Loop algebras & loop-based representations in quantum gravity; subsequent discussion (last part)

0:00 We don't have the functions to say it's non-vanishing except when you're on those points there, but for now let me introduce the idea of the regularization. I don't have to do it for the classical theory, but I might as well do it here, and then I'll just discuss the classical regulated algebra in relation to quantum regulated algebra. These integrals are not invariant in the re-parameterizations of this curve. Observable depends on the choice of parameterization of the curve. Therefore, I can use it to introduce a regularization and I introduce a parameter epsilon and simply prevent by the limits on these integrals, and there are fancy ways to do it, but this will do here, prevent these two points from coming closer to each other than the parameter distance epsilon. And that will prevent the delta function singularities from coinciding. So that's the general idea. If the curve was self-intersecting, that wouldn't work, and I'd have to have a more general procedure to keep points away from self-intersections, but I'm sure you've already seen enough formalism. I don't want to complicate it more. I didn't need the regularization. The definition of T1 is parametrization-dependent, if you look at it. The definition of TZ, the holotomy obviously is not. Yes, it is because I told you in half a sentence that we're putting the distributional weight of the delta function on the first index, therefore the dependence on the second index is not distributional, and there's nothing to cancel the transformation confidence of the GS and the DT. So even before I regulate, things are not in characterization as I discussed. Yeah, Richard. Oh, that's taken into account in the way the thing broke in a joint. That's been done. Yeah, that's what exactly what this is, is the action of those representation matrices and using some S-E-P-P-N-I-D-A-N-E-S.

2:30 Well, you can get rid of it. I don't have to...yeah. Okay, now, this takes a long time to explain, so I'm sorry if I'm pushing through it. As the form of a Tn with a Tm is a Tn plus or minus one because the Poisson bracket always meets one sigma tilde and the regularization is always preserved because there's always some fixed parameterization on these curves. And I think given what's happening to time, I skipped the example of a T2 with a T0. I can show it later. I will also skip, but I can show later the claim that this algebra is complete. Essentially, what we're saying is that any gauge invariant local function of sigma tilde and A can be represented either in the algebra or by limits of shrinking curves down. The essential idea, of course, is if you get F from polynomies, if the loops get very small. I'll bring that back when I need it, okay. So that is the algebra. It is a Poisson algebra which has this fixed parameter epsilon in it which has to do with the parameterization of the roots which for every epsilon is closed under Poisson brackets and what I'm going to do is construct a quantum state space in which there's an algebra of linear operators which is isomorphic up to a deformation terms of that algebra. Absolutely, and all of this could be done through Yang-Nor theory, up until the point in Two Transparencies when I begin to study the effects of this analogy. But it could all be Yang-Nor theory. The representation is going to be, as Carlo defined it, so we're going to be interested in any countable sets of parametrized piecewise differentiable non-degenerate loops in the three manifolds, and a loop functional is going to be defined as a map from any set of those to the complex numbers that have, as Carlo said, this box-based structure. Later, we're going to put some restrictions on this space...

5:00 In order to get a better representation of quantum gravity, but for the discussion of the algebra, I don't need to do that, and now I'm going to just simply, ab initio, invent some operators that act on the space. Okay, so forget everything you just saw. I'm going to invent an operator which is associated with every loop called t tilde, means it's an operator on the loop space, acting on a functional a evaluated at some set of loops eta. This definition is the value of a at the set of loops which is gamma together with the set of loops eta. So that's my definition of T0. T1 has a similar definition, but this is not related in the beginning to the T1 that we just saw. T1, which depends on a loop, on a point in space in an index, acting on a loop functional evaluated on loop eta, is equal to the same singular structure function before. We put in a parameter h-bar there, because this is going to be quantum mechanics, These are evaluated on the two loops that I get by breaking and joining these, which I said were gamma-sharp-eta and gamma-sharp-eta-inverse. So this defines a linear operator on space of functionals of loop on Arvinicio. Its action happens to resemble what I wrote down before, but we don't need what we wrote down before to define this operator. I showed you the classical T2, so let me give you the quantum T2. It's defined in a somewhat different way. So here's a curve gamma. These have two, are defined as functions of the curves in two points. I have this nomenclature that I like a lot, which is that these points are hands, and we act on the loop functional, we evaluate it to curve eta, the result is going to be zero, unless the curve eta and gamma intersect twice or more, and the points x and y are where they intersect, and then it's called, these things are called hands, and they're grasping onto the loop eta. And when they do, the resulting action is h-bar squared times one of these singular functions times the sum of all the different ways to break and join loops at the two points, the a's evaluated at those things.

7:30 And there are four different ways to break and join at those two points. By breaking and joining, I mean I'm making a transformation from that to that. And that, this is a transformation, by the way, that in graph theory and math theory and combinatorics is universally important. So this defines these operators T2, X1. And this is the way the sign convention that I mentioned comes out. Now, the statement is that these Tn's, I can go on and define Tn's. And they have a commutator algebra which you can go and compute, and the statement is that it's equal to the classical algebra plus high order terms in h-bar, particularly a tn with a tm is h-bar times a tn plus m minus 1, and this is just the classical Poisson algebra, plus h-bar squared, if it acts twice, times a tn plus m minus 2, and so on, up until all the n's or all the m's act. Absolutely, absolutely. This is a defamation, it's not an isomorphism. It's an isomorphism on the small algebra C0 and C1, otherwise it's a defamation. Yeah, yes, yeah, sure. If you go through the whole thing and the dimensions of these singular functions and everything like that, it's all... Let me do that in the discussion, because I'm a bit worried for time. You always divide by 1 over h-bar when you increase the limit. I'll skip an example, but I'll just show it. A T0 with a T2 has first of all the h-bar terms where you break and join singly, and then the h-bar squared terms where you break and join simultaneously, that's the two things. I'll come back and show that. All these things can be worked out in great detail.

10:00 Let me, how much, how much time, I don't know when I started. Five minutes, okay. In addition, there are some, in order to get a representation of quantum gravity, we'd like to impose some further conditions on these functionals. And these are the conditions that we impose. First of all, at t0, we impose that taking the inverse of the curves, And we impose that these functionals should respect the basic spinner identity, which is written that way or that way depending on how you like to think of two spinners, and that leads to a relationship among these A of loops. And we impose that the functionals of loops should be reparameterization invariant, which... It's not inconsistent, I won't go into details, but it's not inconsistent with the fact that the algebra of the operators is re-parameterization dependent. Now, given that I only have a few minutes, let me focus on one issue, which is what we mean when we say that we have a solution to the Dictymorphism constraint. Then maybe we can discuss the Hamiltonian constraint later. Now let me begin by contrasting with the metric representation where we believe we understand this problem. Let me first try to convince you that we really don't understand much about this in the metric representation. In the metric representation, yes, we can write down lots of dictyomorphism invariant functionals which are formally annihilated by the generator of dictyomorphisms, but as Chris was saying, we have no classification of this space. First of all, we don't really understand what the space looks like. Secondly, if we'd like to understand The relationship of quantum gravity to topology, then we have to understand how the space of diffeomorphism invariant states depends on the topology of the three manifolds. And this raises an issue that we know nothing about really, which is representation theory of the diffeomorphism group and its relation to topology. Certainly it's very obscure to try to get at these things in the metric representation because the fundamental topological invariants that come into the theories of three manifolds like Thurston The homotopy and it's very hard to get a homotopy with this. Similarly, we can formally write down operators which are Dickey-Morkism invariant, but they have the same problems and they all involve very singular operator products. So we really know that we can do this in any real sense. Now, what I want to discuss is how can we set up and...

12:30 So, how do we solve the diffeomorphism constraints in the loop representation and what do we mean by saying that we've done that? So, first, we have to define the action of diffeomorphisms on the loop space, and since we have the action on the manifold, it's straightforward. We just define the action of the diffeomorphisms on a loop functional by the action on the argument. From this, we can define, given a one-parameter family of diffeomorphisms, what we mean by the operator, which is the generator of diffeomorphisms, which we will take to be the definition of the diffeomorphism constraint. There will be a space, presumably dense, of loop functionals on which this is defined, although we don't need it defined on the whole space. Then finally, we can define an action of the diffeomorphisms on our regulated loop operators in the usual way. When we have an operator, we move the function, we act with the old function, we move it all back, and that defines what we mean by moving the function. With those setups, one then goes to work and one shows, first of all, that the commentator of the generators of diffeomorphisms on the spaces on which that's defined, on the dense subspaces defined, and one shows that the transformations of the operators is what you think they are, and most importantly and, of course, surprisingly, the regulation is preserved. The reason is that the regulator depends... On the parameterization on the loop, and if I have, say, some loop operator, here's a picture of a T3 with T insertions. Then, if I map that by a dictymorphism, the parametrization of the loop goes along, so the operators, the insertions, are at the same parameter points where they were before. Therefore, if this has a regularization on it, such that this vanishes because these two are too close together, say, in the parameter of the loop, it will vanish here. So we have a regularized algebra of operators with quantum gravity which is complete in the sense that the classical limit is a complete set of operators and which is regularized without breaking diffeomorphism invariance.

15:00 n is the Tn, whether it is the number of insertions. For each n, they transform among themselves. Just some comments on this and I'll stop here, okay? We then solve, as Carlos said, we look for functionals A which are annihilated by the generators of dickeymorphisms and they're constant on the orbits of the dickeymorphisms, so therefore they depend on the dickeymorphism equivalence classes of the set of loops. Which are generalized knot classes and link classes. And again, this is a structure which has been very well studied. So if I can summarize, what we want to claim about this is first of all, because the loop representation gives a complete representation for a regulated operator algebra for quantum gravity, we claim that this is the general solution to the Dickey-Morphism construct. Second, up to issues that we don't understand, and Chris always bothers us about this rightly, about whether this is an irreducible or reducible representation of the operator algebra, but conjecturing that it's irreducible, we then could characterize the class of Dickey-Morkes and his various operators as being operators that live on linear operators on the space of linear functionals over the generalized link classes. All right. We have very clear and close relations to the mathematics that people use to describe three topology and we can go ahead and study the dependence of the diffeomorphism in various states on the three topologies because the homotopy group is there in the algebra. So it's easy to understand. And four, the spirit of non-evaluance turns out because of recent work of Von Jones and other people to have close relationships with two-dimensional conformal field theory. And so we can use this to make contact with other branches of mathematics, which are of current interest.

17:30 So, I think I'll stop there. I'd like to show one last provocative slide, really about the last remark, which is the following. There's been so much excitement in the last years, and rightly so, And string theory is wonderful because string theory brings together a lot of deep and, for us physicists, new areas, and for most of us, areas of mathematics and lets us do new kinds of things with quantum field theories that we were never able to do at the non-attributive level. So, for example, there's the question of classification of two manifolds, the representation theory of the one-dimensional dimorphism group. These are related to conformal quantum field theory and string theory. It turns out, due to this recent work I mentioned, that the representation theory of the Brave group, which is closely connected to the theory of non-invariance, is connected to the representation theory of Diff-S1, and there's been a lot of recent work on clarifying these connections. And let me suggest that to quantize general relativity means to understand... At least partially, something about the representation theory of the three-dimensional dithymomorphism group, in the same sense that the quantized string theory depends on information about the representation theory of the one-dimensional dithymomorphism group, and let me suggest that in the future, this is a very optimistic suggestion, we may be able to fit all these things together. By connecting the mathematics that we need to understand quantum gravity with this mathematics in which there's been so much work connected with string theory, i.e. the classification theory of three manifolds, We want to understand that and understand its connection to the classification of two manifolds, and indeed there's much work by Thurston and others about the relationship between these things. We want to understand the representation theory of the three-dimensional different workers in the group, and I'd like to conjecture that if there's anything... And all meaningful about the fact that we found solutions to the dictomorphism constraints and they're classified by knock theory, it's because the representation theory of the three-dimensional dictomorphism group can have something to do with knock theory, and so the problem of solutions to the quantum constraints and general relativity, I would hope in the future, if there's anything to all this, will turn out to be connected mathematically to these other areas. And so...

20:00 Quantum gravity and string theory, while initially very different and probably physically different, although it's hard to know anything at the matriculative level, may share a common rich heritage, if you will, when we understand them, of mathematics having to do with two- and three-dimensional manifolds through representation theory and anthropology. Thank you. When you're fixing them, what is the fact that string theory is one loop? Well, string theory is understood at least to one loop, and I don't know anything about higher genus Riemann services. I don't know about higher loops, but let's assume that some of the experts are right and it is finite. General relativity makes no sense perturbatively at all. Now, these solutions are gotten completely non-perturbatively, and When I go to discuss what it means to solve the Hamiltonian constraint, let me just write an equation down, what we mean is that we regulate the Hamiltonian constraint with some regulator, so it acts in a well-defined way on the representation space, we act on some functionals, we evaluate them on some roots, we study the limits. All of these terms are defined by the positive power of the regulator, which is the sense in which the claim is made that the Hamiltonian constraint is solved. So let me stress that because what's being solved is a constraint equation, this equation is well defined without regularization. After the limit, the resulting operator is not well defined on the general representation space. It is, however, well defined on its kernel. Because it's zero on the kernel. The kernel is defined by the limit of it. So the situation, when a theory is defined with constraints, we have an option, I would claim, for the construction of the state space, and I'll say something about observables in a minute, I'm not making that claim, but for the construction of the state space, we have the option to non-perpetually define the state space via regularization, and we don't have to renormalize it. And the reason is that we don't need the operator to be well-defined on anything other than its kernel.

22:30 In a normal quantum field theory, we need the Hamiltonian well-defined not only on the vacuum, but on the whole physical space, and therefore we have to know how to renormalize it. So that's the first claim. That's the statement. Now, whether the fact that we can find solutions to this is spurious... Or as an indication that the theory really exists non-perturbatively. We don't know. To have some solutions, especially if we have time to discuss it, the solutions that we have, is not to have solved the theory. It's very likely that the solutions we have are some small degenerate sets, if there is really a full physical space of solutions. And even to have that is not to say that there's an inner product and a physical operator algebra such that you can get finite expressions by taking expectation values. So we're not making the claim that we have a finite theory. We're making a claim that we've made some progress in characterizing the solution space with the constraints, and we have a new toy to play with, this loop algebra, which lets us do things in a fashion which is both regulated and diffeomorphism-proven. I think that only time will tell whether, you know, whether there's anything really in the long run. I'm just conjecturing if there is, there must be some deeper mathematical reason for it. Yes? No, we can do everything for asymptotically flat manifolds. The way one does it is one We've stretched the space of A's in the self dual representation so that things fall off just like in the metric representation. One then wants a suitable definition for the loop space representation. Presumably, loops are either fixed in infinity or the symmetry group reduces to something like the rotation group in infinity. And one then studies equivalence classes of states where the dichromorphism group is restricted by the symmetry of infinity.

25:00 And I think this is a possible thing to do when we're in the middle of it. In the end, what we hope to have is a space of physical states and some operators which are defined in terms of surface integrals and infinity, which will both be good physical operators and have an intuitive physical meaning because we know about how to measure things in infinity. But it's just not done yet. It's rather technical and I hope we can move on to something more. I don't have a strong opinion about summing over topologies. I want to emphasize this problem for a given single manifold and a given single topology is a mathematically well-defined problem, although I'm very interested in summing over topologies, let me take a hard-nosed point of view and say for a single topology it's enough to know whether the problem has a solution. The answer is really the same as the answer to Stanley's problem. What I've done so far to describe so far what we've done is a prelude I believe to doing the problem in the asymptotic response case and I have I know of lots of observables that are going to be different workers in covariance or invariance and commute with the Hamiltonian constraint that I'd like to measure and the test of the formalism in the end is going to be are there loose functionals which are annihilated by the constraints such that I can define physical observables like the energy and the momentum and the angular momentum such that I can define that in a product That satisfies the reality conditions, which I know Carol is about to raise, such that, in the end, there are finite expectation values in the physical states of physically interesting planets.

27:30 But that's what we've been trying to do for decades. Well, the only thing I can say about that is that the... You've had us youngsters who have only been trying this for a few years have a hand at it. And if in another few years... It seems to me that either quantum general relativity makes sense or it doesn't. I don't have to convince you that it's worth trying to find out whether it makes sense. My attitude is that it's possible we'll show that it makes sense. It seems unlikely, but I'm very willing to grant that. If possible, we'll show that it makes sense. What's interesting to me is that in the course of trying to do it, we've discovered some new mathematical structures and the general relativity... Now, I'm going to try to push very hard that there's a simple, well-defined problem and we can solve it. Does general relativity make sense? If it does not make sense. But in case it doesn't, I still think it's worth trying. If there is a space of physical states, in the asymptotically flat case, such that there is a large algebra of physical observables that I can define and an inner product, such that the expectation values of the physical observables using the inner product are well defined, you know, things like zero, you know, five, earth, you know, things like that. That's, I agree. One has not really addressed physical kinds of questions in this form of formulation. The difference really has been that canonical approach people were trying to understand conceptually. However, one can go back to something like 3D and so on, where one also does some mathematics which is somewhat formalistic, and yet at the end of it, at least you get some physical prediction.

30:00 It is my belief that, particularly by looking at the general relativity, the canonical calculations, but through this connection, one can repeat some of those things. These are all things that I've not really talked about yet. Maybe this is by a number of CP problems that was supposed to be the beginning of such kind of dialogue, such kind of discussion. We're trying to at least make some predictions about, in the same sense that, I mean, I don't know exactly what criteria that one can hold against. I mean, if you look at QCD and look at what exactly are the predictions, where are the numbers coming from? What I would like to say, just as one learns something physical from a CP problem is strong interactions. At the same level, we will learn something here. The second thing would be the study of anomalies, which really hasn't been done here at all. For example, in Yang-Yen's case, with the ZQ anomaly, again, it's an approximate structure. But nonetheless, with the ZQ anomaly, one can actually make some predictions about the idea that, in fact, leptons are coming up next. You only will have even a number of those. The left-hand is the part of the Yang-Yu sphere, so there is actually a prediction of it. I think potentially there is a prediction from the Yang-Yu. Again, the idea is that if you look at the couplet of the formulaic series, the A, it is similar. I'm not saying that one can just copy whatever was there in the Yang-Yu series, but I think that there are some issues which are open, and I think that they should be answered. And I think there are some concepts now, you know, which one can actually be answered. That is really the hope. These are the things that have been done and I'm just waiting for the real light to break through. I can't be disappointed in the famous slogan of Edward Bernstein, the movement is everything. Well, that's the end of the course. From my point of view, because I think my attitude has changed since the last time. We talked about it. In the metric representation, I think it's fair to say that in some general sense, one understands the effect of imposing distance morphism, but I would like to claim not really.

32:30 We don't really understand the full import of the distance morphism symmetry. And people have been saying for a long time that if we could really use the distance morphism symmetry, then maybe we could learn some non-trivial things about the short distance threshold space and so forth. I think that, although it wasn't, I mean, when Carlo originally suggested that we use this representation, I don't know if that's what he had in mind the first minute, but it seems that using this representation, we can get a non-trivial structure out of asking what are the different morphisms in the grand states, and the structure has a richness. Mathematically, both intrinsically and in terms of the connections to other areas of quantum field theory, it's very suggestive. And the real reason for me to go on, which may not convince anybody else, is to try to investigate the relationships among these mathematical sciences. And that's just that later on what they're going to want to evaluate for empirical findings is the intergrate with respect to some measure I have with Helder. And that means, for example, the intergrate of our space with Newton. So it's going to become important to know what functions are measurable in that measure, for example. It's not going to be, when you're working formally where A and A is just sort of any function of A to N between A to C, but eventually you're going to come up against the question, is A really smooth in the measuring that we need to integrate on the data? And we have a simple example from string theory where we have some strings, we can make observations quite analogous to this, but actually we know that if we go further, it must be algebraic to string. And of course, we would like its advantage on the solutions to these constraints, but in fact it doesn't, because we were careful to define anomalies in the algorithm. So, and in fact those anomalies depend on what kind of space we add the operators on.

35:00 And the way we choose in order to fix the anomaly, we have to restrict the space the operators act on. And the way we do that is that the states have to be integrable in some measure that we're going to use to evaluate expectations. So it seems like there's a whole new scheme in giving down what we mean by these strain operators and ultimately the fact that states are annihilated by their constituents is supposed to translate into a statement about a certain invariance of a group and that can only come about when this other ingredient is produced. Can I respond? I think you're asking, or I would answer it as the answer to two separate questions. An inner product that one would construct from this point on, and we have ideas and no particular good ones, such an inner product cannot involve an integration over loop space, and the reason is very simply that in the absence of a background metric there is, and I don't know a theorem, but I can't imagine how it can be, maybe Chris knows, so I imagine that there is no diffeomorphism, invariant, covariant, quasi-invariant. The idea of imposing an inner product from this point on is to use strongly discrete but infinitely discrete structures. We have a countable basis and we're going to have to use it to get anywhere. So that's the first Until Carlos showed up, we were wallowing around in integrations over loop space and integrations over the space of A's and trying to make these integrations make sense with respect to diffeomorphisms, and in the absence of a background metric, which is what we need to do non-perjurberant quantum gravity, there simply are no such measures. Just as far as I know. So that's the first point. The second point, I think I, well... Other people may have something to say, but one can define these things, and well define the regulated operators, one can study their algebra. And one can study their algebra without imposing inner products on the unconstrained space. Now, that algebra may have high order terms in h-bar, it almost certainly does. Okay, those terms will have to annihilate the same things that these annihilate, but they may be there. One conjecture that Carla develops about them, which I believe at the moment, is that they're going to have to do with, they're going to close on things which shift along the loops, which reparametize the loops. But that's an important problem to be studied. And it may be that we don't know yet how to regularize this in ways to...

37:30 As to describe the most general, the real general space of solutions. I have no doubt about that. But I don't see, if you're saying yes in string theory, you need an inner product to do these things. In string theory I understand why you do. Here I know that I don't have one. I'm, before I impose a constraint and I'm forced just to study these things. But I... I don't see why I can't. If I'm careful I don't make any mistakes and I show that that really is bounded for every class of functions and so forth. I think that if you go back to what you are actually asking about the answer, because you are actually not going to get it, you are on the right-hand side, you need to get that. That's the reason why we don't want to know the story. Now, the answer is clear. In the moment when you show just the conditions of the results of the study, that means that, at least for this setting, If we move on, as you were saying, you were saying, um, if you think, you know, that the problem is only in the first half step of the graph, that, that's exactly what it is still. That's, that's a set of states, which are in the middle of, like, all these states. They're going to be in the middle of the state, uh...

40:00 Well, I have many questions, but why don't you answer, Ivan, then you know that I will ask. What does it have to do with the real world? Real is the technical sense, as opposed to complex. The answer is that we don't know how to answer this question. At the moment, the question becomes, I think, the following. Once you have some physical observables, then you say, here I have some physical observables. I would like to introduce a new product on the physical space such that those physical observables, whose classical limits are real, are permissive. That's one criterion. I would like that new product to satisfy. I don't know how to connect. I mean here, abstractly, we have some physical observables. I can write them down as operators on bin classes, but I don't know what they classically correspond to. Furthermore, as Carlo has stressed, we don't, in the context we're working in, for a closed universe, in the classical case, we don't know any physical observables explicitly. So we can't apply the criteria until... I think, until we go to the asymptotic and flat case, then we'll have lots of observables which we understand are classical correspondences, like energy, magnitude, momentum, and the idea will be, is there some inner product such that these things are all emissions, and... When we get there, the answer may be yes or maybe no. When you are applying that extra-thin string, the relative condition, in a different way than you apply the super-Hermitian string or the super-Hermitian string, you may apply it only in the mean or in the sum. So this is the same thing that happens if you take a one-dimensional quantum mechanics, if you take another, a micro cylinder, and you consider the complexified case, and you take coordinate q plus i t and q minus i t, and you quantize this with a representation of the algebra of this object that we usually call a cross, a double, okay, that exists here in the complex domain.

42:30 Now, if you find an operator for A to A to A to A to A to A to A to A to A to A to A to A to A to A to A to I'm sorry, will you do a bit of math here, because it's been a formerly weird album, the opinion that was just given to you. The only thing I've got for you is that you look at it. Some years ago I came across a series of papers, some of which are called encyclopedias, where you would give a Russian mathematician a public school placement about the representation theory of what he looks at. He proves himself, and he shows every representation he looks at in the film. One thing which is sort of confusing... In fact, is that every class of the subspace, many subspaces, which are taking into themselves, might be looking more into the carrier and what it's going to do. If you talk about carriers on the roof, you know, you'd use them a lot. But of course, there are other classes of subspaces, which are obviously different from one another. These can only be mapped between one another. And actually, there's all sorts of things you can think of. Each one of those classes, you'd think, would be in the same class, which would support all that information you're looking for. Now, although from the point of view of erasers and things like this, you might be able to get people to know the loops, and ultimately, that's what we need to be able to speak to. But from the view of the representation theory, if you want to do it, I can make a few of you do it. Are you saying that the readers pick out this very, very small part that would possibly work in terms of the logic of the loops? What do you think I'm trying to say? I'm sure it's not the case for every representation you've seen. I'm going to say the ignorance. In string theory, you only use a small set of the representations of the first one, but in discussing with mathematicians that I've done for a little bit, it's very clear that the representation theory for three-dimensional diffeomorphism is wide open.

45:00 If you like, we're making a conjecture that there is a class of diffeomorphisms. Which are constructed on the link classes. And those classes and representations are interesting. But you know much better than me how much real mathematics has to go between such a statement and, first of all, a real rigorous statement of such a conjecture, let alone a real demonstration of it. So I'm going to completely ignore some of this. I mean, I really am impressed by all this, and it seems to me there's an excellent problem that even I want to try to do it, and that is the problem I discussed yesterday, which actually I think to a very large degree is a well-defined physics problem, and so it deals with equilibrium, and that is an extremely simple thing, but it was extremely necessary for us to understand that. There is a natural way to sort of represent in the old canonical variables what I meant by equilibrium in effect, and one can see in the new variables, There are a number of ways that one should be able to do the same problem, and the representation of what it means to have equilibrium will look quite different. In fact, it may look quite a good bit better and make more physical sense in this way of doing it, though it is the same problem. And since the reality conditions are simpler in the Euclidean form, and since the Euclidean form is not ad hoc, it's perfectly justified for that class of problem. I would suggest that an answer to the question about physical problems, where you won't get completely mired into really hard, important mathematical questions, is this very problem.