Nature of Time — Further Discussions

0:00 A function of t, but x is a prescription which maps the time interval into space. And in a function space, x is just a point. And this is a domain of integration of the path integral. And one of the reasons function integration of path integral in physics didn't make progress for a long time is that one was not focusing first on the domain of integration. Now, given the domain of integration and there's lots of function spaces, which one I choose depends which physics problem you want me to solve. of Feynman, the first path integral he computed, the path was a function x from an interval t, a, t, b into an x was in the space x, a, b. But given, so this was a space of path which all went from the point A to the point B in time TB minus TA. That's a very special path and a very special space of path. But once given the space of path then there's various volume elements one can define and which volume element you choose is really a skill of, so that your computation will be easy afterwards. There's no one magical volume element. You choose it in terms of the physics problem you want to do. Now, given the volume element, then you can define a norm and the whole story of integration goes through, but then at that point you compute the integral over your domain of integration of your volume element and whichever function integral, functional you have to integrate. Now when I say you choose it with skill, it means that there's some term of the volume element which I can insert in f of x or some part of f of x that I can put in the volume element, So that's where the skill of computing functional integral goes.

2:30 And after all, we do know, what do we know about integration of ordinary integration? We know integration by part, change of variable of integration, and a good table of integrals. That's about all we know when we start computing it. And this is the same thing in functional integration. for function integral, integration by part, change of variable, which is a totally different sense from change of variable in ordinary integral. And a good table of integrals, that exists. There is a good long table of integrals. Now, let's, since the focus of this conference is on time, let me then just say something about this. Because this is a probability amplitude to go from A at T A to B at T B, and this, when Feynman started, we know that, so then it did define the functional integral, that means this is defined by this, so functional integral was defined by the physics as a as a representation of those probability amplitudes. Which is one of the reasons that function integral also slowed down, because a function integral is a mathematical object. You can do a whole course of function integration without even mentioning in a physics. It's as a mathematical technique as much as in the ordinary integration. But since classical physics is the limit of quantum physics, then the functional integral that Feynman worked with was the exponential of the action functional on that path. And this was a space of path going from A to B. So, it's a subspace of the big space of parts, it's those which go from A to B, in time, Tb minus Tb. So, at this point, well, two things. First of all, the boundary can be...

5:00 Let me backtrack. First of all, this is a function of B, Tb, A, Tb. and as such a function it's a solution of a Schrodinger equation given an initial wave function. So one of the interest of functional integrals is that they bring together the one solution of the PDE you're interested in. In other where there's enough information on the path integral so that it does select a particular solution of the Schrodinger equation. Now, at this point, time is a convenient parameter for the path defined on a fixed time interval. Now, without discussing the philosophy of slicing the time interval a little bit of time and looking at this at the limit of an infinite number of slices that are infinitely slim. Whether conceptually, philosophically, historically it's a good thing. For functional integral, it's a very, very bad thing. And the reason for that is that if you do it through time slicing, your pass integral looks like the limit of an integral over rn when n goes to infinity. And function spaces are very, very rarely such a limit. So this is the first thing I've learned about time, regardless of what it really means. Now, then yesterday I mentioned three examples of functioning the role in which we manipulate the time as a convenient parameter, a convenience for the problem we want to solve. So I first, I first review the fixed energy transition is also known as the elliptic PDE or the first exit problem, first exit time.

7:30 I'm going first to give you briefly how probabilists have solved elliptic PDE using pass interval. At this point, a pass is not just a function of time, it's a function of time and a function on a probability space. Capital Omega is a probability space with a measure. and X maps little t and little omega into a point in Rv. So we either look at X of t, that is a random variable, so you need a point of omega to know a path, or we look at X of omega, which maps a time interval into Rv, and there it's called a sample path. Now X of omega, in the case of elliptic PDE, maps a time interval, usually taken at t equals zero, but that's rather irrelevant, to the first exit time of that path. In other words, each path is labeled with omega that is a point of a probability space. And the first exit time is you choose an initial point and look at all the paths and they're killed so to say when they arrive at the boundary. And the time at which the path arrived at its boundary is the first exit time. And the path integral for a probabilist is an integral over the space, the probability space of omega of a function X of omega, which maps this time interval. So each path has its own first exit time. So first exit time are path dependent. To apply that to physics, there was a couple of technicality to do. one of them that probabilists work with probability, whereas we work with probability amplitude. But that is really not a big problem.

10:00 I mean, historically there was such an uphole when Feynman wrote something which looks like a Wiener Integron and people got into a complete dizzy by saying, oh, but there's an I. and then the condition for a Lebesgue integral do not apply when they get right, which means that in a Wiener integral, you can define a volume element that is really a measure in the sense of Lebesgue. In the case of Feynman, you can define a volume element that is not a measure, but that's no big deal. I mean, it does prevent you from doing no integral. Now, to move on to physics, We also have to recognize that in probability time is directed, whereas in physics most of the equations we work with are time reversible. So one way to use, one way to get these ideas from probabilities that use them in physics was to introduce a map from a time interval O1 cross a fixed time interval TATB into R. Mu is a point in the interval one and T is a time, if you want, in the interval TATB. Also written T mu of T, these are two identical expressions. I want to focus on the time evolution. So what it does, tau mu maps a fixed time interval into a time interval which depends on tau mu, mu of tA into t mu of tA and tB into t mu of tB. So that a path, x mu, whose first exit time would be t mu of tB, maps the time interval t mu of ta to t mu of tb into wherever I want it to be, I mean a point between a and b. And of course x mu at this point would map it here, x mu at this point would map it here.

12:30 So, in the context of path interval, we now have set up paths with their own exit time. map we can replace that sequence first finding variable time interval etc we can remove this sequence of mapping by X mu composition T mu as a mapping and saying that at TA it would be at A. And this new function, which has now been introduced, will also be one of the function that the path integral will integrate over. Now here, I have paths paths Which are characterized by one parameter, namely mu. If you ask me why didn't I call it omega? It looks exactly like that. Why do I call it mu and not omega? It's because omega belongs to the probabilities and if I used omega They think that I wouldn't make any difference between a Wiener integral and a Feynman integral and that I'm not but aware of all the problems that the i, square root of minus one, introducing the problem. So that's just to save peace with the probabilist. This is there for me. Now I can, at that point, what I thought I would do, I would go through the transparency and then be open for discussion at which point down he was dealing with the blackboard and then I can answer more technical questions on given that principle of this kind of passing the goal, how do I even do it? But in terms of understanding time and probabilistic time, this is the way I can handle it. And if you want more detail, that's all in the way. Okay, so the other problem I looked at was the Poulomb problem, and this is basically

15:00 a reviewer with just a little bit of a compliment of what I said yesterday. Now, I've been teaching long enough since 1944, that I know that teaching requires repetition. So there is some repetition with yesterday. This, in term of time, show that you can both change, let's say, a path in R-D into a path in R-N, they're not in the same dimension, so it's not a change in R-D. of variable, it's not a change of variable in finite dimensional space, it is a change of variable in infinite dimensional space. So, you have a path X, which at time T takes a value in our D, and a time T. Then you have another dynamical system with a path Y, which has, quote, time U, maps Y of U into our D. and the transformation shows that this system is equivalent to this system or other. Given a system here, one can find an equivalent system here for the couple of DNN which are D1248 and N1231. And that was used to do the first solid pass integral for a particle in a Coulomb potential, a one over half potential, Newtonian potential. It was used to rewrite the functional integral for a particle in a one over half potential in R3 into a pass integral over a harmonic oscillator in R4, which was very easy to do and regular at the center of attraction and that produced the energy level of the particle in the Coulomb potential. But from that simple example a lot of work has been done which shows that there are space

17:30 transformation, coupled with time transformation, which produce equivalent system. So again, what is time for me? Well, it's the convenient parameter that I use to do my work, and I'll do it either this one or this one, depending on what I want to do, but the whole story of coupling a time transformation, which is often a complicated mapping, but feasible, explicitly feasible, but coupling the time transformation with a space transformation to get the equivalent system is one case of infinite dimensional space that corresponds for infinite dimensional space to something very simple i i could even mention that linear transformation on finite dimensional space is so trivial that sometimes they're practical but they don't really teach you much i mean y is not linear in x only if x is equal to a y and that's all there is to it whereas a linear transformation in an infinite spatial space, there's a whole collection of them, including expanding in the larger functions, and I mean, they're totally different, they're still linear, but they are totally different tools, and it's the use of those linear transformations, which very often makes functional very powerful. Now, one of the functions that I've computed, which I like a lot because it brings a lot of things about it. Do you know what is glory scattering? Glory scattering is actually at the origin of cloud chambers. Glorious Catherine is, it has a proper position of the sun and the cloud. You see your image in the cloud and you see a halo around your head. And it's a very interesting phenomenon because actually nobody else sees a halo with you.

20:00 And Whistle was in Scotland and was seeing those. Sometimes we can see it from a frame, sometimes we see it. And we got very intrigued, and we constructed cloud chambers, but then cloud chambers were used for particles and high energy and so on. Now the cross-section of gloria scattering is not a trivial problem in a lot of work, but through a pass integral, I can give you the cross-section in terms of elliptic functions, explicitly and much more complicated calculations to be done. So we get it into a J function and a different one depending on whether or not the wave. I mean I'm describing it as a very common effective wave and visual thing, but it works on gravitational waves, it works or as I say, this different elliptic function, j sub s. that is having used time as a continuous parameter. And the last example I gave was past going back a little time. I think there is a misprint, it should say a particle, anti-particle problem, sorry. And then it is also known as particle going backward in time and also the so-called proper time formalism. In that situation, x is a path in Minkowski space, so it has four coordinates and let's say that x0 would be the time in a certain sense and then x1, x2, x3 would be the space coordinates. Now I take a parameter which let's say s, and I use it to parametrize my path

22:30 in Minkowski space. So I still have time as x zero over s, but in terms of the parameter I'm using, I'm using that parameter in R+. Now, paths going backward in time were introduced so I was told, and I didn't have a reference, but I found a reference since, but only very recently, and I've not read it, moreover it's in German, and it's from 1937, the Soviet Union Zeitgeist for Physique, and I think that's what it does, but I'm not absolutely sure. But for sure, it was really produced by Stuckelberg, but that was during the war, communication between what was going on, I mean, scientifically speaking, between Switzerland and the U.S. And moreover, Stuckelberg had a serious mental illness, so sometimes he was hospitalized, so it was a difficult situation. We have mentioned it to Feynman, but did not produce the formalism. But Feynman did produce the formalism going backward in time, the paths keep moving, but a part of it, X zero of S decreases. And he used it to, he more than used it, he needed it to create quantum electrodynamics. And then he has a very nice explanation to show why antiparticles bring together quantum mechanics and relativity. So you could say that's a situation in which we had to loosen up our understanding of time to consider parts going backward in time to get a good get-together of quantum mechanics and relativity, which culminated, of course, in quantum electrodynamics. Now, his argument, Feynman's argument, why you need a particle going backward in time. In other words, a positron is an electron

25:00 going backward in time. Now, why we need them, this was the argument. In the early days of the Dirac equation, the negative energy states were beginning to be thought of as positrons, but were thought of as holes in a sea of electrons. So Feynman said, I'm going to require particles to have positive energy. If I do so, and go through a Fourier transform of the positive energy, that shows that the propagation outside the light cone. And if you have propagation outside the light cone, then from another frame of reference, it looks like a particle going backward in time. So here we have to think of time as going backward at time or for propagation outside the light cone. So this is the end of the . And I'm glad to answer your question. I have no idea what's that going on. Five minutes or something. Five minutes. Well, it's up to you. I mean, I need questions. I don't know, Russian Yes, yes, yes, but I'm not looking for, I know, I'm not talking about It's a function of integration and it looks not very nice, so I'm sorry, it's a long time to speak. I can also say that I've had a good which came out last month.

27:30 And it's called function integration, action and symmetry. Because there was a feeling too much at the beginning of feeling that to compute function in physics, you needed an action function on it. And that you needed, maybe other terms, but you needed at least an external action of the action function on it. And function regulation and function regulation were really thought of only in that context. But it can be done entirely with symmetries. If you create a symmetry on the space, enough information so that I know what I'm doing, then if you ask me a physics question and you have to tell me in which space I'm working, then I don't need the action functional. I can, with the symmetry of the possibility, construct a possible, all possible, which we answer, whichever, whatever you want me to do. But then if you say, okay, it is, if the action functional to which it corresponds. Well, I can do that because I can do a sort of WPB approximation. And then once I have the answer of the WPB approximation, it looks like a square of W determine the exponential of an action functional, not an action functional, but then I have the action function and out of that action function Sorry, do you state that you have developed a kind of new method? I mean, I've been at it since 1951, so better than 100. No, no, I know your name, but I can't trace anything. No, one of them is to start with equations defined defined by the vector field that generates the symmetries of the space you want me to work with. And from those equations, I can construct a particle. I've used it for instance, I'll tell you where I've used it.

30:00 there's been a lot of discussion of what little parts take their value in the manual manual. Well, actually, I can use it. The parts do not need to take their value in R&D. They can take their value in the manual manual, they can take their value in the multi-day connected space. Well, anything. Just say that to both. Now, in the case of past states we have had in the 1924, there's been a lot of discussion, and I'm going to start on it, because I think it's a great theme on that. The point is, it depends what you want to do, but anyway, to get the bus in the boat on the data writing in Ramanian 94 that I'm comfortable with and can produce things with, I worked on the trend model, because at the moment it's for me to work on, and then project on the big space. And on the frame model, I take an equation which is equivalent to choosing a direction. This is the equation I start from. All I need is a frame model, a choice of direction. I write the line of the direction, or rather the equation of the direction, which tells me how I leave the vector of the space to a vector, to an horizontal vector space. These equations I can use as a stochastic process. I can write a functional table, and then I can get a functional table. I look at it, I look at the Schrodinger equation itself, and I find that itself, the Schrodinger equation without any additional human curvature, that's where the type of slides are. He puts an additional term in the Schobinger equation, but he has very document with it. And my answer is, if you want an additional term in your Schobinger equation, then go ahead and do your facetology the way you do it.

32:30 But at this point, I have no reason to put an additional term in my Schobinger equation. and whether it would experimentally ever show the difference, I don't know, it's so new, I mean, it's such a story, I don't think. I mean, it might some way. But before you leave, these are different certain equations, and the fact is that by different functions they are. Each of them, I mean, some, the one of ours comes from the Schumann division principle. Mine comes from the and for that. So, sounds very interesting. Oh yeah, wouldn't matter for 50 years, we didn't even try to write it for 50 years. So this is just a general question, but I will view the relationship, the path to the operator. Oh, basically there's two different forms of organizing and the operator of organism, I don't know, I mean, it's a matter of taste I guess. Looking at the assumption you make to work out the operator of organism, the assumption you make to write the function table, or the technical computational power of one or the other, and sometimes the problem you want to solve. Right, yeah, I'm sort of, I was just, I can tell that there is that practical aspect that I was wondering if you were looking for. Of course, I'm supposed to be able to tell you when I write a functional rule to which intruder is going to an expectation value of a certain arboretum. So once I've computed my function visible, I can't tell you what he would have done. So in other words, if it's easier for me to compute the function below than the expectation value

35:00 of the algorithm and the initial type of state. Well, of course, I have no experience in the context. So you don't regard one in state that one model is more general than the other? Well, I can say that by the way, it was really good at that a lot. Thought, at least for one of the birthday, that functional integration was much more powerful than the chemistry of formalism. And there's a big book of 1,149 pages to it. Is that the cause of the variance? Why did he assert that this was so? Because we found a concern. I mean, actually, we've written a very few people together, which we've written one on the Museum of Government of I thought I'd come, I mean, you see, you have to have a very dynamic in the first class, and the second class, and the third class, and the third class, and the third class, and the third class, and the third class, and the fourth class. But it was really the way of handling those strings that he started to. That was one, and the other one was I didn't discuss about the Pryos bracket because the particular application of Pryos bracket are the usual, the usual bracket is the classic. Pryos. Pryos. Pryos bracket. The Pryos bracket are on the particular case. The Pryos bracket also is used to a different case. and then there are issues about climate set-up. Well, the fire's blackout, I could see the legal government, and lead to a much easier formation of concerns. So, I think this is what it is. Well, um, yeah, I'm wondering what I just wanted to do. Oh, maybe I'll make my answers really quick. I'll give a chance, but then I'm like, I'll just say yes or no. I'll just say yes or no. Um, oh, for example, my book, Cambridge University of Christ.

37:30 Okay. Cambridge University of Christ, the authors are Gartier, so they're going to come to terms of other math. Gartier and Louis Maret, and it's called function integration, action, and symmetries. And it's in English, actually. Most of it has been edited by the student of Langley in English, but then there are a few additions which have been quite similar. Okay, one more quick question. Actually, it's time to come. While I'm standing up here. Yeah, anyone's right here. I hope I didn't damage it, it fell down. Let's see. There it is. Very good. Yeah, that's all right. I'm not sorry if I didn't do that, but I'll be around and I'll be glad to answer questions. Okay. I'm sorry, we'll give you a moment to hold the computer up there, so perhaps, here, let me leave the microphone. God, I'm so sorry to quote this all. It always seems to be something that comes up. Fortunately, I think the problems have been overcome. I will try to shorten this presentation so that we do have some time to try to collect our thoughts for a little while. So I'm going to follow up with observations I was making yesterday in my popular-level talk, and I wanted to provide some more details today. And I thought I would do it in terms of a very specific cosmological model. So I'm hoping I can use that to illustrate some more general points of how we propose that one could go about constructing observables in a ordinary context in general activity.

40:00 So here's an overview of what I want to do. I do want to talk about this enlarged symmetry group that I mentioned yesterday, which does not actually include a global translation in time. Then I want to talk about choice of intrinsic time coordinates and use of these coordinates to produce a non-trivial evolution in time. And then I wanted to try out on you a procedure for introducing time-dependent Schrodinger equation in a general relativity context in Albany at the semi-taskal limit of this very simple model. So, first, a little bit of formalism here. So, I'm going to consider pretty much the simplest non-trivial and fibrillant all-tricing model that one can contemplate, and that's one for an isotropic expanding universe. This is the metric. One thing that I've done here from the very start is to extract dimensional units from everything in terms of Planck So this is the plant length here, so that all the variables in here are dimensionless. And this will be convenient later on when we undergo the plantization procedure. The fundamental variables in this month are simply the expansion factor, which depends only on T. And you'll notice I'm leading in the so-called lapse motion for reasons that I began to explain yesterday. I've written down the expression for the Hamiltonian, the Lagrangian Hamiltonian system. that I had in NASA's scalar field, phi, that's only on time also, here's the corresponding of the grange written in terms of these dimensionless coordinates, Hamiltonian, canada Hamiltonian produced in the usual procedure, given by this expression. And one can see immediately, in the presence of the lab solution, that one gets an immediate primary constraint, which is given by this expression. So we have restraining relations among the moment associated with both the A and T fields in the expansion factor. Then, of course, you have the other primary constraint associated with the absence of time derivatives of the lapse function.

42:30 I should have indicated the notation here on the dots, of course, represents the derivatives of the time coordinates. So this is the model. Why do you have approximate constraints? So I'm actually sort of following Dirac procedure here and thinking of this as a weak equality instead of it, but it didn't realize I just want to use the ordinary equality. Okay, now the fundamental idea that I want to kind of devote a little bit to the expert group is the symmetry group, which is evident And in all of these generally over in my, here I'm just showing you a very simple case. So the question that one has to address, if one wants to try to realize variations in canonical coordinates that are caused by coordinate transformations, if one wants to be able to recover those variations in the phase space, Then it turns out, not all such variations in configuration velocity space are projectable on phase space. And this is the fundamental idea behind the origin of this enlarged symmetry group. So it turns out that there's no way that you can conceive of a function on configuration velocity space that is a function of the time period that will elapsed, there's no way that you can conceive of that as the pullback of a function on phase space. You can say that, I think, hopefully I can convince you this in one simple sentence, because it turns out that if you had a function that you was a candidate as a pullback from velocity, that means some function f, say, of n and p sub n, or that's the momentum associated with the last function. Now, normally you would write those functions, those momentum as configuration velocity variables, and plug those into the phase space functions and thereby

45:00 by obtaining a function on configuration velocity space. But if you, but piece of n vanishes identically, and therefore there is no way of pulling back function of piece of n back to configuration velocity space. The result is that no functions on configuration velocity space that depend on n dot can be projectable to phase space. This is a fundamental idea, and we might be able to delve into ask some questions about it, but the result is that the simple translation in time, which would consist of t prime is equal to t plus a constant, is therefore not implementable as a symmetry transformation . One is forced to add a dependence on the lattice. Now, actually, this decomposition is one that's very well known and was actually first introduced of Iraq, because this is simply the statement that, in this case, when we're doing a little of the time transformations, that the infinitesimal change in time must be proportional to the normal, to the time only agent. And this expression here is, in fact, the one component of that normal vector. So in that sense, this is a result that's very well known, but we would maintain as explicitly that if you carry out a variation of the lapse that results from a from a coordinate transformation in this case coordinate transformation of time then you can see explicitly if you work out the geometric variation that's caused by that coordinate transformation you get an end dot term this one end dot term there they cancel and the resulting variation does it depend on the time grid of the lab structure. So this is the crucial point. The global translation in time, time evolution, is not one of the symmetry transformations when you're working in phase space as opposed to configuration velocity space. And additionally, I've written down the expression for the actual generator of, the canonical generator of this transformation. You can almost read off, but you can read off here. that the variation in this that's engendered by this generator is simply ,, which is the result we have here.

47:30 So this is the full generator. And of course, this generalizes to more complex situations. Given here just the simplest situation I can think of. I should have mentioned before that some of the things I'm going to describe here are actually composed a part of an honors thesis that was taken, that was performed by Judy Draghi from Austin College a couple of years ago. Her name is Allison Schmidt. The name should have appeared in my program. Sorry, I didn't include it. It's okay. Can you go back? Sure. I just want to understand what you did. What, is it, T is the coordinate time? T is the coordinate time, that's right. T prime is? T prime would be an infinitesimal, a new coordinate time. So this is an infinitesimal coordinate transformation. T prime is the new coordinate time, T is the old one. Okay. But you see here explicitly that I should have written out, N, both N and Psi here, depends on T. So this is a coordinate transformation, but the particular functional form of this coordinate transformation depends on the elapsed variable that already exists on a mammal, on a space town. Nothing. Right, so you are starting with space-time. I started with space-time with some existing metric on space-time. And then I discovered that I can only implement as a symmetry transformation, coordinate transformations that depend explicitly on the labs. What is the step on INDOT? INDOT? What is the significance of your mark on INDOT? I point out in the first line here that you cannot match up functions on configuration velocity space that depend on any dot with functions on face space, or in more precise mathematical terms. There is no function on face space that's pulled back under the genre map to configuration velocity space. Why did you make that demand? It doesn't appear anywhere here. Oh, it doesn't. But it does appear, it would appear. If I didn't have this independent term here, then the exclusive variation, the variation that wouldn't be engendered by a coordinate transformation that didn't depend on capital N would be this one right here.

50:00 And I'm claiming that this variation cannot be realized as a canonical transformation. Okay, here, a little bit of mathematics, I'm just, fortunately, in this situation, very simple model, we have exact solutions. In fact, I can write down the exact solution for any choice of time coordinate. This is the way in which the expansion factor depends on time. You notice here that this arbitrary motion, lambda, that appears in the Hamiltonian, appears And I also write down the general solution for the scalar field. Also, the general solution for the lapse, which also depends explicitly on whatever choice you make for lambda. So it's good that if I vary lambda, if I change the Hamiltonian, then I'm going to do gender corresponding changes in all these variables. So the idea now is, once you can't see that very clearly on it, is that I want to give you an explicit illustration of the way in which one could implement intrinsic coordinates in this very simple model. The suggestion I'm making here is that, I can't quite read it here, is that we choose to let the square of the expansion factor represent the time. Very simple idea, in fact, fairly closely related to Professor Riddler's observation that people use the density as a clock. The density would be a different choice, but that would be essentially, in this case, one over a cube. But we can make any such choice. The one restriction you would imagine would be that whatever choice you make, It should be with a variable which has a monotonic increase or decrease in working time. And that's certainly true for the expansion factor. It is monotonicly decreasing. So it would represent a potential candidate for an intrinsic clock. Our objective now will be to relate the value of that intrinsic clock to the other variables. Very simple idea, an old idea. But in a way, it goes all the way back to Einstein. It was rediscovered, implemented in non-performalism originally by Bergman and Kumar, rediscovered in some sense by the Belly.

52:30 So that's a fairly long history. But I want to show you several different ways of implementing this choice of intrinsic time, One is, one could imagine that we think of that intrinsic time, which I'm calling you capital T, as a new coordinate, certainly is a new coordinate, and one simply makes a transformation from that, from the old coordinate, middle T, to the new coordinate capital T. So, in fact, I'm putting this intrinsic time coordinate on the same footing as the original parameters in several different ways. So then, since phi is a scalar, this mainly produces the new function phi of the intrinsic time as a particular functional form. It also produces...