Round table and general discussion by participants of last day of meeting

0:00 Given black hole, again, one has the idea that in some sense there is a black hole. Space-time is well defined and 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, one has no handlers. It's difficult to forget everything. And one does have to depend on some formalism. Carol this morning pointed out in some discussion, That, as I actually discussed with Gary, that in fact you talk about things like ground state and so on and so forth, and isn't it that you're hanging on to some formalism in some blind sense, and I would like to suggest that one has to hang on to, 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 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 in quantum mechanics, everyone could say, well, these points or brackets is all a formalism. You're hugging onto that. You're saying that it has to go to commutators. This has such a different conceptual structure in classical mechanics, of course, because that's where we are arguing from. And what I would like to suggest is that we should use mathematics as... As a framework to guess where we want to go. Given this sort of preference of this way of thinking, one would like to understand then, how do we go about doing various things? Well, right at the beginning of the conference, Maria Gelman and Jim Hartle's work and so on, they reminded us about measurement theory. And again, I would like to explicitly recognize that all these questions are profound questions, deep questions, but they are questions in the context of a background space-time. So, one of the things that one would like to understand is, whatever that fundamental physics is, which doesn't have background space time, how do our notions that we traditionally use in quantum mechanics arise?

2:30 And it seems to me that there are some proposals, and one proposal was, for example, this Tom Banks proposal, which has been re-examined recently by Lee and Julian Bauer, and made more precise. And the idea there again is that well really this business about Schrodinger equation and so on is a certain approximation in which one of the degrees of freedom plays the role of a classical clock and then we can in other words to begin with everything is on equal footing so to say and then there is a certain approximation scheme or at least a beginning of an approximation scheme which will actually tell us how quantum mechanics as we use and love it sort of arises. But now the criticism, for example, that one can make, and as Lee and Julian Barber have made, is that, well, but that's not semi-classical, and what do you mean by just in an approximate sense? We all know that quantum mechanics needs a notion of time, quantum mechanics needs a measurement theory, quantum gravity needs a notion of time, quantum gravity needs a measurement theory, and so on. And now I would like to go back to what... I was just going to object. I just wanted to sort of re-quote Gary on this point. That's exactly my feeling, that in fact if you do go to quantum gravity and if you take this viewpoint seriously, that we don't really know what space that, what space, there is no space, there is no time and so on. There is something fundamentally different at that stage. And therefore I would like to suggest that there really isn't a need to make such an interpretation. So the question about, I mean... I'm not saying that therefore there is no problem. I'm just saying that there is no need to solve these particular problems in absence of a space-time geometry. Of course, the burden of proof is on us now as to what are the concepts, what are the issues, and what are the problems. Now let me sort of try to give a few, I mean, what I'm going to now say is just meant to be in a way of illustration. It's not that I feel that this is the way it is going to go or any such thing. As a way of illustration, something that I happen to know in some detail and more concretely, is a work that Carlo Rovelli and me have been doing in the canonical quantization program. If we take that, just for the sake of making this argument, that as a solution, as a partial solution to the problem of quantum gravity, then the picture that emerges is something like, well, the Hilbert space of states is going to be a space of function nulls on linked classes. That's what it is.

5:00 Now the point is that one can just say that, but now what we have to do is to understand how to do physics here. I mean, if you just give me this space of functions in these linked classes, where is physics? What do you do? In the spirit of what I said before, the proposal would be that we have to understand the mathematical structure of this space, we have to understand what operators exist on this space, what are the operators, for example, that will map functions which are supported in a particular link class to functions which are supported in another link class, understand what the algebraic structure is and so on and so forth. Once we have understood the background mathematics sufficiently, The proposal is that one might actually then be able to ask questions of physical relevance. Now the attractive nature here would be that, for example, if one has really formulated in terms of link classes and so on, at one stage one might actually get rid of even the background manifold. This is only an example of the direction that one might actually follow that I'm trying to give here. The second thing would be along what Gary just told us this morning about trying to do string theory, for example, without a background space-time. And then, again, that is not the end of it all because we still have to then worry about what are the physical questions in that phase or in that sector of the theory in which there is no classical geometry. I mean, we have to explain things in the phase in which there is classical geometry, but we also have to explain the other phase. And it is at this stage, I think, that there seems to be some difference, fundamentally conceptual viewpoint difference, between, I fear, this is just sort of this morning's and last night's thoughts, between what is happening, this last bit that I'm about to say, there seems to be a difference between what may happen in a background independent string theory, and say, for example, if one follows this program, or such a program, or this program about theoretical quantization and so on, in the context of canonical quantization. The difference, it seems to me, is the following. I feel that if one takes this idea seriously in canonical framework, then one would like to say that the microscopic structure of space-time is, in terms of these linked classes and so on and so forth, and the description of the geometry, the metric as we know it and so on, would only be an approximating, such as a mean field theory or something which is...

7:30 Something that is averaged over and so on and so forth. In other words, even today, even in the universe as we know it today right now, if one went and explored the microscopic structure, this is not talking about the big bang, 10 to the minus 43 seconds and so on, but right now, that microscopic structure really will be genuinely more complicated. The question is, what are the complications? And in one case, the string case, the idea would be that the complications are... The complications have to do with accept with more and more higher massive excitations of the fields and in the other case the complications have to do with totally loss of continuum and really going back to things like functions on loop spaces or something. Well, this is food for thought, but I'm not in favor of skipping lunch here. This is the organizers, or at least an organizer, suggested I bring these transparencies along to give a talk. By the standards of this conference, more concrete than the conceptual problems that the conference is talking about. It's within the context of the old canonical quantum gravity program. And there was a question of closure of the gravitational constraint equations. I should mention this is a problem that I looked at with Ian Jack. Richard Woodard and Samas have a published paper in PRD with results very similar to this that are based on earlier work that turns out to have been done years ago in Woodard's thesis. So there's a lot of overlap in what we did, and certainly our results are, I think, in essential agreement.

10:00 If you don't have the commutators as well-defined operators, can you speak of there being a problem in the commutators not closing? And there has been a standard formal computation of the commutator for the gravitational constraints that has this form. With the operator ordering given by the red parameter beta, if you have different order, if you restrict yourself to doing different ordering simply by shifting pi's and h's, But you can pull the G-A-B-C-D across here and you can do obvious shifts in the ordering, then all those shifts can be simply stated by saying that you put different powers of beta in the H over here. So this is a way of talking about the obvious alternatives for the operator ordering of the constraints. And if you do that and go through a standard computation of this... And I keep saying standard because I'm going to tell you that there are many computations that give many different results. A standard computation gives a commutator of h smeared with two test functions, m and n, that looks like this, where ua is up to a sine m times, well, partial m, so you see it doesn't depend on the metric, minus m, partial m. Now, this is the momentum constraint over here with a particular ordering. Well, Ian, Jack, and I set out to translate Ashtekar's ordering of the constraint equations back to the ordinary metric variables, and when one does that for the For the Hamiltonian constraint, the translation is unambiguous. The formal translation is unambiguous in the sense that the Hamiltonian constraint only involves a sum of either well-defined operators in the metric framework or operators that are products of pi and functions of the metric with

12:30 With no derivatives of either pi or the metric. And for that module of operators constructed by formal sums of that form together with a formal delta, a formal delta that you define as the commutator of pi and h at a point, there is an unambiguous, you get an unambiguous formal algebra. So, you can formally translate this, translate Ashteker's ordering of the constraint and it comes out to be this particular choice of powers of H. And if you go through the standard computation, that's not one of the ones for which the constraint closes. So, there are two different seemingly natural ways of computing the commutator of the Hamiltonian constraint that There are a number of ways in which you can define a formula. A formal algebra or a formal module of operators at a point once you have derivatives of both p's and q's appearing. I think that's the way to say this, and in particular, if you start out and look at this, let's suppose you start with this expression which looks like zero when x is equal to x, you just take h inverse of x minus h inverse of x times this thing, this vanishes formally. So if you want to say that you split the points apart and say that the coincidence limit of this operator is, the coincidence limit of what appears here is zero, then you run into trouble. If you say this first term is zero and the second term is zero over here, Now I'm just doing two different point splittings of del b pi a b h times this on the left of it. Those two different point splittings give different results. They differ by a delta function times something symmetric in x and y. So once you have derivatives over here of the pi together with an h standing to the right of pi,

15:00 you can't do an unambiguous point splitting of this expression. That's the reason for the ambiguity in this formal computation. Now, let me look though at this particular example. The ambiguity has the derivative of log h, where h is the determinant of the three metric with respect to a flat metric that I used to define the point split. So if you want to split points, you have to tell how you're going to compare a vector index at one point to a vector index at another point, and you have to compare them by some parallel transport. So if you introduce a flat metric, then the flat metric doesn't disappear from the answer at the end. It's not a covariant description. There is an alternative to this. There is a covariant point-splitting prescription that you can use. If you do use that point-splitting prescription, then there is no operator or ordering ambiguity. All orderings of the Hamiltonian constraint of the form that I put in there lead to a closed commutator algebra. I don't know whether that has any real significance or not, but it's a fact. So let me just show you the curiosity. All right, the curiosity goes like this. Although you don't have a background metric, you can, in a neighborhood of any point, in a neighborhood of any point, you can pull every metric that's near a particular metric GAB, near in the space of metrics, you can pull it back to a metric H tilde AB on the tangent space to X. Okay, just use its own exponential map and pull each metric back by its own exponential map to the tangent space. Alright, when you do that... So that means that in computing pi AB acting on psi or HAB acting on psi, you can regard psi as a functional of metrics on the tangent space at x. And the tangent space of x, of course, has its own flat connection, so it doesn't have a unique flat metric, but it has its own flat connection that you can use to parallel transport the metric's tensor indices.

17:30 So this partial A log H that appeared in the anomaly when we calculated the commutator of the constraints, this can be expressed on the tangent space as the difference between the two connections, between the metric connection and the natural flat connection on the tangent space. All right, now that difference is of order r in the separated points. It goes to zero. Any two metrics have zero, give zero connection when pulled back by their own exponential map at that point, right? That's just the meaning of Riemann normal coordinates, the metric in its first derivative vanish at that point. So that's all this is saying. So if you do that... So this is really a generalization of using Riemann normal coordinates for doing point splitting of things on a curved space-time background. But here you have each metric. You use its own exponential math to pull it back to the same tangent space. And then you have this flat connection on that tangent space. So I like it. I think it's a natural generalization of that thing. And I just wanted to advertise that. When you do that, because this is of order r, this derivative, partial a log h, vanishes. Well, that's what the anomaly was. So the commutator, regardless of the ordering of the Hamiltonian constraint, the commutator formally vanishes then, when you use this as the point splitter. So all orderings close. Then I have a final comment. All orderings closed, but again, but this is just formal orderings. I haven't regularized the operators. I can go through the formal, I can go through the formal algebra, but what does that prove? At every step along the formal algebra, even though I've, even though I've defined the operators themselves, the steps along the way are not well defined, so I don't know that this, this is a curiosity. I'm not, it may or may not be, I'm not expert enough to know whether this has any meaning, but I want to throw it out to you. What? The point splitting is the regularization. So along the way, so certainly the regularized operators make sense.

20:00 They, then you don't get closure, right, of the, you get closure to order r in the point splitting, and now you're going to throw that away because it's order r, formally. Someone else who's more expert has to tell me what that means. This regularization depends explicitly on the state on which the operator is acting, isn't it? Because it depends on the matrix, which is the argument of the state in which you are acting. The regularization parameter is dependent on the operation. To implement this, this may be very beautiful and maybe it's important, but to implement this in terms of real honest regulated operators involves expressing this exponential math, which is some very nonlinear thing in terms of regulated operators and making sure that there are no coincident delta function singularities. Wait, I don't understand. If I'm given a functional, let's suppose I'm given a smooth functional of matrix G, and now I want to look at this functional in the neighborhood of a point X, then there is a well-defined, then there's a smooth map that I can use to take. Thank you for your attention. So I get a metric on the tangent space that just looks like some g0ab, some g0ab at x plus a third racdd, yc, yd. We've heard a lot about time and quantum gravity over the last two years. The point of view adopted towards time in the restricted case of mini-superspace models in quantum cosmology.

22:30 The various equations I'm going to be writing down have been written down previously by these people, and probably more, although the point of view that I'm going to be putting across is not necessarily the same as that expressed by these people. And I restrict the three metric in such a way that it is described by a finite number of functions, q alpha, then the Rubin-Witt equation, corresponding to this system, is of the following form, is del squared plus a potential. Now, typically, when one solves the Rubin-Witt equation, the sort of behavior one finds in many superspaces is that in certain regions it will be exponential in behavior, and in other regions it's going to be oscillatory, loosely speaking. And the oscillatory region, well, first of all, the exponential region, people normally regard as being some kind of classically forbidden region. The wave function is claimed to correspond to synthetic euclidean in four geometries, therefore classically forbidden. In the exponential region, sorry, oscillatory region, we can use the WKB approximation, and write the wave function in this form. And then putting that into the rule of width equation, we find that S is the solution to the Hamilton-Jacquard equation, and C satisfies this equation, the current transformation. And the interpretation of this oscillatory region is that the wave function is peaked around a set of trajectories for which the momentum is just equal to grad s. These are actually solutions to the classical field equations. Now, we can introduce some parameter, suggestively called t, such that... d by dt is just grad s del grad, so this thing is the tangent vector to these trajectories. It is in this way that time emerges in this approximately classical region. The time emerges as the arbitrary parameter with which we parameterize the trajectories p equals grad s. Also, re-parameterization of variance emerges as the freedom to choose this vector if you put some function, say, n of t in here for that function. Yes. Also, an important point is that time, indeed space-time, only emerge as approximate concepts and appropriate in the facilitator region.

25:00 You can't say anything about them in the exponential region. And indeed, the very existence of these approximately classical regions at all will probably depend on the boundary conditions. Now, that's the situation one has in which all of the variables, all of the mini-superspace variables, become approximately classical. Also of interest is the situation when one has regions of superspace where some of the variables become classical and others do not. And this sort of situation... All of this arises if one considers perturbations about new super space. So one would have some background, let's say, gravitation variables, Q and alpha, which become approximately classical in some region, and maybe, let's say, scalar field, or scalar field fluctuations, which may never become classical. Then, the appropriate Willow-DeWitt equation is the original one, stealth, gravity, and potential, plus the matter and Antoninov vector, which are dependent on phi. And we can get approximate solutions to this equation by making the following answer. We write the total wave function as the mini superspace wave function times some matter wave function. Now, supposing we look for a solution in the region where the q's become approximately classical. Well, we'd expect that Seidel is going to give this form the WKB form. So, putting this in here, and then the whole wave function of the Relibert equation, once again we get the Hamilton-Jacobi equation for s. So once again, this big wave function is going to be approximately p to n, these classical trajectories, p equals grad s. But more interestingly, one gets for psi m, the matter wave function, the following equation. It's just hm operating on psi m is i grad s dot z on psi m. But we've seen this combination before. It's just d by dt, the tangent vector, in the no superspace trajectories. So Psi-m obeys this Schrodinger equation along the classical trajectories. This equation, I think, was first derived by DeWitt and then subsequently by Banks and numerous other people. Also, we can introduce an inner product.

27:30 So what we've recovered here from quantum cosmology is the familiar formalism of quantum field theory in curved space time. So only in a certain region of super space, a region where the gravitational field is practically classical, do we recover all these familiar notions of time and the standard formulation of quantum field theory. Pick a string you can D by DT. You can D by DT vector to the manifold. You don't know what T is. You don't know what the other part is. On Monday, what classical behavior really is, classical behavior really emerges as a result of an interaction with the system with some environment. One has to consider reduced density matrix obtained by tracing out the environment and look for incoherence. And then we can say that a system is classical. So one would expect to have to do this sort of thing in quantum cosmology. However, as Wojciech pointed out, the universe has no environment. However, what one can do is regard some of the variables, some of the degrees of freedom in this environment, and the rest as variables in which we look for classical behaviour. The only example of this sort of thing I know of in quantum cosmology is a calculation that's been given by Tse. He considered the reduced density matrix obtained by taking a wave function like this, a mini surface-based wave function, as perturbations, and regarding the scalar field fluctuations as an environment. So he considered this reduced density matrix obtained by tracing out pi, and then the results turned out to be the following. He did the calculation for the specific example where the q is the background, which is a scale factor on a homogeneous scale field.

30:00 And so on. Well, in the paper I saw, an explicit expression was not given for this, but it claimed that this actually goes to infinity, and therefore one does indeed get the coherence, this density matrix becomes diagonal. I've not been able to reproduce this calculation yet, but what I anticipate happening is that in... In certain regions, certain ranges of q, these background wave functions will be oscillatory. In certain ranges, they'll be exponential. Likewise, this thing over here will be small in certain regions and large in certain regions. And what I would anticipate is that the regions where this thing omega becomes large and therefore decoherence occurs corresponds precisely to the regions in which these backlink wavefunctions become oscillatory. And therefore, hopefully, we'll recover this notion that the wavefunction becomes classical when this thing becomes oscillatory. So that's all I wanted to say. Okay, just a couple of quick questions. Just one remark. I've been working on a semi-classical path integration approach, and it looks in my approach like while one can define, can take certain quantum variables to a classical limit and can see explicitly what happens, one generally doesn't find such a nice decomposition, and in particular that form is directed at density matrix doesn't obtain the final thing, so you couldn't write a Schrodinger equation down for the density matrix in a general semi-classical context. So that if it's true, as you know, that if one of the variables is strongly peaked so that it doesn't have interference with itself, namely the one that you're treating semaclassically, then probably something like that can exist. But if it does have some superposition character, then there's some much more complicated thing that you can still do with it.

32:30 I just wanted to make sure I understood it. Initially you had this E to the IS and you had the wave function, you know, family clusters. Is this decoherence supposed to pick out one? I've only discovered this result very recently, so I'm not sure the size will spread over the whole superposition of wavepack. Even at this point, you'll have the WK being a finite family trajectory. Yeah, I think they're finite family trajectories, but what this result is saying is that these different trajectories are not interfering with each other. A few years ago, quantum gravity was a really exciting field because people had a feeling like we were really on the verge of some breakthrough when it turned out that formulating quantum field theory in curved spaces gave kind of... There's a picture of black hole thermodynamics including quantum effects that was remarkably consistent in a way that really couldn't have been anticipated. And since the quantum theory itself was discovered when people tried to understand the thermodynamics of a box of radiation, people just had the feeling like really we found the key. Now recently in the past several years, I don't know. That's just died down completely. I think the reason is that people didn't know what to do with it. And I just want to make a suggestion about something that maybe we could do with that, something that hasn't been tried and unfortunately it's a vague suggestion, but it's better than no suggestion, so I want to make it. Okay, so what happened when quantum theory was discovered was that in order to apply Boltzmann's idea of entropy, Planck Thank you for your time, and I look forward to hearing from you in the next lecture.

35:00 Put another way, we should be trying to formulate physics in a discontinuum rather than a continuum, so this is sort of an expression of something that's been mentioned just a tiny bit during the conference. So for the purpose of what I'm going to say, what I'll mean by a discontinuum is simply that rather than the quantum state space of a physical system being infinite dimensional, I'll take it to be, the principle is that it should be finite dimensional, so we can count orthogonal quantum states. So I'm not specifically saying anything about space-time manifolds, I want to say it in a quantum language. So in particular I want to say that within what we think of as a finite space volume, the state space of any physical system that could be there is finite dimensional. The dimension of the state space depends on the space-time metric, simply because to know what volume I have, I have to know what the space-time metric is. And vice versa, you would think. If there's some microscopic discrete theory that really underlies the continuum theory, then the space-time metric must actually be some way of describing something about this discrete state space. And finally, is that the microscopic theory, this finite theory, must be consistent with the dynamics we already know about for the space-time metric, and also with what seems to be true about quantum field theory and curved space times, in particular black hole thermodynamics. Now, by itself, that's a completely obvious statement. Of course, anything, any sort of fundamental theory underlying what we know has to be consistent with what we know. Let me try to give one example of where this consistency might actually enable us to constrain the possibilities of the finite theory. Okay, and that's the following. Suppose we accept this idea that the state space is finite dimensional and that it depends on the space-time volume. And since the space-time volume is dynamical in general relativity, that must mean that the dimension of the physical state space is also dynamical.

37:30 So already we come to this conclusion that probably the ultimate discrete quantum theory, if that's what we're going to find, will not be formulated in a fixed Hilbert space a priori, or a fixed state space, but the state space itself will be dynamical. Now immediately you come to a question, suppose I have a state in some finite state space and now I add another dimension to the state space, another basis vector. How I go from the state in the first state space to the one where I've added some new dimensions, and I can consider the first distinction that arises is, do I go to a new state in the new state space, which is a single vector in the larger state space, or do I go to some mixed state because I have to somehow arbitrarily decide? How I extend the original state into this larger space. Now I fooled around with little models and I could see that I could do it both ways. I could imagine rules that would take me from pure states to pure states in the larger space, or from pure to mixed states. But see, here's where we can put in a clue that we have from black hole thermodynamics. Which is that, I mean, although this is something that's not accepted by everybody, there are indications... That pure states do go to mixed states when you take, when you couple quantum field theory to curved space, space-time manifolds. And so it seems to suggest that maybe at this stage and just trying to make the first decision about what we should even think about doing, we should, we should be looking at rules where there is a movement from pure to mixed states when we add dimensions into the state space. Black hole thermodynamics, when we try to imagine what discrete theories could underlie physics and really try to, our goal should be to arrive at something like the generalized second law of thermodynamics based on a discrete theory and that maybe that's enough information to get started in constructing a discrete theory.

40:00 Yeah, this is very stimulating. I wanted to mention something. I think I remember a paper by Sakharov, and anyway, also Bryce has some comments about maximum energy that you can. Now, it's a space, space, it's universally spatially closed. There is a maximum energy, but then the number of states. So there's a physical very sound. Yeah, if you could somehow take into account the fact that there's a finite energy available in the universe. I remember with great pleasure the last Moscow Geoconference, 16 years ago, and I think I can remember this from the beginning of the lecture, and I think we should finish this with one of the world's best organisers, John Stafford, having had an incredible and tremendous amount of work, so I would like, on behalf of everybody here, to thank you all, I hope very much, it's a high probability. I think you did your job well!