Spacetime covariance in canonical relativity / Arrows of time and generalized quantum theory

2:30 I have switched him to addressing him then a little earlier. Now what is in fact nice? The outside of the house so that nothing is stolen from both intruders will come from the other. It became the temporal rest that I would like to follow this way and I would like to look today not as the other speakers by looking into the future and talking about their most recent. And I would like to discuss the things that we were both interested in and the things that we worked on together and that we thought about together. So my talk is a retrospective, and only the last third of it, if I ever get it, will be looking. Of course, one never remembers these things without going to the records.

5:00 On 24th of May, 1973, there was a seminar at that time that I gave at King's College, and you were at that seminar, and that's the first time when I remember that we talked together. Of course, shortly after that, Price David came with an idea that we may write a book together. You know the story, of course, but... Let me start with it, and let me start ten years earlier before I met you, with the admonition which I got from Bryce at the Landshut Summer School. So, Bryce started with a third model, which is of course well known. A chief goal of these lectures is to develop and tailor... Within which the quantization of fields possessing infinite dimensional invariance groups may be carried out in a manifestly covariant fashion. The requirement of manifest covariance means that we shall adopt an overall spacetime view from the outset and ignore canonical formulas. What is forbidden because the covariant one and the canonical one, and they are totally different, of course, that it is not canonical gravity as going an ancient picture. And it comes from the fathers of generalism. In fact, it comes from David Hilden and from Hermann Weyl.

7:30 It's the concept of the space-time to be split into space and time, and to be split into them by structures put in from the outside, by the foliation of the space-time, by the leaves of instantiating, and then by the congruence of lines which brought in the spatial reference frame. And these things brought in my head, thought in, about as a tenuous type of matter, which in fact doesn't influence the gravitational field, but provides the orientation within the space, is the basis, what is called at that time, one-dimension oriented matter. Now we have a mapping which goes from the event in space-time to a moment in time. And that mapping, taking the event in spacetime and sending it into a moment in the time manifold, is what provides us in spacetime with what we call the instants. So these instants, in general, are supposed to form a variation. And an instant consists of all the events which are mapped into the same moment. Another instance corresponds to another point, and of course the time procession is how we go. What we have is of course now spaces which are mapping to the same point form various bird lines.

10:00 There are many bird lines, of course, and they form a congruence, and this, of course, which goes differently. From the time and from the space, the embedding map, which is used all the time in canonical gravity, when we try to take the space-time events and pull everything down into the split-like structure. And it is then split, of course. So let me go to that relation between the canonical gravity and general covariance. What I showed you is the map which goes from the space-time into the time and here is it. What we do is that we take the ordinary Hilbert action, which is invariant under the film, and we take that action and look at it from the point of view of the pullback on the time.

12:30 We use geometry from the legendre to the variables as complemented by the conjugate momentum and then our ordinary equal time course on them. When we vary the multipliers, we obtain the constraints. Of course, those constraints, which are the limitations of should, have some properties. Now they form what people call the Dirac culture. The only trouble with this algebra, of course, is that it is not, that it contains reference to the objects which define the dynamics itself, to the matter. And that's the... Now this is, of course, what was the state of the affairs when I came so luckily to visit you at my sabedito at Imperial College. In one of the nicest years, it was in 1983, into the question, namely, you had a that describes the kinematics of four canonical groups at the same time a farcical situation, and it reminded me a situation in the old 19th century comedies, when there is the bride, there are even, it doesn't necessarily mean...

15:00 It is not necessary to say that we will succeed to you. It is possible to introduce the space-time design of physics. ... was then split. The first half is the basic idea on which we then try to implement. Namely, what happens in a parametrization is another ancient game. Of course, for finite dimensional systems, it goes at least as bad as to Chekhovian. And for the field theories, it was first discussed by him. So I, what is here, shows that you have the field and given space-time, so the metric of that space-time is fixed.

17:30 Observe, it's not invariant. If you have something fixed and you try to modify it, then of course you own it. The space-time is pulled back, we take the invalid variables, the components, and we are adding them to the remaining variables. This is called parametrization. And then we open up a new genre of transformation. And we arrive at the action which depends on the pullback of the field, the conjugate momentum, the embedding and the conjugate momentum, and a set of multipliers. Multipliers which tell you how to go from one point of space to another point. If you look at them, they consist linearly of the momentum and then the object which you can recognize as the energy density and energy flow in the space-time. This is the energy momentum denser of the field. It is of course being expressed in terms of the canonical data.

20:00 It is projected in one of the indices to the normal and it is a function of the variable 1. And now we are following the evolution in a spatial direction. We do it by having this Hamiltonian smeared by the mathematical. And that's what the action is built. And also the embedding, which carries it. And at the same time, it evolves that field in a dynamic way. This all is burnout. It isn't, however, the way that Dirac introduced the parametri. Now, what I want to say and what is the basis of those two papers. By one and you form the Poisson framework. These key things are in fact commuting structures. This happens for the prescribed set. So what is the trick? You take a vector field which is sitting on that manifold, but you then restrict it to the impediments. And that respective field becomes a function of the space-point, a function of the invariant. Now it no longer vanishes, but becomes the same object still by the Libre,

22:30 where the representation of the space-time differential appears. The fields are not taken as they are. They are dynamically evolved to the human value. So, what this object does to the basic variables is that it produces the dynamically evolved field. I would like to use this language by the field sitting on stage. So these are not generators though. The generators of what is wanted. To have that count of the kinematics for the later objects which in fact are generated.

25:00 And I think that all the distinction between what I call perennials and other people observe is based. The transformation vector perpendicular and parallel to the surface, the instant of time as I told you and he did the same thing and he then algebra which is far away representing what's the geometric meaning which simply tells you this you are using a chart of in the space of a chart that is a holonomic chart. However, you can introduce an unholonomic basis in the very same space, and that is this matter of projection, and of course what we need in order to go somewhere is to unproject, to reconstruct, or if you want, do what you have done, the projection, and try to come out to the space-time is if not. Here is to the same thing for canonical graphs. We know somewhere implicit variables and the metric variables are becoming. They do not know about. You have a double set of concepts.

27:30 Now to return to the holonomic basis, you need to reassemble and out-project the variables. But the question is how? Do not know what the unit normal is and what are the tangent co-vectors, functionals of that, because the space on which the whole thing in the parametrized theory is based, the very aim is to reconstruct are the variables which somehow give is that you fix as much as you can by imposing the coordinate conditions.

30:00 We chose the Gaussian language. Temporarily suspended, because you are very, very young. So what you are doing, as singles, and then you pray. Now the basic thing is that the chance still varies. Weird objects, like space tanks, and they correct both the dynamics and the gravitation today, which is not bad.

32:30 You impose these. I would like to read the final book. So, Althusser, the second, is familiar with the ideas of random roots in string models and two-dimensional membranes in non-Aberian gauge theory. Let's turn to the theory of random three-dimensional hypersurface. One is irresistibly reminded of the famous axiom of Maria the Prophetessa. One becomes two, two becomes three, and out of the third comes the one as the fourth. Now, it has an interesting history. Of course, we had those things discussed as choice in between. But when it came to the idea to impose them at the conclusion of the paper, with time, the G. I want to keep my scientific reputation and I do not want those words to come to the table. I prayed the devil's advocate and I told him that of course they will read what you are saying.

35:00 And moreover, if anyone sees that sentence, he will blame it on me, for your reputation will be conserved. The interest in Maria the prophetess was actually precise. Ever get that last page of the second page of those suspended constraints? And what is, in fact, the type of physics which lies behind it? It lies that only later, when we work with child's story, equations do become anything else but minus one. And there is the mixed complex, which Fisch fixes. In the Dirac-Adien version, you say that the lex function is equal to unity and the shift vector is equal to unity. The basic question is to impose them after the variation, after the full set of linear games. De-parameterizing the theory in the Dirac's way by imposing the second class constraints and then dealing with the scheme that includes.

37:30 But there is another way to do it before. In that case, variations of the metric are limited and it means that the Einstein equations get modified in the usual classical mechanics. And you know that if you have a conspense, say a particle moving on a fixed surface, that you can realize the game in two different ways. Use the generalized coordinates and you get what people call the Lagrange equations. Or you impose the conditions by Lagrange multipliers. Now in the second way you get the forces. Let me show you what happens. The action which imposes the conditions on the variables. These are already the parametric variables of the parameters. Bring this condition in the covariant form. You project in this given as part of it. You do the same thing with the second constraint. And you have here the space piece. This action is invariant and so you can vary these variables that are what you get. Now this is what you introduce the space which happens. What you obtain from this action by varying the spacetime by the energy momentum tensor of a specific material system which you know.

40:00 In the motion, which is irrotational. But in addition to it, there is this. Fortunately, in my very early, and what you get here is simply what is taught to the system, not to attract it from. Namely, you count the modification of the Einstein's idiom, that source isn't there, in fact that dust is all. In the ordinary age, then these mark into those momenta which occur in the version of the theory which we needed to represent.

42:30 I'd like to say that they are the same things that in other times is very unphysical. Can you really take something that is simple and just break it with the irrotational piece into the definition? Of what is the for-nausical position into the clutch coefficients, these things, and then you start essentially what we did again in the previous week as the gauge fixing condition, namely, not the U of the potentials, but I want to explain the predation. The little t is simply the proper time measured on a clock moving along the dust trajectory.

45:00 And the x's, which I have here, are the objects which provide the width, the co-moving coordinates. And also, they are the mass density and momentum density along the dust pipelines. All is the motion taking along the dust curve lines, parallel to them. You get the definition of energy density. And curiously, you see that it doesn't depend on its own. It's a square of a quadratic expression. That quadratic expression being the square of the... And then, the algebra. It was quite surprising, namely, that you have the way of combining the standard constraints into a quadratic combination, which generates the anti-anisku lexis.

47:30 Needless to say, there was this puzzled feeling about it in the end, if there is something that you can do with it, because if you abolish the dust, This is a quadratic expression which doesn't mean anything in space-time. If dust is there, if you remove the dust, you necessarily stay. But it was a curious aside, and this is the decision. I wanted to tell you that there was another time. We tried to unfortunately, it was the only time, it was the second contemplating.

50:00 Then, somehow, they published separately in different trusts. My version was published at the Proceedings, which is very difficult to get. It's an obscure and geometrical problem. It was published under the crisis. Looking back, it was the best possible decision. My version. So, we, I did independently a paper with Chris after that first. A joint paper, a collaboration. And this is in fact a directory which is sitting on my system in Utah, in which Chris would pose and depose, and that directory carries the name EJU.

52:30 It is still there if you want to send the files there, Chris. If I cannot discuss it, namely Chris changed this. He changed them in fact what was just to get in I believe that gym, to leave the things which works in them, which goes to the face space that is entirely based on four-dimensional geometry and the four-dimensional moment. And they are complemented by only one pair, namely the time. And I think that within that scheme, I know what it means to implement the true different of this other thing.

55:00 Now these are ADM action and the history action which I have. The very same action on the extended history phase space. And they are expressed merely in two different, the history-induced, the ordinary, instantaneous points. And what is wonderful, though variance isn't lost, but it is rather gained with the introduction of each auxiliary structure which can be freely varied in the action. And that's the usual story of Kretschmann. When you introduce, you gain. So the theory is invariant with respect to space-time diphermorphisms, but also the diphermorphisms of the history of phase-space formalism is truly invariant under all of these diphermorphisms. And each of them is implemented by a simplectomorph. The Hamiltonian function is invariant. So, what is all this? And why did I talk about it? It doesn't impose any, what is called, problems. They are completely free, those projections, perpendicular, perpendicular, perpendicular, parallel.

57:30 Now, the different morphisms act on histories, the virtual ones, not only on the actual histories. And there are many diphermorphism invariant objects one can constrain on the history of space. Now why is this? The constraints in the Dirac theory impose you with the condition on the state function that it shouldn't depend on the lapse and the shift. However, it's nothing to do with the diphermorphism invariants of the speed. Jim Hadley would like to take the de-thermomorphism partitions in the way that explicitly use, for example, the L'Extension. Now, this is nothing prohibited by the de-thermomorphism. The constraints not generate them and they have to do with what is going on. Namely things that you can in some sense interpret within the quantum, so it gives you much more freedom, both cannot be in the deco you would like to return back. Here we are with a gate. There is that sense of the time that goes inexorably from the past.

1:00:00 You can roll this eternal return verb, in which, euphorically, the snake is biting by risk to look at this in that spirit, namely that it's not necessarily the fact that there is some physical properties, which are very physical.

1:02:30 Mass distribution and the moment. Yes, but the for-velocity is not the primary variable. You cannot vary. You vary the Klebsch coefficients. And of course, there is a large ambiguity how you choose the Klebsch coefficients. You can introduce many more of them than you need. You take a minimum set of them. I'm curious to hear your comment on the problem of time in relation to these recent developments. First of all, is it fair to say that the problem of time, in your view, is essentially how to reconstruct time from canonical data that only lives on a spatial surface?

1:05:00 And if it is fair, or even if it's not fair, how do you feel that the problem of time stands now in view of these recent developments that you just summarized on your last slide? It is extended. It is put in by hand. It leads the action with a temporal interval. That structure which is there can be arbitrarily married and it leads to valid equations of it. So in that sense, it is and the problem of I never knew was the problem of many versions.

1:07:30 When you do it with the harmonies, you get the best way which are sitting, now when you do it, you get the time, and it's really very difficult to this, what is it? It's to get into there with respect, agency.