my main intention in this talk is to try to argue with the people who don't feel that it's a problem and feel that this is just a matter of detail. To focus attention on very, very concrete terms. I mean, one sees in papers a lot of wave functions of the universe written down, and I would like to know an interpretation of it. What does it mean? And I mean this in just the very direct, concrete terms I'd like. To understand if I were living in this universe, of course that's my own dynamical variable that should then be included in here, what does the universe describe in everyday language, what the universe would look like to me, and that's what I mean by all that I'm asking for in terms of the interpretation. The types of approaches to giving an interpretation, I think, primarily divided to the two classes that I have written down here, that is, one attempts to interpret the psi of A as pi as some sort of conditional probability, that is, we fix one of the variables, and then it's supposed to give us the amplitude for all the other variables.

40:00 The greatest source of uncomfortableness with this type of answer is that I can't believe that this could work for any possible variable you might choose. Indeed, the psi of a of phi is pretty much analogous to the psi of t of x that occurs in ordinary quantum mechanics. And admittedly, I happily admit that if I fix t the time of ordinary shortage of quantum mechanics, this nicely gives me a sensible interpretation of aptitudes for x. I don't think, extensively, if I fix x, it gives me some sort of amplitude for finding p. So, I think not all variables are going to be suitable for things to affect. In a person's practice, that comes out being the case where things are not normalizable, typically when you fix one variable and try to normalize it in the other. So, if this is to hold some specification of a variable, What I want to argue is that what we probably need is some sort of non-dynamical time variable to interpret a wave function, but I'll go into that in a minute. Very quickly, the other type of interpretation that's given is to look at this situation in a semi-classical approximation where the psi is of a WKB type form or maybe is more often in practice in papers at the sum of a few sections. In that case, one gets some sort of approximate interpretation of this wave function by viewing the s in here as a Hamilton principle function, associating some classical ensemble of trajectories there and interpreting the wave function that way. Well, I mean, my main source of unhappiness with this is I'm much more unhappy with an approximate interpretation of the theory than I would be, let's say, with approximate equations of motion when we have an exact interpretation of whatever. If we don't have an interpretation except in the semi-classical limit, I guess I could eventually live with that, but I'd like to see an interpretation that would hold in all situations that might perhaps then reduce the interpretation given in the semi-classical limit.

42:30 I think the key idea really that I'm trying to get across is what I have written up here in the red quotes. The primary role, as I see it, that time plays in quantum mechanics, the role that it seems to me is really needed, is as that thing that allows contradictory things to occur. That is, if we're going to have some probabilistic interpretation of quantum mechanics, which I'm at least seeking in this realm, is extremely important. In the usual realm, we need to have a specification of something for which we can then talk about mutually exclusive events, and therefore there are no probabilities. So, in the ordinary sense, if I fix time, a particle, if I'm doing a theory of an ordinary particle, maybe a Schrodinger quantum mechanics, could not have two positions. If I fix the position of a particle, it could be there two times. What is needed is some notion of time in this sense. I'm arguing to have a probabilistic interpretation. If we use some sort of fuzzed out notion of time, which I believe would be inherent in using some dynamical variable, that is a physical clock, to try to play the role with time, then I think we're potentially at least running into serious trouble. So, we don't have, in ordinary Schrodinger quantum mechanics, as I'm going to elucidate in a moment, we do have given to us a non-dynamical ordering time, and we use that, I claim, in the interpretation of a theory. We don't have such a variable in the standard situation with the wave functions in the universe. A critical question is, can we use some dynamical variable, and we have plenty of those around, to play the role of this ordering time that will then give us these mutually exclusive events? Well, I would argue, this is only a more personal point of view, but I'm very skeptical with one could do that. One reason is I don't see any preferred variable or even preferred class staring at us in the face in the quantum cosmology type context. But another reason is that if we go to ordinary Schrodinger quantum mechanics, we can ask the question, can we dispense with this parameter time t and replace it by a clock?

45:00 At least if we're using Hamiltonian's boundary from below to describe a clock, the answer in a very strong sense is no. Because what we really need is something that, some dynamical variable whose values will keep track of the order of a currency. And then there's the dynamical variable whose values, if we measure it, will be monotonic functions of this t. But if I look at some situation where I've taken a clock and, say, started it out in some eigenstate of its value of t naught, I would like then at a later time for it to read a later value. If I look at what its probability is for reading a value t1, or its amplitude is for a value t1 sometime t later, what I find is in fact that whatever I choose For T1, this thing will not vanish in any open interval of T. That's a simple argument because this quantity, because the Hamiltonian is bounded from below, is the boundary value of an analytic function of T, and therefore can't vanish in any open interval. So for any physical clock, there will always be some amplitude, admittedly very small, but some amplitude that... One second later, it will read minus 10 years as the value instead of some positive value. Well, I think to get the points across that you're thinking, it's pretty critical to separate out the role that time plays in Schrodinger quantum mechanics into two separate roles. This is the point of view that we're taking. The parameter t that one sees in Schrodinger's equation is this type of ordering 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...