Canonical groups in quantisation of gravity / subsequent discussion

0:00 It's absolutely forthrighted by the group theory, and that's intriguing because it suggests rather strongly that what we normally think of as a weak field limit, this sort of field flies suppressed, and we just don't see it, and so you get the delta. As you get towards the strong fields, and some have been marked already, there's a very real sense in which this sort of stuff is somehow compatible with the large captain rather than the small captain. This field becomes more and more significant, and you could even get sort of phase changes from different regimes, so it's an intriguing observation, well, maybe you don't think it's intriguing, I shouldn't say that, but anyway, it's an observation, which is forced upon you by the group theory, but it seems to be what's happening. But this also evinces the density on theory. Well, I think it possibly would. See, what that corresponds to is different groups acting on the original phase space, in a sense, and they may not be universally relatable to each other. I think that's correct. What if you take the variables that take the positivity automatically into account, like the triads? Well, if you do that, well, see the triads don't really do that, because when you even have a triad, you still say they're independent vectors, and for most purposes that's just as bad as saying that they're positive definite, and if you do that you get a very similar group to this. So you can't get out of it by triads. I know I once suggested you could, but that was a slight oversight of my part. Let me move speedily on because I don't want to take up all morning. Just another very quick remark. There's an intriguing topological effect here which comes out of all these things. These canonical groups are not topologically trivial. The ordinary Weyl-Heisenberg group is itself topologically trivial. These groups certainly aren't. For example, they have a large pi zero. Here's the canonical group for gravity. You can work this out. This is that sequence. What perhaps is even more interesting is that pi one is not zero. In other words, the canonical group has a non-vanishing pi1. In fact, the pi1 typically is z2. So that means there should be spinor representations of these groups. It's a weird thought, but there should be representations. If you take a loop of canonical transformations and come back, the state will have a minus sign in it. That kind of reminds me of other things I've heard of in the context of quantum mechanics, this idea of spin from nothing as well. It's obviously suggested that this is something one might do. Can I just take another five minutes, Bill, to talk about quantum psychology? All of this is meant to be a, you know, as I said, it's not really definitive treatment, but I hope you can see certain, you know, some of the speculative things in motion with this.

2:30 What about quantum theory on the space of metrics? I mean, can you really do this? This is the question. Now, the analogue of the problems that you have in Riemannian geometry, preserving things like positivity and so on, are that metrics, now remember now, I don't mean Riemannian metrics, I really mean metrics on a topological set, satisfy these equalities, inequalities, well, very equalities. Symmetry is greater than or equal to nought, so that's like an ordinary mannion metric. This, however, is the highly non-trivial one, the triangle inequality. Once again, it's an inequality, as I said before, so there's nothing you can't just use theorem brackets. So, in a sense, what we're going to try and do is to find a quantization scheme which preserves as best as you can these inequalities, and then the representation theory will tell you, I hope, how far you deviate from that in the quantum theory. This is the point, you see, Gary, like a mannion metric. You start off with those groups, which, as good as you can get, the representation theory takes you very far away. So what one would need to do is to find groups that act on concepts of things like this. Now, it's not a totally trivial issue as to how you actually were set about doing this. First of all, of course, one has to be marked as far as topology is concerned. You have a sort of a gauge-type structure, again, in the sense that there are many different metrics which induce the same topology. There's an interesting observation here that if you have a finite set X, so a simple little model thing, then of course the inequalities up here are actually the same as they are in Regi-Calculus. Now interpretation is quite different. In Regi-Calculus you've got the edges of a simplex, and of course the faces are filled in, in the sense it's a solid space, whereas here you've just got a network of points. But the critical thing is the inequalities are the same, and therefore in that sense a quantization of metrics on a finite set will actually automatically give you a... A funny sort of group through any quantization of reticulculus, as John Friedman in the Injust example studied recently. So that may be a sort of a free gift, as it were, except for one odd, slightly odd remark. In the case of reticulculus, if you had the three points x, y, and z satisfying the equality, one thinks that has been a degenerate simplex. That requires one's quality to change. In the case of metrics, to say that we have three points satisfying this really means they're on a straight Which is, so what? So in that sense, again, the interpretation would be quite different, but mathematics is the same. Oh yes, no, I'm not saying that, but merely that if you, all I'm saying is that the triangle inequalities on Reggie calculus, which you've got to satisfy, will be automatically satisfied by anything which does this.

5:00 Oh yes, no, no, I agree. It's just that it's, no, I agree. I wasn't going to say it's the same thing, it's a subset of them, yeah. But it's, yes, yes, yes, oh yes. No, I agree, just that that type of inequalities is the same. The question is, what is the analogue of, on Q equals Riemann sigma, of these transformations, of the multiplications which preserve this structure? Now you might think the obvious analogue is to do this, take some positive functions, lambda, x and z, and define V goes to something like that. However, unlike the case of Riemannian geometry, although it's preserved the symmetry and the positive definite... There's no obvious way in which it preserves the triangle inequality. Triangle inequality is a much harder thing to handle when you have a real metric. When the points seem successfully separated, it's fairly easy. When they're way apart like this, the whole global topology's been built. And this, in fact, doesn't do the job for you. Anyway, there's another problem here, which is that the thin edge of an enormous, great wedge is that it isn't making sense at all. These integrals have had to be defined in a way which didn't, in itself, depend on the topology of the space. Otherwise, you'd be putting the background and you'd be losing the whole point of doing the exercise. Now, all measures, almost invariably, do, of course, depend on the background. So it's not a very good idea. A much better idea is to use a sort of universal embedding space, let's say, to try and find, if you can, some universal Banach space in which all metric spaces can be embedded, and then handle that. So the idea will be to do transformations on the target space, and this will map the embeddings around, and that, in fact, works. Now, I haven't got details to show you exactly how it works. Let me just give you an example to finish off. Simplest case I can think of where the whole scheme does in fact go through. In other words, here's a quantization of all metrics on a space of three points, and the space of two points, the space of three points however, which if you can think of is like sort of . So what you've got Now, of course, these three real numbers, the d0,1, d0,2, and d1,2, the three distances, they satisfy these three qualities, and you've got to find, if you like, what you've actually got to look for is a good transformation on sets of all three real numbers satisfying these inequalities, which acts transitively, which preserves those inequalities. And then you can push that forward and get a complete quantization scheme. Now, in this particular case, it's clear that what we have to do is to embed

7:30 The next point must point along the first axis with a positive coefficient, and the next point has to point along, has to lie on the first plane, as it were, And then if you were going to high dimensions, you'd go on to the next, now we have a sort of flag space. So you have to choose a flag, a series of subspaces, and require that each one lies in the next one up. If you do that, then you find that there's a unique group transformation law you can write down, which does all you want. It's simply, on the embedded vectors, it's this sort of lower triangular matrices. So if you act like this, on these points, with a group like this, you'll find the gauge preserving group transformations. It automatically will preserve all these inequalities and will take everything around. And this is the final transformation you get. It's rather interesting that what actually transforms, in this case linearly, are not the lengths of the length squares, whether you'd expect that or not, I don't know, but anyway that's what comes out, it's the length squares that transform, and this is the actual group transformation law. Now you wouldn't immediately think of writing that down. But nevertheless, if you do, you'll find that it works. It's of course one of these lower triangular type of groups, so it's not an ordinary compact type of group. Nevertheless, the quantization scheme obviously will go through, and you will end up with a canonical group, which is a semi-direct product, VaR3, with a particular triangular group. So the claim is that if you choose that particular group as a canonical group, you will do the best you possibly could do in the way of preserving the triangle inequalities for three points. So it's no big deal, but you could do it. And there's an obvious extension for an arbitrary set of points. No, no, no, no, no, no, no, no, no. That's really meant to be a passing remark, not a... Just struck me as a possible use of it. So no, no, no, that's a technical structure. And you can do this for an arbitrary set of points. You shouldn't get arbitrary lower triangular matrices. Because the intriguing question is, is what do you do... No, no. No, no, no, I'm really pushing seriously the point I made just now, but I want to take the groups of these more seriously than the... What I mean to say is that groups of this type are groups that you could seriously consider as being quantum groups,

10:00 There isn't any classes of theory, they seem to be of a natural class too, but obviously, as I said, it gets more and more readable. It seems to be mathematically natural. The intriguing point, and I'll finish on this question, is what do you do for a general x other than a finite set of points? Now, Bryce, I know, won't like this. We've discussed this also many times over the years. You have to use the axiom of choice in some way, because the analogue of this series of flags only made sense to add all of the points, not one, two. So what I have to do, if I go to an arbitrary card from Dunbar, I've got to order them at all points, and of course I can always do that by the well-ordering theorem, but that's equivalent to the action of choice, so what that means I don't know, but if you want me to be fair to accept that, then I think it would be possible to develop this type of thing and get a genuine sort of group structure anyway, which maybe you could really use to study putting hats on D's, as well as on tors, as well as just on the manual metrics. This is just to go through the real line. Yes, no problem, it's interesting, obviously. Okay, I finished, thank you. well i've read their paper but i don't think it really fits in at all um they of course are talking about things from a perspective i don't think it's a real connection i actually well i don't really agree well they're trying to strip away practically everything really um they almost end up with just conversation at each point it's very difficult i mean i just don't feel it in the covariant type of approach Yes, but I don't think, no, that's right, but they haven't done the patting, that's the trouble. You can always quantize at a point. I mean, if you have a manifold, you always take a tangent plane and quantize on that. This is what they're doing. The problem is that they're patched together in a way that's meaningful. Maybe they don't have anything about dynamics whatsoever. No, well, I was never right. I can't make... Is there, in the representation of physics, something that recalls it? How do you think of it? To be absolutely honest, I finished this so recently, I don't know. I haven't had time to look at the recommendations. Plus the fact that Imperial College is a rough place to work in these days.

12:30 Sorry, I can't hear. In terms of choosing the representation, one thing we think you would do with the kind of quantized gravity we know of, plus the unexpected things, just like in early days, we can't know about in terms of electrons, but it has to be things that we wouldn't really see at the biggest scale. So I'm just wondering whether... The only thing I can pick up from sitting here is the fact that either we pretty much have the trivial representation... Well, in a sense it does. I mean, that's a very good point. You see, when I said, for example, that GAB, GAB was phi, you know, it was an extra field in the theory. I think that's correct. I mean, again, I don't understand the significance of this, but there's a certain sense in which other fields tend to arise automatically as being needed to completely... In which, in some sense, all those points lie in a one-dimensional space, and your group would have a subgroup which would be the one-dimensional group, the morphism group, and would there be a relationship between the... The dimorphism group? Well, because, in some sense, the transformations, homomorphisms are... Ah, no, you see, that would break the gauge. You see, as I said, I've really chosen spheres of flags, of flag spaces, and each one's had to lie in that, and of course, any permutation would take you out of that. This is a gauge-fixed quantization. I hope the phrase is correct. So I can't answer the question. I fixed the gauge in doing the quantization. Can I just ask it more generally? Is there some sense in which diffeomorphism groups should emerge as subgroups, as we've more generally talked about? That's not what happens in ordinary quantum gravity, because the diffeomorphism group is not to be added. It's not the case that the complete group of representations we're studying is the diffeomorphism group plus the canonical group. The canonical group alone is sufficient. The diffeomorphism group is added on top, like Hamiltonian is added on top. All you can say is the representations which you choose must be extendable to the people. I had a remark. I'm, of course, very sympathetic to your part of trying to insist in topology.

15:00 The right of way theory is topology. I don't... Go on. Right on, right on. I would like to suggest this is sort of a simpler example because I get excuses all of the technical mathematics that you have. It does effectively the same thing. If I write a quantum theory of the boundary for the number of vertices that are on the boundary, then I can imagine that it's matching the space back together and I get a quantum And since I haven't said anything about the metric structure inside the disk, I haven't got theory of quantum gravity because I would have to somehow induce that from the evolvement now, but there's some perception that one has to quantify the metrics and the topologies separately because at least on this disk you can answer Bryce's objection that the index set has changed. This is what you said in your remarks about the ordering. The ordering has changed on the space, but not the disk, so you get two pieces of quantization, one quantizing the sections of the fiber bundle over the disk, and one quantizing the boundary of the disk. We've got the species of any representation of the species. Face-based, you can have that. But they're very sparse. If you weren't the values of the pattern, you wouldn't have it. But that's also true of representation. And that is roughly speaking the reason why the one face-based is the same as one representation. In fact, I don't think that's a good idea. Well, I mean, obviously the representations are classified by the value of the Casimir's, in some occasions, that's perfectly true, but you have to decide, I mean, the question is, what does quantum... See, if we take the case of the two-sphere, the simplest one you can imagine, configuration space, and someone simply says, what do you mean by quantum mechanics on the two-sphere? Now, as far as I know, there's no absolutely definitive, God-given answer to that. All that one can say is that this particular approach suggests very, very strongly that you ought to consider, on equal footing anyway, as we've done some physical elements, the representation of an internal magnetic...

17:30 And so on and so forth. The first is that, that is true if you work in, if you work in studying representations. If you work in a scale of polarizations of holomorphic functions or fragment representations, the way functions are just holomorphic functionals on the one particle. And the one particle space is just a respectable function. I'm not saying there are three functions. In that case, one doesn't have this function. The second is that, even in the case of... Even in that case, the way that one normally constructively goes about doing it, as opposed to looking at the final picture, is indeed looking at field configurations at an initial step of time, which are, say, on R3, which actually have some fall-off and are normalizable with respect to some energy norm or some norm or other, so it's a respectable function space, not a solution space. But you see, that's only true for Gaussian measures or Wiener measures. That's the trouble. I mean, that use of that triple of spaces is very, very special to the Gaussian memory.

20:00 And, for example, I know a lot of representations of these groups, but they're not all that tight. So, I agree with you, but it's only true, I think, for almost three fields. The second thing you did is low dimensions. It may be true for a class of majors, for example, Duncan. For low dimensions, for the exact same level models, the boxes of the two, the right problem, then it is indeed true. Well, that's true here too. I mean, the singularity distribution is very dimension dependent. Of course, I mean, I don't know if that's right. I mean, in lower dimensions you will find representations. My marks are very much at three dimensions and above. Well, you see, I don't really think you can quantize gravity, honestly. You see, the purpose of doing this is not really, as I said, to say this is the way you should quantify gravity. What I'm trying to get is some idea. Is the mic on? What I'm really trying to get is some idea. You see, I really do believe quite honestly that one cannot use differential geometry. I'm trying to get some feeling for the type of spaces which occur in quantum theory by doing it this way. It doesn't mean to say you can therefore use this to quantify gravity, because the constraints, I would say, were themselves only low energy limits or something. I don't want to put myself on a particular model or another, but I would have thought that was true, but I don't see why you should want to quantize, actually. If you can and it works, that's great, you know. I don't see, after all, these days, why you should want to do that. It's really more to get a feeling of what the structure is underneath. That's all. It's a very modest aspect, in that sense. Divorce from the very beginning kinematics from dynamics. Form the dynamical arenas. But you do not worry at all about what possible dynamics to move in. And this gives you a bewildering set of possibilities. Do you think that it is a truthful strategy? Namely, that it is not better with the view of the dynamics to construct the particular representation that can carry the dynamics? Well, really the answer to that is always the same as the last question. Because if I really believed in my heart,

22:30 Quantizing. My answer would have to be yes, because I don't really believe that's possible anymore. As I said, my aim in doing this is not to write down the quantum of gravity, just to get some idea whether it makes any sense to talk about quantum topology as an actual piece of mathematics rather than just drawing pictures of wormholes. So to be honest, my thing is I don't think you can write down, as I said, ordinary dynamics. That's why I've stopped voting about the problem. But if, as I say, Abbe sort of wants people doing this to get the answers out, of course, But Bohr didn't get very far without the correspondence principle. You at least have to have a classical limit there. What plays the role of that to tell you at least a general tendency to write? Well, that would presumably come from representation theory. I mean, there will be some representations which have states, for example, in them, in which the metrics look, you'd expect, or whatever it is, would look reasonably like that sort of thing. You still haven't got dynamics there. All you're talking about is representation. You do have to have something. No, I agree. As I said, I... I freely confess this, I've changed my, if you know what I'm trying to do, I'm not trying to quantize gravity in that sense, but I don't think you can do it this way. It's just too singular. What I'm trying to get is some idea whether it really makes any sense to get rid of differential geometry at a blank length. That's all I'm trying to do at the moment. I absolutely agree with everything you say. I can't even remotely imagine making this sort of stuff compatible with the weird-wit equation. I think it's very, very difficult. That doesn't apply to my remark about the internal symmetry, you know, the sort of spin length. I think that's a bona fide comment separate from that. But as far as topology is concerned, I don't think it's possible. You mentioned that there's a little bit of gravity. Do you know any of the, and also, are there any, for example, finite-dimensional ones that are familiar? Yes, there are some finite-dimensional representations, which of course are not very interesting. I know that, thank you. A fair number are known, but the trouble is that there's a great gap in mathematical literature in the representation of what you call non-compact current algorithms. That's what we were talking about. And to some group, G, which is non-compact, rather than compact, there's almost nothing in the mathematical literature, other than what Goldfrid tried to do. And that's what you need to do to get anywhere. Can you speak up, please? I wonder if I could get you to say some more about this idea. Well, the statement is that if you regularize the way that Clowder discussed, and that is just in itself, of course, the Pety Patel point that you should do that,

25:00 but he considered the regularization of local theories, which this type of group tends to be consistent with. I'm not talking about the local... If you do that, it involves handling products of delta functions in a certain way, which when from the point of view of general relativity is incompatible with the diffeomorphism groups. It's a funny thing though, I mean delta of f is not a, I mean basically it's not a scalar, it's a scalar density of something. So if you start throwing away delta functions in the way that Clowder did, you find you actually change the transformation problem of the diffeomorphism group. Well, the way I've seen it, I mean, the way I know when it comes in, it's a separate field. But possibly, I don't know, I mean you could try, but it's very difficult because unless you're careful you find yourself contradicting yourself. You start thinking root-det-g, but then root-det-g's got the wrong way too because det-g has the wrong way. You probably don't know where you are. You try to find a self-consistent solution. We never actually managed to do that. We didn't look at that. It's very, very complicated. They're much simpler if you have this field on the right. One might think that string theory would be a good example of your methods. Yes, I agree, it's very frustrating, but I've not benefited in any way. Why is the situation different in terms of what you want? I mean, why are you trying to detect the problem? It's not what's appropriate or not, it's just what one can actually handle. I mean, if you, there's no obvious, these groups with these things like Sigmund Fusen and so on, there's no obvious two-curve cycles to start. I mean, there's not any obvious projective structure.

27:30 If you just look at the structure of constants, they're just totally different. It just doesn't work. I mean, we've tried playing with these. But that's just the, why the mathematics given to your methods? Of the string? Yeah, of the string. Well, in a rather trivial sort of way. I mean, do you mean sort of a single string propagating a... Yeah, I mean, you can study a single string propagating on a torus or a sphere. Yes, of course you can do that. It doesn't actually give the same represent... That's an interesting question. It doesn't give exactly the same representation as is common to me. In string theory, because there's a, well I think there's a 60-fold story, but it is actually slightly different than what you get. I mean, there's an overlap, but the groups aren't quite the same. But do you have economical groups for that? Well, yes, I mean, we've looked at that, and as I said, I mean, you can do that. It depends on what the target space is. If you've got strings on toy or something like that. Yeah, you can do that. And the group which you get is not what people normally quantize. What group? Sorry? What groups? Well, you've got the loop group, something like E2 or En. Yeah, that's it. You've got the loop group or the non-compact group. Now some of the representations of this group you can write down, and they're the same as the ones people actually do use on the screen too, but there are other representations that we have to be quite different. So in that sense, yes, there would be a prediction that there should be another way of quantizing from the free stream, which would be radically different from the way it was normally done. But again, I don't know what the representations are, because this group of the games, we've got functions that are non-compact. It's a non-compact cat movie. They're looking at loops. I mean, loop groups of compact groups are widely studied, but loop groups of non-compact groups are so different from the fish. Some representations we know, right, I mean, we found out, and they are more or less what we would expect, but it's clear that it's with the others. Could you say something about the relation between your program and the super space program? You mean by super space, you mean winners? Yeah. But that one would pose a very... I propose a challenge which I've never been able to actually answer. Maybe we should use that for, that part or... It's easy enough to say that I've been talking about, for the sake of argument, the configuration space in Riemann's signal.

30:00 If we followed Wiener, when we're faster out there between morphisms immediately, we get Riemann's signal over Dixit. Now, I would dearly love to apply a book-free conversation to this. I really would make you feel very happy if you thought I could. I really would love to do this. But the trouble is, I'm not actually informed the point is for that, because the groups which you can have action on the, see where you've got a flower bundle like this, there's the remaining signal, here's the gauge for the orbits, and here's the quotient space. Now the groups I've been talking about don't preserve the orbits, that is why they should, so this little crystal will go like that. So it doesn't project down to give you an action on the, you know, super space. What's mean is it's a totally different group altogether. Now we do have a very very non-local group. If you factor out the different morphs, it wouldn't be so surprising, perhaps, if the action evolved by non-local things. It's an old question I think people have worked on many times. What are the natural observables? They are the integral. The integral is the wild squares. And they will be, somehow, the natural things that handle them. That's an actually different sort of question. I wouldn't like to know the answer to that. Yes you can do that. Yes but of course as you know the problem is to find I mean the perturbation which is very singular. Yes I mean the whole structure is very compatible with taking g goes to infinity. The g goes to infinity, the Hamiltonian is left, is beautifully compatible with the natural Casimir operator, the groove. So it's very very easy to quantize. And obviously what is suggested is a one-over-two perturbation theory, but the trouble is, at least in all the representations I know of, these very, very singular ones, the perturbation, which is a three-curvature, is simply too badly defined, it's too singular to make a go of it. Otherwise, I mean, that would be a very nice way of tackling this. Well, probably going to be thinking about this a long time, but it's unsuccessful.

32:30 Subtitles by the Amara.org community You can formulate it on convex sets where the phasings represent the pure states and the mixtures are all the same, and you don't introduce amplitudes. Oh, I'm sorry. I missed the interview. Yes. And it's pretty clear how to generalize a convex set. I see. Sorry, I missed the interview. I didn't really focus on that. I guess Gary will probably discuss this later, but it is come down in such a way that Yes, I, of course, agree with that, although I'd be surprised, actually, if string theory, for example, came up with a picture of space-time which was not different from manifolds and totally different, nevertheless was topological.

35:00 Yeah, but the question is, no, I agree with that. The question is, is there anything in between? If you've got smooth manifolds down here and string theory up there, whatever it's got... And here's the descent from heaven to hell, as it were. Then, of course, you're right. It may be that one goes down here immediately, but what's in between? It's nothing at all the sort of things I'm talking about. I agree that's a possibility. Although, another instance is, of course, energy is a non-logical advantage. It's not that you get no information. Yeah, okay. Well, it may be that quantizing that background is all you get. That's possible. I would like to think that there may be something else you can get. Something where you're still using some topological ideas that are no longer manifold. Thank you for your time, and I hope to see you again in the future. But tell me, sir, that doesn't your heart beat at all at the end of the day. You're wondering, you've got to de-eventualize that. Absolutely, absolutely. Warm or cool, I'm not sure, but certainly it's a matter of some concern and interest. I'm just saying that, like the questions in the past have often been bypassed into mathematics,

37:30 but here is an actual, I would say, it's kind of misguided as a possible model that we might end up bypassing. No, I agree, I don't think it's true. Some of my best friends work in this area. The other thing is actually something which I dissuaded my neighbor from asking the question, and I should read it, but... Wanted to let your neighbor ask it. So I won't! Let your neighbor ask it. I haven't whetted our appetites, I think. This is the question of the currently advocated ideas of... Oh, yes. Zero. Yes, although it was not incompatible. I'm sorry. I was taking value of the zero. Okay, I don't know if the organizers have any announcements before we break. Okay, let's break until 11 o'clock. Hi, I just wanted to say that these conference proceedings are going to be published by Verkuyser and I would really like to urge you all to get your manuscripts in by the August 31st deadline date so that we can stick to our production schedule next year. And if anybody has a problem with that, we'd appreciate it if you'd let us all know early. How did I know that? You're no physicist. Yes, I do. The funny thing is, all of these conferences nowadays, you're invited, no word is mentioned of this.

40:00 Because I suppose it's assumed that you're going to bring a manuscript. Well, we're giving you three whole months. You know that 10 to the 18th words have been spoken since the beginning of language? Not all of them ought to be immortal. I don't think she's responsible. I think you have to get the organizers. Thank you for putting a manuscript together, and I'm going to leave them back here where all the pads and things are, and if anybody has any questions, pass them around. I have some general questions. Our immortal words are already being recorded here by the microphone apparently. What's the relation between that recorded message and the... If you want a transcript of your talk, let us know. Otherwise, we'll assume that you will prepare your talk and we'll only give you... Can we correct all the errors in the talk? Retroactively. Yes, can we correct in the final manuscript all the mistakes we made in the talk? We would prefer you didn't immortalize mistakes if you could. Not knowingly. Alright, well I think we should leave for the special discussion period, further discussion of the proceedings. Our next speaker is Abhay Ashtekar, and he's going to tell us all about how to solve all of the old problems by using new variables. Well, there are going to be a few disclaimers, but the first transparency is... Well, first of all, I should say that... Any questions? That's the answer to physics.