Canonical groups in quantisation of gravity / subsequent discussion (contd.)

0:00 All right, well I think we should leave for the special discussion period, further discussion of the proceedings. Our next speaker is Abbe Ashtekar, and he's going to tell us... All about how to solve all of the old problems by using new variables. It is really Jamal who convinced me that I should speak at this meeting, so if you have any complaints about this talk on your board, you know who to blame. Thank you very much for your time, and I hope that you enjoyed this lecture, and I look forward to hearing from you again in the future. The one that has been, sort of, come up several times already is non-perturbative quantum gravity, quantum general relativity, for example, a viable theory, or do we have to do something very drastic? I mean, we all know that perturbatively the theory has problems, and so perhaps one just has to do something drastic and go to either the string heaven or ten-newmans heaven or twistor space or something like that.

2:30 My feeling is that we do not have any concrete evidence that one is forced to do such things at this particular time, and so in a way all these attempts are sort of made at getting insight at this particular question. Well yesterday Murray told us a story that when he went to the institute for the first time in 1951 or something, somebody told him immediately that Mr. So-and-so had a non-portability theory of the weak interactions, the 4.4 muon thing, and of course it is still to come. And so that was supposed to be a message. I mean, I don't, I mean, I also have a story. It doesn't compare to Murray's, but nonetheless, I thought I would share it with you. When I went out as a graduate student to the middle of my state, Chicago, I heard a big advertisement that certain supergravity theories We are solving the whole world for us, and that was the theory of the world. In fact, we know, not, none of the people in this room, but other people have advocated such views. We know some distinguished people who gave distinguished opening inauguration lectures, titled, is n equal to 8, the supergravity, the final theory of the world, is the end of physics in sight. And so what is, I just want to feel that I've heard these statements enough number of times that one shouldn't take them. And the last thing is really that, just as an amusing remark, insight that I certainly did not have, there was a history of relativity and quantum gravity, such things, a very small conference that John arranged about a year ago, in this very place, in this very room, and there was a discussion about the issue of string theory at that time. And, to my amazement, historians of science just took a sort of blasé attitude, saying, oh, that, that's a typical end-of-the-century phenomenon. And I was not aware of this, but they gave... You gave the millennium, that's millennialism. But apparently this is a typical thing at the end of the century and various intellectual endeavors feel that the ultimate theory and ultimate goals have to be achieved or are about to be achieved, and I never thought of it in these terms, but it's not a serious term.

5:00 Well, more concretely, now these green questions are in the context of canonical quantization, which is a potential non-perturbative way of approaching the problem of quantum gravity. The old questions, some of the old questions are, can we have, can we find some general solution to the quantum constraint equations. If you can find the solutions, then on the space of solutions, one would like to introduce an inner product and make it into a Hilbert space. Are there any guidelines or hints for introducing such an inner product? What about the role of time, which already has come up a few times during discussion yesterday and we'll hear more about it tomorrow. In what sense is it true that quantum dynamics actually comes from quantum constraints? Is there some nice regularization schemes such as, for example, a lattice formulation? And we will see, so these are some of the questions within the canonical quantization program. Then there are some questions which are not properly within one program but have to do with the relations between various programs or predictions and so on. 15 years now, 14 years now, Roger Penrose and Ted Newman have these objects, mathematical constructions, for so-called nonlinear gravitons and edge spaces, and it's an intriguing thing as to what they have to do with anything in terms of gravity or what is their role. Another thing would be that one of the sort of standard criticisms about quantum gravity is that, well, for a long time, it is a non-vertebrative kinds of thing like canonical quantization and so on, is that for a long time we have focused our attention really to formalism, sort of questions of formalism and what about some phenological type predictions such as the ones that were used in particle physics, for example, the theta angles, the CP problem, anomalies, which may give some information about the number of generations and such things. In other words, really... Concrete things. Are there such predictions of quantum gravity? And of course, finally, is there some nice unified way of looking at all interactions? When we treat, say, electroweak interactions and strong interactions, they are treated in a certain way. When we take quantum gravity, the fundamental variables and so on are dramatically different. Is there some unifying underlying mathematical theme behind it all? Then more profound conceptual questions. What is the microstructure of space-time?

7:30 Then how does a continuum picture arise? We know that many theories of physics are renormalizable, the successful theories. In a certain sense, if we go back to the history of the subject of renormalization, then one can see that these were profound insights into circumventing the problem about what was happening at very short distance scale. On the other hand, if we do know about microstructure of spacetime, we should be able to do things from scratch, so to say. And then, what is it about the microstructure of spacetime that makes the various theories renormalizable in the first place? And finally the interpretation problems and the measurement theory. Well, the disclaimer is right here that none of these problems have been solved using these new mathematical variables to describe gravity. However, I hope I can communicate to you that actually some progress has been made in the first three category of problems of here. Or here one has some vague feelings and so far we don't see at all any really insight that we're getting by the use of these new variables for the question of the interpretation. So what I want to do is to focus on some of these questions and then communicate to you what has been sort of achieved and what the status is. Roughly speaking what has happened is that if one uses these new variables then The mathematical problems that we have to solve, concrete ones, change quite a bit and therefore we have some fresh directions to address these issues and there also are a few new physical concepts that have been introduced into the framework and I just want to give a review of that. Well, he told us many things, but one of the things was he outlined a certain program for non-quantitative quantization of things. I'm going to introduce these ideas based on new variables, and then Carlo Rovelli and Lee Smolin are going to tell us more about how to implement some details of this program. And what they are going to tell you is really what is happening at the forefront of this program.

10:00 And in a sort of a curious way, what happens finally is that there is in fact a new algebra of observables that has come up in their work. And this new algebra of observables is the one that one could use in Chris's general program in order to do the quantization. So in a certain sense, this part of Day's program completes a circle. We also have other talks about string theory. What do string theory tell us about relativity and so on. Well, as far as this particular talk is concerned, the organization is as follows. In the first part, I would like to overview the basic framework. I apologize, some people in this audience have already seen it three times or so, but there are some people who haven't seen it at all. Then I would like to sort of address this question about the problem of time. However, in the asymptotically flat context, and I would like to sort of say what the problem is, I would like to propose a solution, and I would like to tell you... In what sense, what we have learned about this problem, what insight we have gained by the use of new variables. Then I would like to sort of talk about two new representations, that is to say, ways of representing the space of states that have come about because of the use of new variables. The first representation is so-called self-dual representation, in which the wave functions are holomorphic functionals of self-dual connections. And the second representation is so-called loop space representation in which wave functions are suitable functions, suitable as to be qualified, suitable functions defined on the loop space of the 3-manifold. So I left the 3-manifold altogether, now we are looking at loop space and that is where one is working. Karlo's talk and Lee's talk would really give much more details about the loop space representation. But I thought in my talk what I will do is the following, I will tell you what these representations look like in the linear theory. Because then one would have some feeling as to what to expect and what not to expect out of these representations. So what we are going to see in particular is the way to do, say, Maxwell theory, just linear Maxwell theory, or linear gravity on loop space. Then, depending on time, I'll talk about two more applications. The use of new variables clarifies this problem and then I would like to sort of spend some time in talking about the possible role of non-linear gravitons and edge spaces in quantum gravity, the thing that has come out of the use of these new variables.

12:30 Let me then begin with the first part of the program which has to do with just a brief review. And the reason is because of basically simplicity of presentation and also then this way we will be doing at once Lorentzian relativity and Euclidean relativity. So what I have is a manifold which is real and I have a metric which is complex. The basic variables that in this new, the basic new variables to describe relativity are the following. First of all we have tetrads on this whole manifold. These are complex tetrads. The mu is just a space-time index and alpha is the internal index which tells us which tetrahedra vector we are talking about. And the second variable, four dimensional variable, is a connection. This connection is self-dual in the internal indices. So this is something that you can talk about without reference to any metric or anything of that sort because if you like on the internal indices there's already a flat metric, so it's not a space-time metric. So, which means that this is equal to i, is equal to i times its dual, if you're doing Lorentzian relativity, or is just equal to, forget about the i, if you're doing Euclidean relativity, so that's, that's the thing that we're, flat, eta mu nu. The integral matrix is, no, the integral matrix is real, it's just eta mu nu. I mean, the, you know, it's real, minus 1 over 1. The metric is then the spacetime metric, given that it is tetra, we can construct the spacetime metric, which is just given by this formula here, as usual. And of course, given the tetrad it also defines a unique connection which kills it. This connection acts both on the space-time index and on the internal index. Therefore, we have the Gestapo symbol for the space-time index and we have this certain connection which operates on the internal index. So, given the tetrad it also gives us all this mathematical structure. I just want you to keep this in mind which we might refer to as we go along. But the basic variables are these tetrads and the self-dual connection.

15:00 The Lagrangian density up here, and this is the Lagrangian density proposed by Joseph Samuel in India and Ted Jacobson and Lee Spolin up here, the Lagrangian density is basically you just take the tetras, you take the curvature constructed out of this A, and you just contract all possible indices. You just contract this, this, and you make it into density by the square root of G formula. It looks very much like the Palatini form of the action. However, it is not quite the Palatini form. The reason is because this A is a self-dual connection, whereas in the Palatini thing it is not a self-dual connection. That sort of is the key difference here. It is a first order formulation in which the Lagrangian independently depends on two variables, the tetra and the connection. No, not yet. It will be spin connection on a solution. Right now, it is just an abstract connection. No, that's it. It doesn't. And that is a surprising feature about the whole thing, that all the simplicity comes by making itself dual, and yet it doesn't over-constraint it. I'll just comment on it in a second. That exactly is a surprising, I mean, a priori surprising thing. If I vary this with respect to the A, then I just get the statement that this A-mean, the A that we have up here, the connection which is in the Lagrangian. This is a self-dual part of the internal connection compatible with the texture. So this is the field equation tells you that this is like the spin connection that is defined by the texture. Sir, that was self, A mu was self-dual with the... This is self-dual to start out with. But the script A is not. The script A is not. It's the usual spin connection. That is the usual spin connection. The self-dual part of it and make the Lagrangian out of that. That's right. I just want to comment here one second while you're looking. The second thing is that you take the variation with respect to E and then you find that in fact that just tells you that the metric G that you have satisfies the vacuum equation. Now one might worry, for example, that maybe there are more equations because, for example, if we are doing real relativity or something, I mean the Lorentzian relativity, then this Lagrangian actually is complex in the Lorentzian relativity. E is real, E is real, but A is necessarily complex because it is self-dual. So just to amplify Bryce's question, so one might worry that when I do the variation, I may get too many equations because there is a real equation which may be like this and there may be also complex equations, the imaginary equations that will over-constraint the system.

17:30 It turns out that the imaginary equations are always just Bianchi integrals and therefore there are no further equations. Therefore, this Lagrangian actually is completely equivalent as far as field equations are concerned to, say, Palacini-formed action. Then what you want to do is to do a 3 plus 1 decomposition and then do the Legendre transform to go to the canonical framework. So when you do the 3 plus 1 decomposition, then you obtain the canonically conjugate variables, and I'm just stating what they are. The first canonical conjugate variable is that you take this A, this is A mu, so you just look at its spatial indices, and here also you just look at the spatial indices, so to say, of ij. Instead of four-dimensional indices, you look at three-dimensional indices. So it is a spatial connection up here. And these i, j indices now are just, well, if it was real, then it would be SO3 indices. Now it is complexified SO3 indices. The second canonical variable is basically the triad. In other words, you take your tetrad, you look at the triad, which is considered a triad out of it, and then, however, what I would like to do, just for mathematical simplicity, I would like to have two internal indices on the triad. Therefore, I just take this internal index k up here and dualize it with respect to the epsilon, three-dimensional epsilon, thus I've got two internal free indices i and j up here, and I densitize it with the square root of the three metric up here, so that this can be canonically conjugated to that. So E, in fact, is basically a spatial triad, except for the density, and in fact, if I take trace of the internal indices ij, ij up here, then I just obtain the determinant of the three metric times the three metric. Now, what about A? How do I understand it in terms of the usual variables that we are used to? This A is basically the spin connection, three-dimensional now, spin connection, ij up here, defined by this three-dimensional tetra. So this involves the space derivatives of the triad up here and plus i times the extensive curvature term up here. In other words, the relation between all variables and the new variables is just specified up here.

20:00 The e-tilde is really like just a square root of the matrix, so it's not a profound thing. The new simplification comes because of the use of A, and what A does is combine some information about the space derivatives of the triads with the time derivatives of the triad. If you give me any solution to the field equations, then this A turns out to be just a potential for the self-dual path of the wild curvature. So these then are the sort of basic variables. What do they do for us? You can now write down the Einstein's equation in terms of these basic variables. When you do that, you obtain the following set of equations. You obtain certain constraint equations and certain evolutionary equations. Let me tell you what the constraint equations are. The constraint equations are of course all functionals which are built out of the tetrads or these densitized tetrads up here, densitized triads up here, and the connection on the three-dimensional manifold. The first constraint equation is exactly like the Gauss-Waugh constraint of Yang-Yu theory. It says that the gauge covariant derivative, which is just defined by this, where g is Newton's constant up here, is in fact equal to zero. So what this constraint tells us, it just tells us that the internal rotations of the triads, triad indices, should be thought of as gauge. Then you have got another constraint which says that if you construct the The field tensor out of the three-dimensional A up here, and you take its trace with E tilde A, in other words, you just contract this, this should be equal to zero. But in the usual language, one doesn't work with this FAB, but one works with the magnetic field, B. B is just a dual of this FAB, so what it says is trace of the pointing vector itself of E and A, constructed out of E and A, should be equal to zero. E tilde, E tilde, tilde is just to say that it is a density of rate one. E tilde A, E tilde B, FAB, which is really trace of E dot, E cross B, this equals zero. These are the constraints of the system. And then you have got evolution equations. For simplicity of writing, I have set shift equal to zero, and that is n up here.

22:30 And the evolution equation looks like E tilde dot is equal to the gauge covariant derivative of something constructed out of quadratic in E tilde. And ADOC is just an internal commutator between FAB and EB. Although this is quite a different form of equations compared to what one is used to seeing in Einstein's theory, this set of equations for the case that I'm considering, complex relativity, is completely equivalent to Einstein's equation. There's no more or no less content in these equations than in Einstein's equation. Actually people, yeah I think it might help. I mean in fact people, Nakamura's group in Tokyo, I don't know how, what programs they have done, but they are using these variables in order to do things, numerical relativity, classical, classical. Yeah, I'll just come to that. Now, I want to sort of look at these formulations, these equations that are up on the transparency and I want to point out a few features of all these equations. The first feature is that our basic variable is e-tilde with upstairs index and a up here connected to the downstairs index. All these equations are just polynomials in these basic variables. In particular, for example, notice that I never require e-tilde with the downstairs indices. As a result, there are some simplifications. First of all, these equations are meaningful even if these e-tilde were not invertible. In other words, even if you are talking about three metrics which are not quite invertible, yet these equations are completely meaningful. In a slight sense, this is a slight generalization of the standard Einstein's equation. This reduces to the standard Einstein's equation when each of those are all irreversible. The second thing is that if I just focus my attention on the constraint surface up here, then the constraint surface of Einstein's theory is embedded, as you can see, into the constraint surface of Yang-Mills' theory. Because what is the initial data for Yang-Mills' theory? It is precisely the electric field and air. So if I think of E tilde as an electric field and A as a Yang-Mills connection, then the first equation is precisely the Yang-Mills constraint equation. So to get to Einstein's theory, what you do is you go to the Yang-Mills constraint surface and you restrict it further. And the further restrictions are these equations up here. And these equations are just polynomial, quadratic and cubic, in the field strengths.

25:00 These have several ramifications, this embedding up here. The second recent result, which came out from Joseph Samuel, again the one who found the action, which is the following. Using this sort of relation between Einstein's theory and Young-Niel's theory, he has shown that the one instanton solution in SU is in fact an Einstein instanton with a cosmological constant. In other words, you take exactly the same mathematical object, wherever there is A, you think of it as Einstein A, wherever there is the electric field, you think of it as a triad, then that automatically satisfies all these field equations in Euclidean origin. And in fact, it satisfies the field equations with a cosmological constant that you can write down. Yeah, it is in the Eguchi-Hanson. This is not the Eguchi-Hanson. It is contained in the physics report, so it is not that significant. Running right through singular metrics? Yeah, I think so. I mean, I think... Changing the signature? No, I don't. Actually, I've not looked at the question of change of signature, but it's certainly true that you could go into the... The matrix advantages, as Chris was saying a while ago, at some isolated points, it is possible that this will actually let you just evolve through those kinds of problems. FAB is the tensile. The tilde means weight one. Weight one. This and this has an inverse term.

27:30 Exactly. If you have all these squares numerically or however, you will have to work that. So FAB is only derivatives of A. It's derivatives of A and only spatial. A is spatial and FAB is spatial derivatives. FAB is just a dual of magnetic field. The last thing is that this is first I thought that well this is some simplification perhaps it occurred because we are you know we have no cosmological constant, we have no matter, if you did put in matter then maybe terms will arise here which will involve lowering of the inverse metric, the lowering of the tetrahedral index or something. It turns out that's not the case. The inclusion of matter that is to say scalar field, Yang-Mills field and just recently Joe Romano and Ted done the Dirac field has been done and Inclusion of cosmological constants also has been done, and Ted Jacobson has done the extension of all these things to supergravity. Again, the self-duality seems to break simplifications, the self-duality seems to give you these constrained equations, these evolutionary equations, which are all falling out. Finally, the question that Bob was asking us. This is all fine as far as complex relativity is concerned, but we are not interested in that. We would be interested either in the Lorentzian or perhaps in the Euclidean domain. What are the additional conditions we need to impose in order to get back to the Lorentzian domain or the Euclidean domain? Well, it turns out that in the Euclidean domain, the problem is quite easy. Namely, what you do is you just restrict your basic variables e to the a to the real, right from the beginning. Then the whole thing just goes through, and you just get Euclidean to be real, everything is real. The conditions are more complicated in the Lorentzian regime. In the Lorentzian regime, in order to get real relativity, the thing of interest to my chapter here, You have to impose, first of all, the condition that the real part, that E be real, the triad be real, which will give you the metric to be real, but then you have to introduce another condition, which is that real part of the connection, which is our variable A, minus the spin connection, which is constructed out of E, should be equal to zero. In other words, you have to impose a condition that A minus gamma is equal to zero.

30:00 The difference between these two things arise because of the I which came up in the Lorentzian versus Euclidean regimes. Let me just explain this with respect to using a simple harmonic oscillator analogy of what is going on up here. Do you also quantize this constraint in a way which makes those computation relations consistent? I'm coming to that. So I just want to give you a feeling for these things first of all. And the feeling for this is that if you consider a simple harmonic oscillator, but instead of considering q and p real, if I consider them to be complex, then I can just define a new variable called z, which is also complex, which happens to q plus ip. Q and P are canonically conjugate, but so are Z and Q, or P and Q, P and Z for that matter, but certainly Z and Q appear also canonically conjugate. So supposing I try to use Z and Q as my basic canonically conjugate variables in a complex case. Then to restrict myself to the real phase space, I would have to demand that Q be real, and I would have to demand that Z be really of the type Q plus IP, where P is real. Or, real part of Z minus Q should be equal to zero. The analogy up here is that E tilde is like Q and A is like Z. And so we have the equation, imagine a part of Q equal to zero, imagine a part of E tilde equal to zero, and then the real part of Z minus Q equal to zero is completely analogous to this equation. In the Lorentzian domain, there is a non-polynomial dependence on E tilde, which comes up here, and this equation is really complicated. Nonetheless, as I hope we'll see... The ability to isolate the complications in this equation is actually useful and is powerful and we can do certain, we can actually exploit that.

32:30 Come back to Carol's question. The program then is a forward. Let me just go to the next transparency in which the program of quantization is then the forward. First of all, we introduce operators A and E and we want them to satisfy canonical computational relations. Then one would like to introduce a representation. In other words, what are the states going to be represented by? And as I already mentioned, at the moment at least, there are two possibilities which look attractive for various reasons. One is that you represent states by wave functionals which depend on A, the A representation. However, since A is complex in the Lorentzian domain, this is, as we saw in the case of the harmonic oscillator, it is the analog of z, which is q plus ip. Therefore, we want this to be a holomorphic function of A, not an arbitrary complex analytic function of A, but a holomorphic function of A. This is like the Bargmann representation of harmonic oscillators. But it's not formal in the sense of delta A bar equal to zero. One is just doing that. Again, I mean, if you do it in the linear theory, which I'll explain, then there is only... But you could, yes, but it decouples, but one can just do it, I mean. But in the full theory, one would like to say what the space of A is and what the structure is of E. Then in the loop space representation, the wave functions are going to be functionals of the loop space, the functions of closed loops on the three manifolds. Then the idea is to solve the constraint equations, which are polynomial, either as operator equations, as equations operating on psi equal to zero, either in this representation or in this representation. Then the program would be to try to isolate time from the variables in some sense and write down the formal Schrodinger equation. And now, up to now, we have just ignored these reality conditions, which had to do with looking at the real section of the complex phase space. And the idea is that in quantum theory now, we would like to incorporate these reality conditions by certain hermeticity conditions.

35:00 And the idea here is that one would like to introduce an inner product on the space of states, such that the operator that we want to be real becomes actually Hermitian. So, in fact, the reality conditions, which for a long time I thought, which Hermitian inner product should you use? There are two guidelines at the moment on the Hermitian inner product, namely the reality condition up here and that this quote-unquote evolution, formal storage equation you get up here, should actually be unitary. Variant theory, even though you're not explicitly by hand put in Poincare invariants. You'll see that in a minute. And finally and most importantly, one would like to find physically interesting observables which operate on these three functions and find suitable approximation methods so that we can compute results of interest. So this is the general program and now what I would like to do is to tell you a few of the problems that have changed their status because of the use of new variables. First is the issue of time. And I would like to focus on the issue of time in the asymptotically flat context. Most of the discussion of this issue are in the spatially compact context where it is more puzzling, more difficult and so on to understand. And quite often it is sort of asserted, because one does some perturbation theory or something, that in the asymptotically flat context we don't really have a problem about time because at infinity we know what time means, what time translation means. The question is really, how do you sort of bring in the time that you know at infinity inside? How do you more precisely, is it true that in asymptotic method context, there is a sense in which the constraint equation contains dynamics? If so, what is the sense? The proposal is the following. I would like to address this question in general, first of all, without reference to any variables. The proposal is the following. Associated with the lapse n, shift equal to zero for simplicity again, it is given by the constraint function, constraint functional, associated with the shift, with the lapse n, plus an ADM energy, which is a surface integral out at infinity. We know that is true.

37:30 Now what I would like to do is to sort of do something which looks naive and dumb at first, but seems to be a useful way of looking at the whole thing. And what I would like to do is to just turn this equation the other way around. I just want to write this equation as C equal to H minus E and then the equation that you want to introduce on the wave function is that C hat is equal to zero. C hat operating on the wave function is equal to zero. That immediately gives you the equation which says that the Hamiltonian operating on the wave function, if you could make sense of all its operators of course, that the Hamiltonian operating on the wave function is equal to minus the energy operator operating on the wave function. If you went to an appropriate representation, one might intuitively think that energy is a sort of dual of time, and therefore in an appropriate representation, perhaps this operator can be written as i by g by dt. And the key issue then is, are there such representations? Can you find them? And this may sound a bit strange, a bit abstract or something. I want to make this more concrete by first of all telling you what the status is in the standard metric representation. How can Hamiltonian be represented by a surface and a glider? Hamiltonian is one that involves only first derivatives. Because I would like to get time out of it. I mean, it's true that one could solve that equation, but I do not know how to interpret it physically. But then because you have the time at infinity, you know that this Hamiltonian... No, that is only an asymptotic time at infinity. I do not know what is going on inside. Price is arguing the way I used to argue. It's a bit difficult to... He's smiling because he used to argue the way I used to. He used to argue against me the way I'm trying to argue against Bryce. The way you argue against me. Okay. The old answer was that the choice of how you've argued was all games. That's certainly true. But I would like to sort of... There's not a lot of wrong answers. It's just not very useful. I would like to know, if you like, in that term,

40:00 is there a nice gauge fixing procedure? I mean, how do you match that all in? It's one thing to say it starts off gauge, but then you know do it. What I'm trying to say is that well, how do you do it more concretely? Let me explain what this problem is. For example, if you use a usual metric representation, then we've got the equation. Again, I'm just writing the same equation up here. So you have constraint operating on the wave function equal to zero. But we know that in the metric representation, I mean, we know that the energy up here is just a function of q. And therefore, it's just a multiplication operator, the wave function up here. So what one would like to have is a representation in which up here you don't have the full metric, but you have, so to say, the variable which is canonically conjugate to this energy up here. The wave function should depend on that kind of variable. And in fact, already in 1970, Kanner has done this in the linearized limit. He has explicitly emphasized the fact... That metric representation is a bad carrier of time and in the linearized limit at least, a better carrier of time seems to be the foregone thing. But you choose a representation in which the wave function depends, this is linearized, so wave function depends on transverse traceless part of the metric, so this is the longitudinal part of the metric, instead of depending on the trace part of here or the conformal part of the metric, it really depends on the variable which is canonically conjugate to the conformal part of the metric, namely the trace of extensive curvature. I'm sorry, it was done in 1960, I did not know that. And published. But if you use this representation, was this also done? You get this equation, it was there. Then in this representation, what happens is the following. We want to impose the condition c hat equals zero up here. The Hamiltonian, in the linearized theory, when you do the linearization, just gives you this kind of Hamiltonian, which is the second functional derivative of transverse stressless states. This is just the ordinary usual linearized Hamiltonian. This object up here, when you've done it all, in fact, in the linearized limit, This gives us precisely a derivative term. Instead of being a multiplicative operator now, it becomes a derivative operator, roughly speaking because this just depends on the conformal part of the metric, and now our wave function depends on pi, and pi is like delta by delta, the conformal part of the metric, and that is why this just becomes delta by delta t times this, where the time now is to be interpreted as Laplace into the minus 1 times pi t of here.

42:30 So if you use this representation in the linearized limit, you are in good shape. If you use a standard metric representation, you are not in good shape. But there is a problem with this. I mean, it sounds like a nice solution to the whole thing. The problem is that while the metric representation exists in full relativity, in the sense that at least you can write down formally the constraint operators if you bend down the representation, already the linearized limit is a bad carrier of time. It doesn't satisfy the ADM Kukash criteria. On the other hand, if you went to a representation in which the metric, in which the wave functions depend on this thing, then you're in good shape in the linearized limit. Unfortunately, this representation doesn't generalize in the full theory. There's no simple way of writing down constraint equations if your wave function is supposed to depend on these variables. What happened with the use of new variables, basically because they... So to say, combine the information both in the metric and the extrinsic curvature is that this problem can be overcome in the following sense. We will look at the A representation now in which the wave functions are functionals of A. Now the ADM energy of course is like the derivatives of the metric, therefore here it becomes like the derivatives of the trial vectors up here. What about the constraint? Then the constraint is equal to the Hamiltonian plus the ADM energy term up here. You get the Hamiltonian to be dependent on A star A times the prior vectors up here. That is the Hamiltonian and this is the alien energy. But now if you want to go to the A representation then E is canonically conjugate to A and therefore you would expect this object to behave like a differential operator rather than a multiplication operator. So the hope is can you find some variable T which is of course constructed entirely out of...