Quantum gravity on the computer (last part) / subsequent discussions (contd.)

0:00 Similar questions apply to any approach we are using, half-integrals or functional integrals, or various approaches, and in particular the obvious question here, I think is a very important one for people who are working in quantum cosmology from this point of view, is can you actually get beyond the semi-classical quantization of the classical background, because that clearly falls into the... Of course, what Rice is doing at the moment does get around this in some way, because Rice falls into the same trap. We discussed this last night. I thought you convinced me that it didn't. Are you convinced that it did? Well, what Bryce thinks is the metric to be flat out of the boundaries. But did we not agree that, nevertheless, you were expanding the metric itself about flat everywhere? So, in that sense, it's... Well, I can support you. Anyway, I think that Bryce's methods also get around... I don't know how to get rid of the fact that... No, well... Anyway, the question, a question which is obviously follows from this immediately from this type of analysis is, of course, GHAT-compatible. Now, the interesting parallel there is the nonlinear sigma model, again, of course, which Bryce has already referred to, and it is an intriguing question, this. I mean, it's a nonlinear sigma model, which I really mean any quantum theory where you have a configuration space, a space of functions, from some manifold into some other. So the configuration space itself is a nonlinear . Can you, in some sense, preserve that geometrical structure in the quantum? It's not a trivial question, of course. To put a hat between the Texas values and the manifolds is a manifestly impossible, and this, I think, is a good parallel, as always. Oh, I'm not demanding it. In fact, they won't. What I want to see is how far will they go. My real, sort of, honest belief is I want to try and set the whole structure to try. You see, we start off by writing g is delta plus h. Well, we know, for example, this is, again, we've said many times, the situation for metrics.

2:30 One knows the difficulty. A metric is, of course, a particular geometrical object. It's non-degenerate, has signatures. I think this is the most elementary model there is in a sense. We know that commutation relations don't work. There's an interesting point about this which is worth remarking, that when you, normally when people talk about constrained systems, which again intrigued me for a number of years, is this, which I've been intriguing by a long time.

5:00 The question again, of course, obviously applies to nonlinear sigma models. If this were a nonlinear sigma field, classification of such, I would assume, absolutely, yes, that's right.

7:30 But I think it's much worse than that in the case of that. Anyway, there are obviously general questions, and this is the aphoros status of toruses to be marked by an ordinary Riemannian drama to write down if g goes to g-hat. Now, you can see the sort of problems that might arise because, in answer to Raphael's question, I mean, my expectation would be,

10:00 but notice, it's not a null separation in the sense of null separations of light cones. This is much more dramatic. You know, this is a different thing altogether. This is still a potential, a lexidian. It started with one kind of topology. What I'm asking is that you can't do the second without having done the first, having done the g-hat. ...having done the ordinary topology of a differential geometry... All of this is done by drawing pictures, and when we draw pictures of manifolds, it's very hard to draw pictures. I just think this is wrong. I mean, it's okay if semi-classical, literally, in my heart, but down at the flank left, there's always these lovely smooth manifolds popping up. No, no, they'll, they'll, they'll, they'll, they'll, I think, given the manifolds, you can cusp it up. But the point is that if you always draw it, even from a larger dimension, embedding spaces, this is a distinct improvement. So what this really boils down to is that I want to claim that any real theory of quantum gravity, somewhere along the line, will involve spaces which are not... Now, this problem has been addressed by Jim Hartley, for example, in the context of Reggie Calculus, where you actually get the problem very, very clearly there. Do you sum over the triangulations of... There's nothing at all unnatural. Again, it's not such a totally wild thing. It sounds all right, but what does it all mean? I mean, for example, can you talk about all topology? Is this really a meaningful thing?

12:30 Fortunately, I anticipated that. Here's the question. First of all, can we even talk about old topologies, the operative word being old? Secondly, do they form a nice space, in any sense whatsoever? Can you seriously imagine writing psi of tau, wave functions of topologies? I have a claim you could say. It's a strange idea, that, of a wave functional, but I believe it is possible. Now, however, there's another question which arises, and it's, why topologies only? Of course, here we have this terrible tendency that theoretical physicists and men are prone to... I've got a book about topology, but there are many other things too, which in some way lie on the edge of this. For example, there are things called limit spaces, uniform spaces. I found a wonderful review paper about 35 different things. This is a perfectly serious question, actually. Why should you feel constrained to talk about topology? Now, the first remark is that all topologies form a class, not a set. You have trouble hearing people say, you can't talk about all topologies. The set of all topologies on a particular set X is a perfectly meaningful thing to talk about. And what that means in practice is only the cardinal number of X. If I had asked, for example, the set of all topologies, say they were different, would we do that? It means you avoid the logical problem. It means if you like the permutation, well, can you put topologies on top?

15:00 I remember many times the writer asking me the question. It turns out there is a way of answering that question. There are several. Recently there's a great mathematical literature on all these who have glued together lots of them. Time is going to preclude me from saying very much, but the first observation is that instead of all topologies form a lattice, say, if you say one topology has more open sets, there are other things you can do. A topology, for example, what is a topology? A topology is a certain class. It turns out, again, in the mathematical literature, studies have been made about putting topologies on closed subs. There's been many topological studies. If you can find a topology on here, then. And the intriguing thing is it is not a closed subs. This refers back to this question up here. You may find that to make the theory mathematically complete, you need to go a little bit beyond the topology. Now, what I said was that there are several topologies you can put on peer vector, which is to each one you can ask the question whether the floor is closed or not. If it's not closed,

17:30 that will then be your structure, which you could then have a well-defined space. See, our floor has no actual reason why you could put a topology instead of all topologies, as I've said. And the third one, which I just mentioned briefly, is the Gromov, indeed, only applies to metric space. It gives you a way of putting in metric It's not actually a talk, it's very much, isn't it? Yes, indeed. What I really want to make is that in spite of one microphone up to your eye, there's a well-defined and unique way, and therefore you really can't talk about distance, and therefore you could start to talk about quantum ideas, if you like, because that's the minimal thing you really need. Well, okay, I mean, that's sort of some of the general background things. How do I set about trying to construct and somehow take into account all these strange ideas? Of course, some of you here know full well the technique which I've tended to favour in recent years is a group theory method, and it's come as a pleasant surprise to me to find that you can extend the group theory method to include... Let me just very briefly summarise without going into the details of what these group theory methods are supposed to be. Here's quantisation, the conventional picture. You start off with a cut of a phase plane. If you start off with a classical phase space, and of course the observables are functionally defined on that space, of course there's a cross on bracket, and the wisdom is that in some way you would associate operating functions in such a way as not to preserve the problem, but to properly term small deformation. However, the problem is that, generally speaking, this only works for subsexual observables, and certainly the straight substitution does. This is a very well-known whole problem. And the obvious, therefore, crucial physical question is how should such a subsexual...

20:00 I mean, in a sense, this is what quantization means. You can't do it to all of them. The sort of question to ask, in a way, is what are the analog of these relations in the ordinary quantum? And the second is, of course, the x's and the p's aren't any old gestures. They generate all other functions. Now, the way they're called polynomials is that they, in some sense, are giving you all observables. Well, I think this is the question you asked me earlier. You can obviously choose coordinate patches like that. The problem is that the vector, the coordinate fields between them. Well, even if you don't have to do that, you still have to do the patching. It's the patching together which is... That would be a real quantum differential geometry, which is simply, on a general phase space, try and replace with some other set, which would do the same job. Now, what doing the same job means is that the algebra has to be small enough to allow...

22:30 So the problem is to try and tie all these things together. Now, since I don't want to give my entire LASIK lectures at this point, the results of doing this... It's all on one slide anyway, but I'll skip it. Let me summarize the results of doing this. Someone can make a very strong and plausible argument. It's one of the few things I've ever done in my career where I think it may actually be true. That's a very modest or arrogant remark, I'm not sure. The set of functions is the group G actually technically on S. In other words, if you can find a group which acts as a symplectic function, if you choose a group which has no subgroup which does the same job, you can possibly hit the van Holen problem. So in other words, there's a perfectly clear way of doing this. For large classes, if it doesn't, then you're... If you can find such groups, then of course you study their irreducible representations, and then you're on the way. And if you find the correct group of G, that's the whole point really. For example, the topological effects, the problem I mentioned earlier, the effects as normally deduced, I'm going to give you some examples of leading up to the cases we're interested in. The simplest case, of course, is Q equals RA, configuration space. Because the group which acts transitively on the phase space is just R2A itself, by the Heisenberg group, usual weight mechanics, so nothing to do with the loss.

25:00 What is held irrelevant is that the method works a very large class of spaces other than this. Which are where the configuration space itself is a homogeneous space in the form of... No, in fact, I wouldn't say that. If you can find two different groups, it would be a genuinely dumbified, illiquid quantization of that system. I mean, that's absolutely correct. It's not the case, even R2n. For example, I know that very strange quantization on the real number. The question about the energy is whether the energy can be added to the family. I mean, when you've got a family of canonical observables, in a sense, every operator can be written in terms of that set. Oh, in this case? Oh, yes. Anyway, here's a class of systems, which unfortunately includes the cases I want to talk about, where Q is d squared 8. Of course, there's not many configurations based on none of that form. So it turns out that all the many metrics are, in a certain unexpected sense, instead of all following the approach. Now, I'll just tell you the answer here, is that you can guarantee this will give you a good quantization scheme. If you can find any vector space W that carries a linear representation of the group with an orbit before that vector space W, Then you can prove that there's a bona fide, a pretty consistent quantization, which preserves the third set of the generators in the elements of the dual of W. The dual of W, of course, are linear forms of W. They induce functions on Q, so they are indeed functions on Q, that's it.

27:30 And the corresponding group is a semi-direct product. Now, this works for a vast class of systems, would be in general if Q was found. Although there are some funny cases where it can't be. It's very intriguing. Now, this group, of course, is a fascinating group because it's utterly and totally different from the bar of Heisenberg's group. Now, nevertheless, it's very important that you know a great deal about, because one could agree with an example like this. So my actual belief is that, genuinely, from this point of view, quantum mechanics in the real world is actually anomalous. Almost literally, I mean, the xp is i h bar, the i h bar is almost like an anomaly. And these really are the standard examples. Okay, so obviously one can do a lot with that. Here's a good example, and we'll work up from here on to the cases that we're interested in. Take the n-sphere. Now there's something which many, many people have written about. The quantum mechanics of the individuals embed SNRM up one, it's an orbit under it. However, the interesting thing about this is if you then look at the induced representation theory, a la Deacon and Murphy, you will find that the orbit, because there's always, you have to look at the orbits in the dual space, what you get amongst many others, well, basically the orbits you get are spheres, so that's good, that you have functions on Q. That was a little group, and of course, just as in the case of induced representation theory, in other words, what you find from this is allotation is almost never the case. Even apart from the Y-mark of distributions and so on, with the underlying group theory, there will always be an index.

30:00 If one can apply this to a variety of the relevant thing I think from the point of view of the quasi invariant measures on the orbit. I wouldn't underestimate these. Yes you do, that's right.

32:30 So why Unifred? How are the labels coming? Here I'm talking about... Oh okay, I missed out that part. We're talking about quantum gravity as a problem anyway. Now if you apply this to gravity you get some interesting results. You can work your way up to a series of models starting off... So all positive determinant matrices on each one of these yields for this treatment. For example, sort of symmetric sort of paradigm for quantum gravity. Symmetric representations of you get some intriguing methodical algebra just to mention that the representations again you have an index that's very strong in the case of gravity but also the indices here. You find the orbits on matrices is ranked less than n so you get sort of the general. This is in the finite dimensional case and you can generalize all this to the The situation there is fairly straightforward. They use a global frame to make the GAB well-defined. Usual C cells are no good. The obvious embedding is that you embed sort of all the manual metrics in the vector space of autism. Group B is a set of all functions from sigma to GL3R.

35:00 You find that that preserves and so you can write down what the canonical group is. The representations of these things, if you do, the question that asked me was what sort of singularities do you get? Easy representations can be found in various places. You get these distributions, or thoughts can be as wild as that. The other representations are concentrated on loops, which I find very intriguing in the context of the work that Abbe and his colleagues are doing. We have a new lab where you get these concentrated on loops. It seems to me there must be a connection somewhere. No, no, no, I don't say anything about that. I can always try that. The problem is that this is the minimal one. This is the one that's down to the three of the van Gogh's points, which is not a trivial point, Matt. You can find all sorts of spaces. The only comment I'll make about the various remarks here, I don't have time to cover, is that the little group here is actually very big, it's enormous, and has very big representations. Oh yes, you can always take the trigonometry, but I would have to show that that was the only physically relevant representation.

37:30 Oh, that's okay, that's what I'll define, no problem with that. Oh no, other than the usual problem of making sense of everything. Oh no, that's fine, I mean, the group theory looks after all that for you. You see, you've got generally, you have generated, like, pi. Well, the answer is I haven't done it. No, I mean, it's what we've been talking about this afternoon. What I would like to see, you see, is a sort of fun with this extra thing as it's on. I think that would be very intriguing. Well, you have to look at that. It's a bit like, who knows, indeed. In fact, it's a bit like internal spin. You thought you always had angular momentum. In the case of a sphere, yeah, the thing is, of course, it's a very simple space. You can argue all sorts of ways if you want to ransack the quantum. Representations from the sphere include... Well, that's correct, that's because I think quantum mechanics is more serious than classical physics, it's serious enough. I think it's much more meaningful to look for a well-defined quantum structure. Well, what do you get with these approaches? The thing about a typical group like this has several different classical limits. That's true. I mean, in fact, I think that's an interesting feature of physics, that certain types of classical things are stacked together. They basically fit together in a certain natural way, possible classical limits.

40:00 I agree. I mean, the whole object of the exercise really is to try and get a world-famous quantum theory. I don't really want to go from here. I want to go the other way around. This is really a way of getting into it. Well, another... if you study... Clowder looked at the problem some years ago, how he would regularize, just after passing the mark. If you do his methods, what I've discovered is rather intriguing. If you try and ask, for example, what is the inverse metric, given you've got a metric, or say, if you've got a triad, what is the metric, and so on... There may well be a requisition problem. G a la clave, what I've discovered is rather intriguing, is that you actually find that you're forced to say that G inverse times G is not delta front, but delta times another quantum field phi. That's absolutely forced on you 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 phi is suppressed, and we just don't see it, and so you get the delta.

42:30 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 somewhat compatible with the last couple of marks. This field becomes more and more significant, and you could even get sort of phase change in 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, it seems to be, well, I think it possibly would, see, what that corresponds to is different groups acting on the original phase space, as I said, and they may not be unitarily relatable to each other. Well, if you do, well, see, the triads don't really do that, because even when you 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 with it. So you can't get out with my clients. I know I once suggested that you could.