Time, a Convenient Parameter / Time & Observables in GR

0:00 Let me explain that, and given that, everything follows. First of all, path integration is a chapter of function integration. So I'll say a few words about functional integration in general, but in terms of discussing time, my examples will be out of path integration. The difference is that in a functional integral in general, the variable of integration is a field. In a path integration, the variable of integration is a path. So let's just look at the particle of integrations. You can look at the path in two different ways. Suppose that I have a path in one dimension, and suppose this is x of t, and then this is the path. You can also look at it as a point x the space of path. X of t is, I'm taking it now, I'm taking it now, but to know the whole of X of t, I have to give you an infinite number of information, that is to say the value of X of t for each band of t. So that means if I need an infinite number of information to know where I am, I am then in an infinite dimensional space. So X of t is in R, but the instruction X, what to do with t, is a point in an infinite dimensional space. Now, fortunately, we have no intuition about infinite dimensional spaces, but throughout the 19th century and the 20th century, mathematicians have made a lot of progress in developing the properties of a lot of interesting

2:30 function spaces, so now, for instance, if we say it's a Banach space, you know, a number of things. Or if you tell me it's a subalief LBQ space, I know I can work with that for function integration. So we do have a lot of information about function spaces, which means that we have a lot of information about the domain of integration of function intervals. And once we know the domain, everything falls. Unfortunately, too often in the early days of function integration, function spaces were considered—this is wrong, so I'm putting it in position one—was thought of at the limit when d is infinite of R-B. In other words, the original definition of functional integral made by Feynman was thought about which would go from a time interval of P-A to a time interval of P-B, and he was dividing the time interval into little pieces, and getting in the holes, suppose there was D pieces here, get it, write it in an integral of D, and say, in the continuum, that would be the limit of R v when e goes to infinity. So actually we did talk, we have quite a bit of talk about subdividing time, et cetera. I'm not talking about the deep issue. I'm only telling you in a very practical term that function spaces are not the limit, are not equivalent to R infinity. They have a lot of structure in them, you simply can't do the function integration or you cannot answer the index in quantum computing

5:00 in physics. Given a space X, with a definite space X, function space, then one can construct a volume element on that space. It's not a measure in the Lebesgue sense, but one can construct a volume element in the sense that one can construct an integral over the space X, a volume element, and then of s. And this is a function variable. Because it's a variable over a function of a function. Given, now in a more technical talk I would tell you how to define the volume element on a given function space. I would take example of function spaces and then use a volume element. Once we get the volume element, we can get a norm, and once we have norms and volume element, then we can produce functional integral. So, now, to go to the idea of time, let me write down the first particle written by Feynman. Well, I want to say two several things, by the way, about Infinite and Feynman, Barthes de Gaulle, which I'm going to do now. About Infinite, I don't know if you know the lovely, lovely little book of Borel that Borel wrote when he was in his ninth day, called Infinite. It exists only in French, but if somebody would translate it, I would certainly find a publisher for it. It's really like a testament of Borel

7:30 and then reading that book I understood why Borel's sets were a good thing. I couldn't answer a question on what is a Borel set, but I had no sense of why it was so good. Anyway, so much for infinity, which will have some bearing when you look at time man in the world over all these sub-division of time. So, the first part of that was to compute the probability-amplitude transition from a system knowing to be at the point in the configuration space A at time PA, and to have the probability-amplitude to find it at the point B at time PD. And the function integral we wrote was a sum over the space of parts going from A to B. So now imagine that this is now in our streets, and say that this was A and this was B. So it was a sum over all the parts in the parts from A to B. and X in B is a subspace of that space X. So you have to learn how to write explicit a subspace. Then to each space, I should have written here capital X because whereas of a bank measure is valid in lots of spaces, there's a different volume element for different function spaces. So when I write without the x as an aggregation, I should really tell you the volume element of which space. So here I should write the x and specify that I'm in the space x-a-d, but sometimes I don't write it, and then because you wanted, and that's a very useful approach, to think of classical mechanics at the limit of quantum mechanics,

10:00 that imposed something on this probability amplitude, which meant that that functional was I over the constant, the classical action functional. So, this was the first possible rhythm by Feynman. Actually, it stayed at a sense, well, no, two things. First of all, he used it to get quantum electrodynamics and to compute the land shift. The pass and the one was such a new technique that it was not adopted very quickly. The way the calculation is carried on from here to the lampsheet is using diagrams. And the diagrams were much easier to construct than to compute that pass integral. So the diagrams took the life of their own and were sort of to be the important things, but they were really a byproduct of this. and actually in a sense that story of Feynman having to show his calculation using diagrams whereas he was doing useful function integral is a little bit like when Newton had invented calculus and could get the solar system with calculus, when he published the Prunctitia, calculus was still not really acceptable so he did it geometrically, so there's no trace of calculus in the Prunctitia, it's all geometrical. But anyway, the point here, so what is time at this point? At this point, time, the value is a poem in a fixed interval, P-F-E-V. And by thinking of, let's call a time-sliced version of par-centable, by always wanting to work

12:30 with a time-sliced version of par-centable, one was basically missing the whole power integration. To the extent, I won't take time on that, but if you have time, we can read the Nobel Prize speech as acceptance of Feynman in which he sent the Nobel Committee, who I mean given him a Nobel Prize for computing through his letter, the the land shift, but the last sentence of the speech is, he talks then about passing people saying, what happened to the old lady? I was in love with my youth, and in the context it's clear that that's what it is. She has become an old lady, she has lost all sex appeal, and the only thing I can say for her is that she has given me a good child. Thank you, thank you to the nobody talking to me. Well, that was a little bit too much for me, so at one time when I wrote in some work of integration, and I put in the margin, she's beautiful in her own right, he said, answer, you're the best of us. But anyways, this is to say that I'm not going to look at totally different path intervals and show you the role of time in these other path intervals. You see, this is one which is a solution of a Schrodinger equation, that is to say a parabolic equation. It's more than a solution of a parabolic equation because it brings together the parabolic equation equation and the boundary condition or the initial wave function so it's a way to integrate there's no more initial value data problem once you work with function integration because it's organically put together but this this means only with parabolic equation when you have enough information to have one solution. I'm now going to give three examples of pass integrals in which the time is a totally different call and it's handled in a totally different situation than a point in a fixed time interval, which

15:00 fixed time interval you supposedly can divide by an infinite number of points and that the limit would be what you're looking for. The three examples I'm going to talk about One is sparse integral solution of elliptic equation, which is what you want when you want not a time-dependent Schrodinger equation, but time-independent Schrodinger equation. That is to say you want something which looks like Hamilton and psi equal to psi. Or other, I can value, I can value, I can value, I can function. This is a solution of an elliptic equation, not a parabolic equation. And we see the time of the variable p playing a very different role. The second example will be the Coulomb problem. How to solve the Coulomb problem with sparse integrals in three dimensions. And that brings into the whole business of stochastic processes and quantum mechanics and so on, transformation of constant high motion and the third example will be about five that go backward in time which are essential for one of his theory of all at least in theory so how do you start equation of fixed energy problems. Actually, the solution of any field equation is very well known for probabilities. It has to be adapted for quantum mechanics for two reasons. They talk about probability, we talk about probability and tissue. Importantly, their equations are usually future-directed, like Hito calculus, the little differential elements stick out in the future, whereas quantum mechanics is time reversible. But anyway, it is, so when I talk the technical talk, I'll tell you how I adjust the probabilistic solution

17:30 of elliptic equation to quantum physics. Right now, I'm only going to give you the probabilistic solution. For probabilists who want the solution of an elliptic equation for a domain, and you have an elliptic equation for a function psi, and the value, you have a domain m, d, and that domain m, d, for probabilities, is a domain now, d. One of probabilities nowadays, I don't know if you use my phone, but let's look at that. You want the solution of an analytic equation for a function of psi, and the value of psi is given on the boundary of the domain. Now, the solution will be a pass integral and c, I want the function psi. Psi at the point A will be different, and I'm just going to write this schematically, as an integral over I hope you will write it with my own location. We'll have to know in context. is equal to the sun over all pasts, and at this point, which are defined as follows, all the pasts starting at eight, and they find up to their first exit time. So each pass has a different exit time. A pass is defined, a pass x, with the time from an initial time, 0, to the first exit time.

20:00 So, the first exit time is a path-dependent notion. Each path has its own first exit time. So that's why the function of regression is a little bit more complicated to write, and I will write it in full tomorrow morning. But what I want to say about time is in the We have time which was a parameter in a fixed time interval. And now we have time which is path dependent and the time depends on the path. and each for each part of the time what I can take to take PA is from PA to the first place of time so again time will parametrize my past but it's a very different parameter than in the first example. And that's reason for fixed energy problems. Because in a fixed energy problem, you want the path or the particle to have a definite energy but you don't care at what time it will arrive at the point of interest. And actually, That's when we use the impulse train system and there is a classical analogue in Abraham and Marsden that use some of the techniques which I'll show tomorrow on how to deal with first exit times.

22:30 But that's one of the examples I wanted to mention. The other example I wanted to mention is Cécile, what do you sum, what's the sum end? The sum is of our old past. But you sum the boundary value where it exits? Where light connects. I write it not quite correctly, but I couldn't. It's a start of a certain volume that I would have to explain, and then P over I, H, and the action, instead of the action, then if I write the action of the integral of the to deep to first exist in time. So that means that I have to define the past as, well I'm getting ahead of myself, but anyway. I can't take that, but that takes me into the whole business of having now part time which is part dependent. So I'm going to produce an adaptation in which I can thrive for each part of its own exit time. So I mean, I'm going to probably see that that goes to certain parts and so I don't know about it and something else from one of these. But anyway, I'm supposed to answer down one way with time. The first example was time was a value within a fixed time limit. Now time is a barrier from zero to a first exit time that is past dependent. Now, the new example, the second example I chose, was the Coulomb problem.

25:00 The Coulomb problem is the first example in which a change of variable to describe the the path of the process is coupled with a change, a transformation of the time. Let me first tell you about the Coulomb problem, which is a very specific case, and show you that it's really an example of a large body of information. Now, the constantine motion problem or the constantine motion transformation. Excuse, the Coulomb problem is explained on Coulomb force with Virgil Porticus, is that it? The Coulomb problem is to compute the problem in quantum mechanics in which the potential is one over half. Yeah, you want to get the Coulomb force, yeah, from virtual particles. I mean I'm doing a particle which are in a one over half potential. Thank you. and I don't want to have singularity at the origin, et cetera. Anyway, the constant high-more-stifal transformation says that if I have a dynamical system which in classical mechanics, that's a classical mechanics statement, in which X of E is in between R, D, and I'm working with time E, then the constant of the first transformation is a transformation from this dynamical system to another dynamical system, which would be characterized by Y of S in an other dimensional space, and another parameter.

27:30 Time is still my parameter. Now, the constant time or shift of transformation works in the foreign case. D equals one, two, four, eight, and the corresponding N will be one, two, three, one. And in the Coulomb problem, it actually gives a very neat calculation of something which is very well known, is that if you have a Coulomb program in three dimensions, you can transform it into a harmonic oscillator in four dimensions, and then you don't have any problem at the origin, at R equals zero. So, this transformation has been used in that situation for trans-web in this direction, this direction, to rewrite the Coulomb problem in three dimensions and how many are situated in four dimensions. And Kleinberg is the first one who actually computed the positive for the Coulomb problem and got the eigenvalue and everything else by using that transformation. But then it was generalised by Blanchard and Cyrul, who showed that, and now I am going into probabilistic terminology, that in these cases and also in other cases, given the process, you can change the process, given the process, you can get an equivalent process by changing it, but changing also, making also a time transformation. So it's called the process through time transformation. So that at that point, my parameter, which which was a term in my Coulomb problem, has become a totally different parameter once I solve it as a harmonic oscillator. Do I still have a few minutes for the electricity pass? Three or four minutes. Okay, so the third example I wanted to take

30:00 was pass going backwards in time. Now in the past way of blackwash in time, you would have X mu and X poly S where mu, X mu S would be the universal space, and S would be a parameter. Now unfortunately I call that parameter the proper time. It's not the proper time, we all know what is the proper time in relativity, but it's a parameter. So, and in that case, you have passed, if I put time this way, when I say that I will have x zero for the time, and the other one will be x one, x two, x three. Now, suppose I'm plotting x zero, and I've passed going backward in time, and parts when I was in time I've been in time as the same equation as a positron and this is the kind of parts which are necessary and those parts are necessary if you want to write the kind of possible that you need for quantum and actually in Feynman's lecture for the Dirac Memorial, which is a lovely little book, he has a very good argument why those paths going backwards in time are necessary to weld, that was his work, to weld relativity and quantum mechanics. Was this a reconciliation with Dirac? He expressed grief and somehow insulting to her. It's memorial, I don't know, later, later, later. Ask me afterwards, maybe I'll understand the question. But anyway, the argument is, in my days, when I found that nobody remembers those things, positrons were holes in the sea of electrons. and but that was a very unsatisfactory proposition and the bus going backward

32:30 in time is a very clean solution so you don't have any more holes in the sea of electrons you have faster than that was in time and we show that I can extend up that more detail in the technical section where you do leave those parts going back on it in time now I'm not talking about time as a concept and through that, to be it's only a parameter that I use in my equation. What is time for me in my equation? The parameter which makes my equation simple, easy to solve, and practical. I'm sorry, I don't have a deep understanding of time, but you asked me what I thought of time, here it is. Would you entertain a question? Oh sure, I thought I'd use that for my time. No, that's it. Okay. All formulation, of course, is covariate, right? Pardon? The formulation of the path integration is covariate. It is given in terms of invariant quantities, right? Not in terms of the coordinate system. No, of course not. The path is the path. whether this guy is a pass on his partition coordinate or his polar coordinate, a pass is a pass. And the literature which went down to say, I learned to pass it, it was just polar coordinate by adding the potential and doing this and doing that, it's simply by not understanding the pass is a pass regardless of whichever coordinate you write it to it. But as a matter of fact, if your potential is in polar coordinate and you want to pass in polar coordinate, through pass integral, I can rewrite it. I mean, I can tell you what to do. Then it's better, I think it's better to speak not about the timing of T sub A and T sub B, but something between two events. One event is the event A and the other event is the event B. and so on, because when it should be important in such a way. I know, but, okay, I talk about events other than space-time, when the question where and when are relevant. For instance, if you are detected and you want to kill an alibi, you need to know not only where, but when, or if you're dispatching trains.

35:00 But if you want the path in three dimensions here's a parameter so I'm I'm just a craftsman I mean I'm using tea as a parameter which I don't want to use is whichever whichever problem you give me and if I'm speaking to the law I'll tell you how I'm trying to handle that parameter so the parameter called time which is in a fixed time in the original One human finding one becomes a path-dependent one, becomes one going backward in time, becomes changing the dimension or everything. So it's, whatever, it's continued. Well, thank you. I can see. Yeah, it's fine. Oh, that's a good question. The observer will always come from the system. Thank you. Thanks to get ready for our next and final presentation. Jumping jacks, jumping jacks. Actually, Cecilia is asking probably the most profound and pertinent question of the afternoon,

37:30 and that is, who's going to keep time to leave? So I'm really going to try to be I'm trying to cut back a bit, but I wanted to say I'll have opportunity tomorrow morning to give you a more technical explanation of the issues that I want to address in my college this afternoon. Let me see if we gather a computer out yet. Dr. Paul, can we switch over? Yes, let me do that. Well, with a little bit of tongue-in-cheek, I've entitled this The Internal History of The Frozen Times is supposed to be a play on Stephen Hawking's short history, right? The joke fails, I have to explain it. So let me give you first an overview of what I want to talk about. So my focus in my talk is going to be on general coordinate symmetry. and sort of building to undertake arbitrary transformations in positions and times. And I'm going to first talk a little bit about Einstein's resistance to this apparent symmetry of this emerging theory of general relativity. And then I'll address a very perplexing problem that people began to deal with in the 1930s and that continued to deal with, having to do with what does it mean to pick a particular choice in time, and then to evolve that our

40:00 information about the future. I'm going to tell you a little bit about the role that Peter Bergman and Walter Rackett played in the story. And ultimately I'll get to this notion of frozen We'll have to give a little bit of historical background on that. Then I'll tell you about, in a further analysis of the symmetry that actually was performed by Bergman and Omar. Then I'll tell you a little bit about work which I've done in collaboration with Larry Shepard, who actually is at UT Austin. Unfortunately, he was unable to come today to the Health Basin Center. I'm disappointed with that. And then we'll finally address this issue of the ontological significance of time in the sense that we're, and hopefully I'll be able to say a word or two about implications for quantum gravity. So I first want to say a little bit about Einstein's whole argument. He wasn't that concerned in 1913 when he first, in one of his first attempts at a theory that incorporated gravity, when he determined that his candidate theory exhibited a kind of indeterminant. And the argument that he brought forth is known as the whole argument, and I'm trying to sketch it here and that he imagined that he plotted time in that direction and say one three dimensions of space in this direction and then he had arbitrarily selected a to choose this particular curve that's representing an initial time surface in the space and he asked himself then what would happen if he were to alter the coordinates in this hole, sort of space-time hole. And he determined that, in fact, if he were to alter the coordinates within the hole, of course he wouldn't change any physical variables in the exterior region, but he would change them in the hole. And this would have a disturbing consequence, that even though he made no changes at the initial time, he would get new solutions in future time. And if, in fact, his theory was co-variant, we say, under general coordinates,

42:30 which means that the laws take the same form in arbitrary coordinates, then he was forced to the conclusion that the same initial conditions at this time would result in two distinct solutions. to him was a very disturbing consequence because he wanted to believe, as I think most of us do, in a form of determinism, that there's some unique evolution that follows from our knowledge of the world at some fixed time. And that seemed to be contradicting by this example. So, as a consequence, he actually abandoned his originally quite good attempt at general and worked away for another three years, until finally, three years later, he returned to a version of this theory with his ultimate genocereal relativity, which then indeed exhibit this covariance under genomic coordinate distributions. And he actually came to terms with this general convergence by implementing an idea that actually has emerged several times already in our discussions today. That idea he formulated in terms of particle coincidences. I'll draw you a little picture here to explain how that works. Hopefully it works. So he said, well, let's suppose that in addition to what other kinds of variables we might have in this region of space-time, they could be gravitational, they could be electromagnetic fields, whatever, suppose we have two particles. And so I've traced out here two particle world lines, a red particle and a blue particle moving forward in time. And he said, well, it does make sense, and every observer would agree, where those particles cross, where they collide, they are this one-dimensional picture. The particle, this is coming at the left, that was coming at the right, they collide right there. And no observer would dispute the fact that this collision had occurred. And so he said, well, that collision then uniquely fixes an event in space-time. And then one could actually inquire as to what other values were assumed by other physical

45:00 variables at that particular event. And that would be a significant observation about nature. So this is the beginning of this relational view of general relativity that we've heard much about and of course we'll know quite a bit more about it as I've received. So I'm in a almost successful animation here. It didn't quite work. I'm not going to repeat it so you can't detect where I went wrong. But what I tried to do was to show that these two trajectories remained the same outside this area, but they were altered inside. So I want you to imagine that I'm actually thinking of positions on this paper as representing fixed coordinates. And as a consequence of having performed coordinate transformations, has actually moved in the interior of this hole. And so now, Einstein has convinced himself that this is not a problem because he says, well, that point there is the same point as I had previously. That's where that collation occurred. And so I can then look and see what values other physical variables assume with that particular space-time point. So in spite of the fact that we have general covariance, we still can make precise general predictions. And of course, we haven't violated this notion of determinants. Because presumably, whatever value that field had at that point of collision would be the same amount of what portals we chose. And it's interesting, this whole argument emerged at the same time that Einstein created the general theory of relativity. And in fact, it's been occasionally rediscovered and reinterpreted since then. I don't want to turn to this problem of posing initial conditions in general relativity. Well, in fact, I'm going to start with a theory which is like general relativity in the sense that it exhibits the same sort of symmetry transformations. In this case, the theory that's most relevant has to do with the development of formalism for dealing with general relativity was the theory of electromagnetism. I'm showing

47:30 you here a work that was performed by Leon Rosenfeld. Rosenfeld actually gives me occasion to mention that other name which had not been mentioned until the last talk, actually in the previous session, Neil Bohr, he probably recognized Rosenfeld as having to be very co-author of a very famous paper dealing with the publications on the campus in the poor Rosenfeld paper. But Rosenfeld was very, he was a young student, he was actually a young collaborator Wolfgang Pauly in Switzerland at the time, and he was aware that people who were attempting to put together a quantum theory of electricity and magnetism, they had encountered a substantial obstacle, and that obstacle is related to this determinism problem that I mentioned that had arisen in electromagnetism. And apparently it was Wolfgang Pauly who suggested to Leon Rosenfeld a particular procedure that could be used to avoid the problem in electromagnetism and then by extension in the general theory of relativity. So the problem that we're addressing here is how do we formulate electromagnetism in this case but then also general relativity so that we can track the evolution of relevant physical variables from some initial time. In fact, in a formulation that lends itself ultimately to constructing a quantity. Those of you who are players in this game recognize that we call that a canonical quantization, but the procedure that we ultimately want to try to put together is one in which we look at values of fields and positions and the corresponding momenta at some initial time and then try to track the evolution of those variables in time. And that's the issue that first had to be tackled in the case of electromagnetism. And Roosevelt then put together a very impressive quote, one that's not adequately appreciated, in fact, in 1930. And I'm looking to hear the first page of that article. It's called On the Quantization of Wavefields.

50:00 And in fact, this was a very impressive first attempt at a very early stage at unifying electricity, magnetism, and also gravity in one single theory. In fact, it suggests to me a parallel with the Apollo moonshot. It went amazingly far in 1930 and then was subsequently forgotten and then rediscovered and reinvented 20 years later. But before I get to that re-invention, let me just point out some of Rosenfeld's connections. Here's a picture of him with Senator Heisenberg. By the way, I thoroughly enjoyed a conversation recently with Professor DeWitt, who actually was a friend of Lea Rosenberg. And I heard some very interesting voice. And you may want to ask her about him also. OK, so let me now go on to . What was Rosenberg's formalism? He was actually the inventor of a formalism that we now call constrained Hamiltonian dynamics. And he actually produced an algorithm which showed how one could employ the symmetry of general relativity to deduce that one was not actually free to prescribe initial conditions to positions and momentum when applied to general relativity, or that there were constraining relations among them. And in addition, and this is something that has not received Goddard's attention, he was the first to show that consistency conditions actually lead to more constraints. It's actually a discovery that is most frequently attributed to Bergman in Iraq, 20 years later. Well, let me now tell you a little bit about Peter Bergman. Peter Bergman actually was my patient advisor at Syracuse. He collaborated with a very beloved and respected thing at Syracuse University. And it's a very interesting story and I just want to tell you a little bit about it.

52:30 Because he was born in 1915 in Berlin. He actually moved with his mother to the private world in 1922. His father was in fact a very famous scientist. is probably better known in chemistry than Peter Bergman was known in physics. He was the head of a Kaiser Wilhelm Institute in Germany, at the same time that Albert Einstein was head of one of those institutes in Berlin. And it's very likely that Max Bergman was a friend of Bergman. Even at that time, there is actual correspondence that comes from later on. Peter Bergman actually went to Charles University in 1936. He graduated in 1936 and amazingly, immediately following his graduation, he went to work with Albert Einstein in Princeton. He was just a young man and he worked with Albert Einstein on unified field theories from 1936 to 1941. After that, he went to Syracuse University. He worked there until retirement in 2002. He died, actually, in 2002. He was actually the recipient, with John Wheeler, of the first Einstein Prize from the American Institute of Physics. I think Justin deserved the award. Here are just a couple pictures. This is a famous one. Peter Verben at the board of Bernal Einstein in Princeton. Oh, there's another famous one. This is Bergman, Barman, and Einstein. I'm walking just outside of Poland, Mercer Street in Princeton. I can never resist showing you this. Most of you have already seen this picture. But it turns out, one day, I was walking with Professor Bergman in a place I loved visiting in the 1980s, I think it was, and the guy, a friend in the background, said to my wife, picture. My friend recognized immediately the parallel scene and said, someday, well,

55:00 completely wrong. But in any case, it's a fun picture. So I want to tell you a little bit about Peter Burman's work. From the very start, when he joined Circus University, he immediately was concerned with this attempt at merging quantum mechanics with general relativity. And in fact, I think he was one of the few people at the time who really was in a position to be able to make headway in that endeavor. In fact, I'm showing you here an excerpt from a letter of recommendation that Albert Einstein wrote for Peter Bergen in 1954, and he essentially says, well, there are present only few that have penetrated deeply enough into both theories of climate and individual activity to be able to undertake attempts such as this. And I think that's true. In fact, it's true that in this era, this was in the 1950s, early 1960s, Syracuse University and Peter Rivers really wanted the, if not the major, research location for general relativity in the world. Most relatives actually passed through Syracuse in that time. So what did they do, or what did the certain Bergman and his collaborators do? Well, as I mentioned before, the task that one faces if one wants to undertake this canonical quantization procedure is to put general relativity in the form of positions or fields and corresponding momentum, and so one can then promote these physical barriers into quantum mechanical operators. Of course, that would only be the first couple, but we'll still be faced with substantial conceptual and technical problems, none of which have been absolutely solved up to this date. But that's an economic procedure. To be contrasted, function integral decision, which was described to us, although, of course, there are consistent relationships between them. Now, one very important focus of this group, of the Bergman and the group, was this insistence that one wanted to have a handle on how the physical

57:30 variables, in this case, gravitational variables, and, of course, not momenta, how they changed changed when one conducted changes in coordinate systems. That was the focus from the very beginning. And they made substantial progress in this direction. But in a way, their work was rendered, perhaps, less consequential by a breakthrough that Paul Dirac made in 1958, which he succeeded through a fairly simple change in variables and putting general relativity in the form which we enjoyed earlier, which these constraints that arise take a fairly simple form, dynamical laws are consequently simplified. Now actually, Dirac and Bergman had been independently developing this formalism of constraints and dynamics and at least those of us who are associated with the Syrian school always refer to this formalism as the Durak-Bergman procedure or Bergman-Durak although that is not a terminology that has universal usage unfortunately because as I've been discovering as I have been developing this history, Bergman and his group did make substantial and important contributions, very much on par with those of Dirac, and in some senses, superior, as I hope to show you in just a moment. So one thing that did occur as a consequence of Dirac's simplification of generality that he caused everyone in the world to abandon as serious physical variables quantities that we call the lapse. In fact, more generally, lapse and shift functions. These are things that, in Darwin, one can change arbitrarily. The lapse actually determines the time elapsed on some classical clock, taken along this world line, there's some function that one can use to calculate how

1:00:00 much time has elapsed. And the argument was that one could arbitrarily change how you rule the constant time services in space-time with a corresponding change in the laps. And And as a consequence, people argue this lapse function is insignificant, it has no physical significance. And I view this actually as a significant detour that I'm hoping that in the very near future will be corrected. Now let me say a little bit about Frozen Time. Actually, we don't need to read this entire correspondence here, but this is a, I've been fortunate to be able to have access to Professor Bergman's papers of the journal and the archives of the University. I've come across some of this correspondence between Bergman and Dirac. Actually, this part is simply an expression of congratulations to Dirac for having achieved this break. But perhaps more interesting, this letter of 1959, is this question which Dirac poses to Dirac. I discussed your paper at a Stevens conference yesterday. Two more questions, which I'd like to submit to you. To me it appears that because you use the Hamiltonian constraint to eliminate one of the non-substantial field variables in the the final formulation of theory, your Hamiltonian vanishes, and hence all the final field variables are frozen, constants of motion. As far as I'm aware, even though this is an unpublished letter, this is the first instance in which this, that I've uncovered, in which this deduction is made from this formalism that existed at the time, that since the development in time itself appears to be a symmetry, one can view it actually as a simple redefinition of time by a constant translation, which would be a symmetry in the original form.