Time in classical & in quantum gravity

0:00 Analyze. I mean, there may have been some changes. I have a sheet. I'll start it at once. Those who want to go to the airport, those who want to go to Boston, and those who have space, who are going. This session really has been organized by Carol, but since he's the first speaker, he thought somebody else should keep time. So it is my painful duty to keep time. It's your painful duty to introduce me. I think I got this right this time. So there are three speakers. As you see, it's a bit crowded. The idea is to have about a 10-minute discussion period after each talk. So we can have short discussions, short questions, short comments. And longer remarks can be postponed until tomorrow when there is a whole discussion session with Chris Isham. Karath. Thank you. A couple of years when I taught a course about time in Utah, my favorite bed book became a book of mottos of the sundials, those profound wisdoms which you find engraved on the sundial faces. Now, I offer this as an ultimate consolation to all of those who will become impatient with me in the middle of the talk. Now, what then is time? In classical theory, I know. In quantum theory, I know not. And this is the summary of the talk. In any case, in classical theory, I know that there is a spacetime as a primary quantity, which is not, in fact, differentiated into space and time. It's the spacetime manifolds that we start from. Space and time are only auxiliary structures. Space is a reference frame, a congruence of lines which identifies points, and time, of course, is a foliation. I can label the leaves of the foliation by the time function, the leaves are the level surfaces of the time function, and I can label the word lines by some co-moving coordinates.

2:30 So in this way I introduce time and space into the space-time which doesn't have structures a priori. Now the point in spacetime is simply identified by the intersection of an appropriate word line of the reference frame and an appropriate leaf of the foliation, and I can take those two functions, invert them, and have then one parametric family of the embeddings. Now I would like to stress that in all the theory these structures are auxiliary structures put in mathematically, but that in quantum mechanics we can think about them as being realized by some kind of the classical objects, so there is a Plenty of classical objects floating there, providing the background for the measurements, the points of the reference frame being, for example, realized by some coordinate fluid which is put into the spacetime. Now, the first thing which is important is to keep track of the causal structure, which is different in the Newtonian spacetime and the relativistic one. Because there are infinitely fast signals in Newtonian physics, there is only one foliation of the spacetime which is causal. That foliation is unique. If I label it, I introduce the absolute time. The time is absolute, but the reference point still can move with an arbitrary acceleration, and therefore space is relative, even in Newtonian physics. In the relativistic physics there is that system of the light cones which guides my surfaces of the foliation which also guides the work lines of the reference frame, but both time and space is relative because I may draw the foliation in different way and I may draw the coordinates of the space frame in a different way. So in Eddington's saying, we still leave the father time, his hourglass, though we do not let him decide with which he cuts the space-time manifold into those reefs. Well, if I try then to respect the causal structure in the propagation of particles and fields, I try to solve the Cauchy problem.

5:00 I try to give the initial data on the surface of my foliation and little evolve them to different leaves of the same foliation. In Newtonian physics, the foliation is unique because I have absolute time. In relativistic physics, I have different foliations, say the red one and the green one. However, the evolution is foliation independent. If I start on a given surface and give you the data and propagate them through the green foriation and the red foriation so that they ultimately coincide in the future, the propagation of the data should not depend on foriation, on what foriation I am using in the process. Now this is codified in the canonical theories in what is called a parametrized canonical formalism. There, I have the Hamiltonian given directly on the spacetime. I am not assuming from the outset that there is any split between the space and time variables. I simply give the Hamiltonian as a function of the coordinates and the momenta of the spacetime quantities. Now, this is done for particles. There is a similar scheme existing for the fields. And, in fact, this Hamiltonian is expected to propagate the particle along a space-time world line. This is done as ordinarily through the Poisson bracket, but the Hamiltonian is multiplied by an arbitrary function of what we call the label time, which gives me the trajectory parametrized by an arbitrary label. If I choose this function called the lapse function, then I am choosing the parametrization of the trajectory. I am not changing the trajectory itself. Now, because this function is arbitrary, the Hamiltonian which propagates the fields is simultaneously constrained to vanish. For that reason, some people call it super Hamiltonian, reminding that memento mori, which is so much associated with time, namely how easily a super quantity can be turned into zero.

7:30 Now the things to remember are these. Time and energy pair is necessarily included in the phase space. However, we do not know what is time and what is energy unless we introduce those auxiliary structures which split space-time into space and time. In fact, what the super-Hamiltonian constraint does is that it defines energy in terms of the true dynamical variance. There is a statistical version of the theory which, instead of propagating the coordinates and momenta, propagates the classical distribution function. In fact, that propagation is expressed by a single equation, namely that the function commutes with the super Hamiltonian, weakly modulo the constraint, and if you work out the consequences, you find that it is just the ordinary Liouville equation. Now, the classical evolution of this function, again, is foliation independent. Now here I would like to formulate the problem of time in quantum mechanics. The question is the following. In the Dirac constrained quantization, I want to take the super Hamiltonian, turn it into an operator, and say that the physical states are annihilated by it. This is the counterpart of the classical equation over here. And then two questions arise. The first question is When I talk about the evolution of the states and introduce the splitting by the auxiliary structures, is the quantum evolution of the state foliation independent? And second, can I introduce an inner product or the norm such that this inner product is conserved as I am following the foliation? So, these are the two problems associated with quantization of the system. Now let me show you how the things work in a Newtonian spacetime which has a privileged foliation in it. I am giving you the super Hamiltonian without assuming a priori the split among the space

10:00 and the time variables. It is some quadratic function of the momenta and the coordinates which are hidden in all the potentials given here. Phi is an ordinary four potential which gives you the forces acting on the Newtonian particle. U is the reference frame which you are using. In fact, in case of the Newtonian particle, it is necessary to use it in the super-Hamiltonian from the outstat, and I will tell you why later. GAB is the spatial metric of the leaves of the... yes. UA is a timelike vector in the sense that after we introduce time, we shall see that it cuts the leaves of the absolute time. Yes. G is the metric which is degenerate with the signature 0+++, and it satisfies certain integrability conditions. Now my message is this. Where is time in this theory? We do not see it, but we must infer it from the geometric structure of the constraint. It goes roughly as follows. The metric is degenerate and therefore they are degeneracy co-vectors. They fill a ray, and due to the integrability conditions, there exists such a function which generates one element of that ray as the gradient, and this function is the time function, which labels the leaves. The product of that U with that particular covector over here is positive. This is your assumption of the timeline character of the vector here. You scale H down by this. You scale the coefficients down to the lower case coefficients. You introduce the co-moving coordinates of this vector field. And then you write down the constraint in the form which we all of us know. In fact, this is the energy associated with our choice of time, and this is the ordinary Hamiltonian. If you impose the constraint in this form on the state functions, you get the ordinary Schrodinger equation. However, the task is slightly different. How to get the Schrodinger equation, in fact, without performing this geometric split?

12:30 Now you can proceed completely geometrically in the spacetime which is given to you. I wouldn't like to go through the procedure in detail, but only highlight that I am working with the spacetime quantities which carry the spacetime indices. You have an alternating symbol in spacetime, you hit it by the time covector, you get a three index tensor, you normalize it with respect to the metric, you then obtain the Levy-Civita symbol on the slices, and you factor order your super Hamiltonian in a certain way, inserting in between the momenta operators that alternating symbol. You must also appropriately order the linear terms. This is Newtonian so far. You apply the constraint on the state function that gives you the physical states. If you have a vector which is lying in the spatial direction, you can normalize it with respect to this tensor and you will find out that the inner product of the state functions which satisfy this equation, in fact satisfies the conservation equation. It is conserved as the Newtonian time is going on. If you turn this theory into the physical language, you just end with the ordinary quantization of the Newtonian system. Now, I would like to return to the problem, why should I have introduced that vector field, which is describing the observer, into the system? In fact, the Hamiltonian that I gave you is a quadratic function of the quantities involved. But I can easily imagine that I change my reference frame that in fact I am passing from one vector field which is normalized with respect to my time function to another such. In that case the two differ by some covector the index of which was raised by the degenerate method.

15:00 However, I can introduce a compensating transformation of the potentials. Such transformation, that when I return to the classical Hamiltonian, it remains the same. And this transformation gives me the fictitious forces, which then become encoded in the additional terms, which are generated by this change of the spatial frame. The Hamiltonian which I give you is invariant under this gauge transformation, and therefore as you solve the constraint equation for the state function, you find out it's the same in the new frame as in the old frame. However, if you adapt now the coordinates, the co-moving coordinates, to the u-bar rather than to u, you find out that the detailed structure of this function as the function of the coordinates involves changes. Now, this has amusing consequences. This is a problem which I call how to quantize Newton's apple in Einstein's elevator. You have a Newtonian particle which in the gravity free space is described by the plane wave. Now suppose that you observe it from the frame which is accelerated with respect to the inertia frame. In the space-time diagram, the foliation is the same, but instead of the straight lines which correspond to using the Galilean observer, you have then the accelerated lines which express the point of view of the observer in the elevator. Now of course we know what happens. What is done is that there is a homogeneous gravitational field in the elevator and therefore the eigenstates are no longer the plane waves. They will be rather airy functions. But there is this transformation between the state functions here and what it tells us in effect is that the Airy function is a relativistic transform, a Newtonian relativistic transform of a plane wave. You have a similar thing in the problem of the apple which is falling in the tunnel through the Earth, the gravitational harmonic oscillator, which you can observe either from the framework of the Earth or from the co-moving system of the particles which are providing the Gaussian frame.

17:30 In the Gaussian frame, the motion of the apple is the plane wave. In the frame of the earth, it's of course the set of Hermite polynomials, and you again find out that this equation is kind of a transform of the Hermite polynomials into the plane waves. Well, this then shows that there is no problem of time in non-relativistic physics, the foliation is unique, the problem of space is amusing, but it is not insolvable. It simply gives you the relationship between the descriptions of the same situation by two different observers. Now let me pass then to the problem of the relativistic particle. Here the constraint is given as the quadratic function of the momenta. Of course, the statement that the expression vanishes tells you that the particle is on the mass shell. The metric here can be an arbitrary curved spacetime metric. Now the time function is arbitrary up to the limitation of the causality. And, in fact, its gradient gives you the normal to the surface of the constant T. That normal, if you want to normalize it, must be divided by the norm of that gradient, which is the usual lapse function between the two surfaces of the foliation. You introduce the projector onto the leaf of the foliation, which, after all, is the spatial metric, And you notice that when you take the momentum of the particle, the classical one, it should be future-pointing. This product should be negative for every future-pointing vector. This allows you to write down this constraint in the square-routed form. In fact, the quadratic constraint is equivalent to the new constraint, which depends on the choice of the time function.

20:00 Now, classically, if you are using various time functions, then this set of constraints is highly redundant. The Poisson bracket among any two of them closes into the old constraints, the square-rooted ones and the quadratic ones. And also, the square-rooted constraint is in fact equivalent to the squared constraint over here, the quadratic constraint, because this commutator closes. However, the objects here, which I foresee, are structure functions of hideous complexity. They contain the rational functions, they contain the square roots, though they are non-vanishing on the constraint surface. Where with a choice of space, if I introduce on top of the foliation also the reference frame, I'll be getting then the constraint written down in terms of the canonical coordinates and momenta with respect to that particular split. The square-rooted constraint again can be written in the form in which pt is isolated, and I can try to impose it in this form on the state function. Now the story goes as follows. The piece which is connected with the shift piece of the constraint doesn't matter. The problem of space, in fact, is trivial. And therefore you can adapt the reference frame to the foliation, you can take it perpendicular to the leaves of the foliation, and then you try to define what this operator means by means of the spectral analysis. You introduce more or less arbitrarily the product into the space of the functions which satisfy the linearized constraint. It's the ordinary Schrodinger product with the measure which people usually take to be associated with the labs here and with the determinant of the metric on the hypersurface. However, it's not vital. You can choose different measures here. Then you introduce the momentum operator which is self adjoint with respect to this measure. You also introduce the operator conjugate to the energy which is chosen so that the change of the measure is accounted for in the conservation equation.

22:30 You order the square of this operator. You order it as a Laplacian corresponding to this metric n2gab with respect to the measure introduced. You add the potential term coming from here. This operator then is her mission can be extended to a self-adjoint operator. It defines the spectrum, it defines the eigenfunctions, and then I define the square root by the spectral analysis. It can be done in effectively the same way. Oh, I thought that you mean the labs. External fields, when, no, this is done in the time dependent situation. The square root can be done whenever this operator under the square root is positive definite. And if you have PA minus AA coming into this equation, it still will be positive definite. Now, this is not the ordinary Klein-Gordon quantization. It's the quantization based on the particular choice of the foriation. And let me tell you what the problems are with it, just in the... Is the measurement time-dependent? Yes, it is time-dependent. It doesn't... It's still through the controller... You know it instant by instant. And you get the inner product, which is conserved in time. Now here are the problems. You have that hideous structure of the constraints which correspond to different foliations, which in fact are closed classically but with complicated structure constants. And you find out that you cannot preserve the algebra of those constraints in this way when you plot dice. So, for example, if you try to base the quantization on two different foliations, as I indicated here, they are no longer compatible.

25:00 It means if you start from the original surface with the state function prescribed and devolve along the red foliation and the green one, you come with a final result which differs depending on what foliation did you use. And moreover, the quadratic constraint is not compatible with the linear one. It means that if I try to impose the constraint in the square-rooted form and have the time dependence, then the quadratic constraint is not satisfied by virtue of the iteration of the square-rooted one in the time-dependent case. If the metric in spacetime has a Kirin vector field, it means I can define a privileged foliation. I start from a single leaf and then I retransport it along the foliation. In that case, on that foliation I have time independence and the quadratic constraint and the square-rooted one can be made compatible. Now realize that this doesn't select the foliation uniquely. I can start with a different initial leaf and have a different foliation and on this again the evolution is consistent. However, I do not have any means to pass from the red leaf to the green leaf here by going by the amounts given by the Kirin vector here. So this is the solution of the problem of reconciling the quadratic constraint with the square-rooted one. It's not the solution of the problem of getting a foliation-independent quantum mechanics. Now I feel that something drastic should happen, otherwise I run more than three minutes over my time. But in any case, I should at least tell you the next piece of the story. Everyone knows that the solution of the previous problem is traditionally solved by saying that you cannot maintain the one-particle interpretation in the case of the time-changing gravitational fields. And thus you pass from the space which describes the particles to the space which describes the fields.

27:30 Let me have the space which again has time built in it, in the form of the embeddings, and which has the momentum which is conjugate to this many-fingered time, the embedding, embedded into them. I have also a true field, say the scalar field, described by its coordinate and the canonically conjugate momentum, and I am asking about the dynamics of the field as I pass from an arbitrary hypersurface to another arbitrary hypersurface through an interpolated foliation. The Hamiltonian of the system now contains the many-fingered energy variable, conjugates to the many-fingered time. It contains the Hamiltonian, that Hamiltonian being the projection of the energy momentum tensor into the direction normal to the correlation. Now the objects here satisfy the ordinary Averian commutation relation. In fact, in classical theory they commute. This is what ensures the path independence of the propagation in the classical theory. There are other versions of the independent foliation relations, one of them being the familiar Dirac algebra which is obtained when you project those super Hamiltonians perpendicular and parallel to the embedding and then commute these. You get the algebra which is characteristic of the gravitational field as well. Now the would-be quantization consists in the imposition of the quantized constraints turned in some way into operators on the state function, but at this moment the problem of the path independence arises again in the quantum context. You want to preserve this algebra, which means that you should order your operators as well-defined operators on some Hilbert space in such a way that the commutators yield zero.

30:00 Unfortunately, when you do it the ordinary way, say in a flat space relying on the annihilation and the creation operators with respect to the flat space-time structure, The Heisenberg variables. You find out that there are Schwingert terms occurring in the algebra of the components of the energy momentum tensor, which find their way into the algebra of the operators which you order in this manner, which then turn this relation in quantum theory into a relation with the anomaly. Now I feel that I should be cutting at this moment to get to geometrodynamics at all, so let me give you only the very end of this part of the story. The story is that if you use the factor ordering which I indicated, namely the factor ordering based on the Heisenberg variables, That it yields to the anomaly in the algebra of the operators which I would like to associate with the momenta canonically conjugate to the embeddings. Therefore you cannot identify them with the momenta. However, in the space of the embeddings, the anomaly which appears there is an exact two-form, you can generate it as a curl of a potential in the space of the embedding, and if you add that potential to the candidates for those momenta, you find out that they commute. And then you can consistently carry those constraints on the state functions. That addition, in fact, is impossible to write in a spacetime. It is an object which essentially depends on the embedding use. However, it gets you a consistent set of constraints, and that set of constraints gets you the evolution, which is path-independent. So here is the rough summary of this part of the study. If I have a field theory on a flat background and I tried it only for a massless field on a two-dimensional background, here are the conclusions.

32:30 The Dirac constraint quantization can be consistently carried whatever representation you choose. The Hamiltonians, which generate canonical evolutions along all possible foliations, are represented on a single function space. The evolution is foliation independent. It doesn't depend along which foliation you are evolving your state function. You always come with the same one at the end if you start with a given one at the beginning. And after the constraints are imposed, then the space of the solutions is endowed with the Hilbert space structure. In this case, the spacetime was flat, and I do not know how to do it unambiguously for a curved spacetime. But remember, if the spacetime was flat in the case of the relativistic particle, I couldn't achieve the path independence, even if I had all those Kirin vectors. In the field theory, I can do it. Well, here is finally what happens in geometro-dynamics. There I have the three geometry which is evolving along a given foliation and the conjugate momentum, the familiar super Hamiltonian and the super momentum, the super Hamiltonian being composed of the kinetic piece, as people call it, quadratic in the momenta, and the potential piece depending on the potential. Now, there are similarities and differences from the previous examples. The similarities first. There are constraints. These constraints are the results of a diffeomorphism invariance, though they directly do not generate the algebra of the diffeomorphism group. Hamiltonian is a linear combination of them, and it evolves the gravitational data from one space-like hypersurface to another. There are commutation relations in classical theory which ensure that this is a foliation-independent evolution. The differences. The canonical variables which I have are no longer cleanly separated into the kinematical sector which would tell me what is the hypersurface and the dynamical sector which would tell me what are the dynamical degrees of freedom.

35:00 Second, the formalism resembles in this respect a parameterized relativistic particle which is moving in a given space-time. It's not space-time this time, it is super-space. But there is no clear division between the kinematics and dynamics, between space-time and the dynamical data, that should be read from the geometry of the constraint itself. However, the potential which occurs here is an indefinite potential, and therefore the trajectories do not respect the light bounds in super-space. Finally, the constraints are given to us only in the projected form, telling us how to go perpendicular and parallel to the surface, but there is no easy way to reassemble them and arrive at an abelian structure. We have the so-called Dirac algebra. And in this split, the gauge associated with the change of coordinates on the surface is clearly separated from the dynamics which is given in the perpendicular piece. Now, what then is the method by which we may try to find time from the geometric structure which is present in front of us? Well, the first suggestion is that we should introduce those foliations and the work lines of the reference frame by hand. We introduce them as matter fields. We use matter to measure what is the gravitational field. And that was initiated by Bryce in his paper in which he introduced the fluid of clocks in order to provide the framework for the measurement of the gravitational degrees of freedom. One uneasiness with this scheme is that we presume that gravitational field is more primary than the fields. In fact, it is universal. It should serve as a clock for the fields rather than fields serving as clocks for the metric variables.

37:30 And therefore, the other program is to introduce what is called an internal time. Some set of functions on the phase space of the theory, on the g's and t's, which would serve as the space and time coordinates in this scheme. Now, because of the quadratic structure of the constraint, it is easy to assume first that this set of functions does not depend on the momenta that is entirely extracted out of the metric. This type of time I shall call the intrinsic time. The intrinsic time has however difficulties. Even in the simplest context of Hamiltonian cosmologies, you find it is difficult to make the theory independent on the particular choice of time which you introduce. If you introduce the omega time, the volume of the universe, you get different quantum dynamics than when you introduce one of the anisotropic parameters. I would like to tell you that Hamiltonian cosmologists are an extremely bad framework for checking this idea of the path independence of the quantum evolution. In any cosmological solution, you are fixing the foliation by the assumption that the leaves of it provide the homogeneity, and therefore the foliation is fixed. What you are changing when you are choosing different intrinsic times is only a choice of a laboring of a fixed foliation and therefore in the cosmological context you can never pose and you can never solve the question of the path independence of the evolution. Well, the other choice is the extrinsic time in which you allow the extrinsic curvature to enter into the scheme, and for the first time now I know how to carry this scheme in a path-independent way, and in the way in which all the operators are defined properly, at least in one simple midi-superspace model, and that's the quantization of the cylindrical gravitational wave. I know how to do it from one leaf of the foliation, in that way, to another leaf in the path-independent way and in the rigorous framework.

40:00 Just the Einstein-Rosen way, so that I have a governing linear equation. Excuse? Oh, it depends also on g. You usually make it to depend on the trace of the p or something like that, but it's the presence of p which is emphasized at this point. Well, the intrinsic time assumes that it is a variable which is associated only with the matrix. Well, with the internal time, well, that's the generic concept, of course, but in this case, I have an internal time. I would say rather that intrinsic is a subset of internal. Whenever you admit the extrinsic curvature, you have what I would call the extrinsic bank. Can I trespass by five minutes? All right. Now, I would like then to pass to the other proposal, which is the fixing of the foriation. That foliation can be fixed and then preferred with respect to the dynamical evolution by some set of conditions. And of course the best developed scheme is that of Jimmy York who uses the surfaces of the constant extrinsic curvature and then tries to decompose the data into the two degrees of freedom and that variable which serves in the role of time on that particular foliation. Now again I would like to tell you that this scheme does not directly face the problem of the path independence of the foliation. It's applied only to one foliation. You cannot change it easily and compare two different evolutions. To do that you would need to do something else. You would need to reconstruct that particular value of k which is associated with the maximal surface. By being reconstructed from the data, which is given on an arbitrary slice, that may be in principle possible, but before you have such a reconstruction formula, you cannot change this scheme, easily at least, into D1, which would allow the discussion of the foliation independence question.

42:30 That those variables which you try to use as the candidates for the internal time should satisfy certain requirements. If you take, in fact, a given point in a classical space-time, and then you construct the candidates for those variables from the data on a given hypersurface. In order that this point in space-time be uniquely determined irrespective of the hypersurface, the variable which you have should be such that it is unchanged under the bending of the surface or the tilting of the surface. Now, such variables exist. For example, if you take the square of the Weyl tensor. For the space-time, and you decompose it into the gravitational data on the surface, which gives you some quadratic combinations of the extrinsic curvatures and the intrinsic geometries, then this combination is such that it is unchanged under this operation. And in fact there exists a form of criterion within the Hamiltonian theory itself, which tells you whether the variable satisfies or doesn't satisfy such a criterion. However, it is difficult to construct the variables in this way, which would have the nice splitting properties when it comes to the study of the basic regression. Well, let me give you the last idea here, which is based on trying to find the symmetries in super space, which are similar to the symmetries which you have in space-time expressed by the Kirin vector field. What you try is to construct a variable which is linear in the momenta, like a Kirin expression is, with some expression which is a function of the geometry.

45:00 And constructed in the way in which the expression commutes with the super Hamiltonian and the super momentum. Well, if we let me give you only the result on this, no expression of such a character exists. Super space doesn't have any analogs of the Kirin vectors and therefore you cannot quantize the Wheeler-DeWitt equation as a one-field system. Now, if you want to take this seriously, you can come out with the proposal that you should treat it, in fact, as a particle moving in a time-changing background, that you should come to what I would call the third quantization of the system, in which, in fact, you are trying to introduce the functionals of the geometries as operators acting on some higher dimensional space. Now, my time is certainly over, so let me finish at this moment. Thank you. Well, unfortunately, there is... Now, my time is certainly over, so let me finish at this moment. Thank you. Well, unfortunately, there is the potential term. And that potential term, well, you can leave it there. You can say there is a conformal killing factor in super space respecting the metric, but it scales them incorrectly, the potential term. Or you can do it, like Bryce did it, by trying to multiply through some function and absorb that potential into the kinetic piece of the metric. Good afternoon. Well, although you gave me that title, that's actually not what I was going to say.

47:30 All right. I'm sorry. And that, I think, it is in four-dimensional volume. But that is then parametrizing something else than three-dimensional slices. That's right. But what you're finding is that you get, instead of the same thing, you get a trillion-year table, which plays out to depend on the four-dimensional variety that you get down from your head to your head. But that's an extra variable, probably. And then you suspend, in some sense, the constraint. What you do is, actually, you get all the constraints at once. Out of all the infinity of the constraints, you get all of them at once. And that one loss is sort of complicated by an arbitrary list, and it has a lot of constraints. But you have the ability to completely go various constraints. Well, I think it's a little difficult to explain it in just two words. Why is it that the search for internal time doesn't go into the experience of general covariance? If you try to find this sort of time, you probably do not find any which satisfies this criteria, which is a space-time scale. In fact you cannot find any which are local. I find it highly unlikely that you find some which are non-local. So probably the search of the intrinsic time goes against the spirit of the general covariance. I would say that the general covariance is somehow hidden in this scheme here in which you try to preserve the variable at a given point and the various fields of the surface. Well, these variables, of course, are geometrically privileged ones, which are based, say, on the curvature of the spacetime, and you wouldn't deny that you have privileged coordinates like curvature scales in a given curved spacetime.

50:00 Well, it is because I am using the normal scheme of quantum mechanics in which I am trying to measure quantities at a given instant. I mean, let me state the question. You always add all these entities for the independence of the coordination. In the classical theory, the independence of the source of coordination is coded in the algebra of the... Now, this is the formal way of expressing independence. In the quantum theory, we can translate it in the absence of the anomalies of the source from the quantum theory. So it seems to me that if we add the other two elements, there is no such phenomenon. The invariance of that is going to be easy. Well, the problem is how do you do it in practice? Namely, how do you transfer those classical constraints? into operator expressions which would satisfy the appropriate algebra with the appropriate ordering which wouldn't in gender have a constraint. And there are two messages in this. You cannot do it by assuming that the constraints are self-adjoined and act on a Hilbert space All of the objects in the end, and you get the situation in which the structure function receives the state function. But I misunderstand the question. I mean, the problem of the fact that you have to define a new parallel problem on the state of solution of constraints,

52:30 because the solutions of constraints are literally known, literally, in the natural parallel problems that I am discussing, is the general problem of any DHT system, and you have exactly the same problem in the Young-Middle School. Can I just rephrase this question? This question is basically that you have incorporated the dichromorphism freedom in the constraint algebra. It is true that it has to be anomaly free. Supposing that is achieved. Then the next problem is how do you find the inner product because normally if I try to do it naively it will be infinite. But he says that supposing, by hook or crook, I found the inner product, would you still feel obliged to find an isolated variable called time? That is the question. Well, I would like to know what are then the observables which you measure. And I, for myself, would like to see those observables somehow changing along my work line and therefore to be associated. The problem is shifted to the construction of an appropriate class of what I would call observators, and what I would call observators would probably not be what you would like to call observators, you would like to use the constants of motion. I would like to use the variables which are hypersensitive dependent, though we both of us know that there is a way in which you can translate one language into the other. However, it has also the observational connotation. If you have your constant of motion, you could measure it at an arbitrary time, so to speak. But the constant of motion is so complicated in terms of the instantaneous coordinates and momentum that you have a hard time to design the apparatus which would measure the constant of motion anywhere else than at the moment at which it is designed. Well then, what questions do you ask in the NDA? Not the scattering questions, but correlations between what? Physical flux, anything else? Well, suppose you have the gravitational field, then the correlation between what?

55:00 You might want to know what the Riemann cancer is and the certain coordinate system provided by a laboratory at a certain time. But then you are introducing the external clock. Because the world is full of external clocks. Why do we have to build everything out of field data? Well, it wasn't full of those clocks way in the past, in the early stages of the evolution. I would say that time is probably not a very useful concept. When you start to think about canonical dynamics, what do you do with this matter? He's wanting canonical because he wants the time, he wants to be able to see something he wants. I do not want to see something involved, I feel they are involved, and I want to explain why they are involved. Okay, I think it's a nice point to break. I guess the next speaker is me, and I think Carol is in charge of the session. In that case, I should announce your icons, which I have somewhere.