Time in quantum gravity (last part) / Time in quantum gravity (first part)

0:00 Now, a couple of years ago, I wrote an article and pointed out that this non-invariant combination e squared plus b squared, which many people have noticed before, this combination, is not independent of the lapse in the ship, yet if it's zero on a Cauchy slice in a Ricci flat spacetime, then the Riemann tensor On that slice is zero, and by using the uncontracted Bianchi identity, one can prove that the Riemann tensor stays zero in the domain of dependence of the data. My final remark is to remind you what the vacuum is. We've all been talking about foliation independence, and the last time I heard flat spacetime, a Lorentzian signature minus plus plus plus, was a pretty decent vacuum state. And if you cut it up in a wild way, you have to satisfy those equations, and when somebody can produce me a connection whose natural curvature gives me those, then I'll be totally convinced that you have the perfect variables. Now, our base variable gives half-flat, and that's a fantastic result. Oh, they do. I asked you yesterday, I misunderstood you. Then the variables are about as nice as they can be that way. So one can actually explicitly deduce these equations in real relativity. They have to use that. Now in Euclidean space it's even easier. Okay, so anyway, that's an argument strongly in favor then of those variables. It's a combination that... It's easy to do in differential geometry. It's not easy to do in a symplectic structure. These are many topics. We have time for a couple of questions. This will be marked at the beginning of the lecture. Is that what you're going to do now?

2:30 No, I didn't say that. No, no, I gave you credit for that. I didn't say you couldn't have dynamics. I said that the... I said that the new... Variables, when you used twisting congruence, were not new dynamical degrees of freedom, and their equations of motion could be deduced from the others. That's all I meant. But the orbit I was doing was very well done. Yeah, sure. I didn't mean any more than that. You can certainly do it. I just... Yes, you can do dynamics on the space of orbits. If you want to do a single thing that's stationary, amortizationary, with no dynamics, you could try to do that. Yeah, Bob Garrosh used the metric on the space of orbits and the curve problem, but, I mean, back when I was a barefoot country boy in North Carolina, I figured that stuff out. What was your conclusion at the end of the last? I didn't understand whether to smile or frown at the end of the last. Last transparency. To laugh or cry. No, we should be relatively happy. The point is the following, that we've talked about all these slicing independent things that we try to get in the dynamics of general relativity, classical or quantum. Dirac ADM variables are basically the three metrics from which you build up this curvature and these guys. This is essentially the canonical momentum. And it's easy to see if you take a flat slice in flat spacetime, an ordinary preferred slice in flattened and constant spacetime, that the three metric is just the flat one and the momentum is zero and that therefore these equations are satisfied. You simply know that you have the Koshy data of flat spacetime. But now if you take an arbitrary curved surface through a flat spacetime and look at the ordinary canonical data, it isn't very easy to see that there are solutions to these rather complicated equations. These are the embedding equations for a space-like surface in Minkowski spacetime. Now, Astrakar's curvature of the connection that they use, which their covariant derivative involves a little bit of the three geometry and a little bit of the momentum in the other representation, the curvature of their connection being zero is, as I've just understood, equivalent to these equations, so that their curvature really is the right combination of Q's and P's.

5:00 To tell when you're in flat spacetime, no matter how the kinematics was done, no matter how you sliced it, if their curvature is zero, you know you're in Minkowski spacetime, whereas in the old variables, if somebody handed you these things, you'd have to do a long calculation to see that you were in flat spacetime. Well, and I think the ideas began with the similar, from those kinds of ideas. And then everybody said... The only thing is that in that case you have to solve a non-local problem. That's the problem. The thing is that I think once it was realized, and even Ed didn't realize at first what his derivative was, once it was realized what it was, even the mathematicians started writing papers and said, of course, this is the natural thing to do. I mean, even Kasdan wrote such a paper. He's no slouch. We should have been doing this all along in Riemannian geometry. That's what we learned. 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 state, 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.

7:30 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. It gives the joint probability for a series of yes-no questions, which are represented by projection operators in the Hilbert space, operators who ask the same questions at different times are connected by the Heimel timing, and the fundamental formula is... The joint probabilities for succession in time of such questions is written down here in the Heisenberg picture. These two clearly contain within them the familiar elements of the Schrodinger picture story, the smooth evolution by the Schrodinger equation, and the reduction of the weight packet, 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 ordering of the operators, but also in the way it's written. Any alpha here, that is, any subspace of the Hilbert space from an observable, with 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.

10:00 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 dynamical variable? 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 now. This is the view that history, that 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 wave function 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 in sensitive order projections at a single value of tau. It's perhaps most honest to say that we have direct access to time only through present readings of the plots, and to history only through present value of records. The probabilities, however, of these particular correlations in the present are not the only variables, only probabilities which we need to compute in order to consider these questions. A clock, for example, is a mechanical system whose indicator is correlated with the location of successive space-like surfaces with some high probability. A record is a mark whose present value is correlated with past events, but also with high probability. So just to define what we mean by good plot or record, some shared 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 relativistic language, we have to consider probabilities which involve observations, if you will, which involve different numbers of space-like surfaces. The second view I'd like to talk about is the view that we only need to define time classically.

12:30 And this is the view, therefore, we're dealing with all our observations in cosmology that 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 of 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 sort of decayed classically when certain properties on different space-like surfaces are correlated according to the 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 arise in quantum mechanics. I don't think we're going to get away with doing everything either in the classical regime or in the regime in which we work just on one space-time curve. 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. This is 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 2. The second option, which I wish 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. That is, we want some expression for the joint probabilities, whatever it is, 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 class of a behavior of the quantum system, which is spacetime, because it would be ordered by classical time.

15:00 Now, this certainly can't be a general statement, more than classical spacetime is a general statement. It can only be an approximation which is appropriate to specific initial conditions. Now, sum over histories quantum mechanics gives us a natural route for looking for such a formula. We can, following Feynman, do the sum over histories formulation of quantum mechanics. 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, and we ask for the probability of starting a non-relativistic particle at a position of zero, and we've detected its successive intervals, delta one through delta n, at times t1 through tn, and our joint probabilities can be constructed as follows. We first take the sum of e to the i s over all paths that started at zero, and through the regions where the particle is supposed to be detected, and wind up at some point x n in the final region. A squared, a unit rate of a squared 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 a particle is in a definite state like zero, you calculate the trace of the projections down in the region. The point is, it's the same, right? As Feynman, Hayes, and I learned this morning that John Stachel showed, right? It's just a mathematical identity, but this is the same as that. This equality could be regarded as an interesting mathematical identity, or we could start from here, and this is a mathematical identity, or we could regard this as a starting point for quantum mechanics. Let's see how that would work in quantum mechanics and closed cosmology. In the quantum mechanics and 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, or the rest of the three fields, upon that surface.

17:30 Now let's consider some multi-surface probabilities that would be how we would calculate them according to quantum mechanics. There are many probabilities, but the one, perhaps, 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 the model I consider free, with geometries in confused configurations of d1, phi1, d3, phi2, d3, and phi3, which line regions of super space, which I'll call O1, O2, and O3. So the question is, given certain initial conditions, right? What is the probability, if you like, if the history passes through these regions O1 and O2 and O3 of super space, 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. You first of all consider all histories which satisfy the initial condition C as true to the particular three geometries and the successive surfaces in... And lined up, let us pick for a moment, at a definite point in O3. So you calculate the sum of paths which satisfy those criteria, passing through O1 and O2 in any order, you then square that and integrate that over this final region, then repeat, because we have no way of distinguishing whether this was the final region, that was the final region, or the other thing was the final region, and sum up the answer. Expression for this joint probability based on these surfaces. They're not Earth. So the sum of all the possible works. I only have 20 minutes. Let me finish my reading.