Time & Quantum Consciousness (contd.) / Consistent Histories View of Time

0:00 So our perception of moving through time and forward motion is actually due to the fact that we are confined in the present moment, an ever-present reality, in other words. But that's one response, I guess, that would be, so I mean, it's still occurring, it's just our perception in this progressive present. It seems as if we are moving in a progressive, linear fashion towards the future. Yeah. Did that answer your question? My connection is something else that was said that I have no answers. And that was somebody had the Copenhagen, yeah, the Copenhagen School of Thorpe Gore, saying he only believed in imagining quantum mechanics and emerged into classical reality. He had a classical measurement device. So he might say you can only basically measure time, and that connects with the Wheeler-DeWitt equations, where there is no time, only in terms of parameter, until it emerges in a place of reality. It has no meaning. So the board of the school and the Wheeler-DeWitt stuff, even though they're on the opposite sides of the humanities that were, you know, because Wheeler-DeWitt is the Princeton School of Law. Okay, I'm done now. Thank you. Yeah, and so along those same lines, just talking about the – well, I'm still thinking about the forwards versus backwards things. It's because we have forward-facing eyes, and we identify ourselves as walking forward. answer as to why it's one instead of the other. And I suppose it might require a cosmological answer, which is kind of the direction that I was going to go now, which is, so we talked about the quantum realm and time emerging out of timelessness as superposition happens, but then relativistically, I also think of time emerging out of timelessness cosmologically as if you start at the Big Bang, which is a radiation event, in a state of timelessness, then you have particles freezing out of that state of timelessness. Essentially, time is created with math. So, that's just another idea to play with. As far as the, there's a bit of a complementary relativistic way time emerges out of timelessness.

2:30 and quantum way of time and work is out of timelessness. It is 2.50 now and I could either just start entertaining questions and we could talk about it or I could go into a little bit about why I want to use fractals to describe this realm between time and timelessness. Any opinions? I don't know what you want to do with that. That was cool. All right, let's talk about the fractals. Okay, so basically there are four, five, six, I always keep changing them, properties of fractals that I feel map onto, properties of our subjective experience of time really neatly. And so it looks like it would be a really nice mathematical model to use to describe what I kind of refer to as our subjective topological experience of time. So that if you think of a regular map where you just see the ley lines, the grid lines, as opposed to a topographic map where you see where the resistance is, where the uphills are and where the downhills are. And so if you think of a topographic map of your temporal experience, where is the more resistance? Where does time feel like it's going slower? Where are the downhills? Where does time feel like it's going faster? Anyhow, so a fractal topology of time is kind of what I'm trying to get at. Okay, so the first thing is that fractals are self-similar, recursive, reiterative. You know, they go in cycles, which is very neat because that's exactly what our temporal sense emerges out of our relationship with the earth and with the sun and with the seasons that are all going in cycles. In addition, fractals give us really good descriptions of the natural world, of mountains, of coastlines, of various other things. You know, it seems like it would be a natural extension of this that they could also describe the intricacies of temporal experience. And one of the really beautiful things that they do that I love is that they have, so like an example of a fractal is this basic thing. It's one pattern that's repeated continuously on smaller and smaller scales. And so what you have is a finite space here. And within that finite space, if you continually repeat this, you're going to have an infinite surface area within a finite space.

5:00 And so it's this relation between the infinite and the finite that we experience in time all the time. That our moment that we are in is both finite and infinite. You know, you can never escape the moment. It's really the only thing that is eternal. While at the same time we think of it as this very thin slice of our entire reality. And so in terms of age, as you get older, time goes faster, right? the kind of generally accepted notion of one of our subjective perceptions of time. And one way to think about that is that each year is a smaller and smaller percentage of your life. So at five, one year is one-fifth of your life. At 100, one year is 100 of your life. So of course it's gonna seem comparatively smaller. So in trying to develop a model of time that's more than linear, that encompasses linearity, Because we do need to maintain that aspect because it's part of our experience, but we also want to encompass the different textures of our experience at the same time. I tend to think about time as deepening instead of being this linear marching forward. And so if you have the one moment of your lifetime, you can say that, okay, well, that's one year of my life. And then when I turn two, I'm just going to divide that moment. You know, you've got a wave, and then you've got the next, you know, harmonic of that wave, you know, and three, you've got the next harmonic of that wave. So you're just deepening into this one eternal moment that you've been given. And it's a practical path, basically. It's the same thing. It's a dimension of scale that you're moving through as opposed to an external dimension. And so, and additionally, often fractals appear on the complex plane, so that you'll have a chaotic system that appears in a regular class of reality down here, and a fractal is used to describe it. And once you bump up in the fractal plane and find that equation, then there's a beautifully ordered, infinitely complex image that appears in this realm here, that appears as chaos in the class of the realm. Anyhow, and the role of complex numbers in time, in time dependence in the Schrodinger equation, and in time and relativity in the cosmological sense is huge, and I haven't heard a really good explanation of exactly what that is or what it does.

7:30 There's definitely something going on right there. Okay, so I think that comes. And Laurent Notal is a French-esque astrophysicist who has a theory of fractal space-time, which comes from, I believe, Feynman, or at least he quotes Feynman as saying that quantum particles travel in fractal trajectories. What was his name? Laurent Notal, and of scale relativity also, and so he's trying to extend relativity so that not only does it cover motion, but that it transfers across scales as well, and so that really applies to the notion of a fractal dimension or a dimension of scale or a dimension of complexity that's going on there, and I really like that kind of alternative dimension as opposed to trying to postulate another dimension, you know, out here somewhere. It kind of makes sense as, you know, a curl dimension of string theory and this dimension of complexity or density or scale so that you're going into something or out of something, zooming in or out rather than moving in a straight line necessarily. All right. I'm going to open up for questions, but I want to clarify that with if you often ask questions, I want you to hang out for a second, and if you're normally quiet, I want to encourage you to ask them for a second. Yeah, just one remark on the question. Self-stimilarity is one of the universal properties of factores. Okay. You have factores that are non-self-stimilar. Okay, so then I'm specifically looking at self-stimilar. And my question is, well, you're making an analogy. Yeah. I mean, the idea that we can find structure and fit the amount of structure behind the structure. And with time, how do you get on with this analogy? What do you do with it? I don't know yet. I'm still working on it. You mean how would I, like, experimentally verify it? Not even, I mean, what do you do with it? What do you do with it? Well, for one thing, I really think we need to reframe the way we look at time and as a society,

10:00 because I think that's one of the crucial, why we're destroying the planet, because we can't think on long enough timescales and we can't judge quality over quantity. And so to bring in a new kind of beautiful, depth, complex, accessible version of time where it just makes people think about their lives in a different way could be helpful. All right. All right, I'm reading a book off and on by Lisa Van Loo. Okay. And once I wanted to take my mind of troubles intensely, so I wouldn't sabotage a relationship with this guy by begging or something like that. And I tried to make emeralds tilings for two to five-year-olds, all right? And I missed the part in the first section of Emperor's Mook Mind, he missed saying something about their non-local rules a child couldn't do. Lisa Randall said that eight-dimensional children could do it with loved ones, maybe even ants. So actually, I couldn't address the ants, because they're smaller, and if you think about the superposition lasting longer for smaller systems, so maybe they just don't have as many of those conscious, because in Hammeroff's paper gives the example of a neutron star that is exceptionally massive, and so it's collapsed what happens so quickly that it doesn't have a chance to be conscious, to have that state of superposition and to objectively, you know, reduce... So the hand would happen very long. I have a chance, but I hope I never read one because I don't like the count of demands. Okay, I'm done now. Anybody else have any questions? Yeah? Well, just a comment. I mean, just remind me of the three left. You talk about the well-incident support. of issues regarding stewardship of the planet and thinking about the long term. The guy who founded the Whole Earth Catalog, Steward Grant, back in the 70s, I don't remember. But the Whole Earth Catalog, I would say, is sort of a part of the college movement in the 70s, and the main image of it was the picture, the first picture of the Earth from space, and it really gives people a better sense of the planet as a sort of a finite system. This is not going to do with fractals, but it's different ways to make it a lot of time.

12:30 He's involved in probably building the world's slowest computer. Thinking about the processes, computer computations take faster on super long time scales. It's objective and different experience. I thought that was going to be so interesting. Thank you all. Thank you, Kerry. Thank you, Kerry. I feel my mind is considerably stretched by these observations. We have one more talk. Chris Beagle from Florida Atlantic University is going to talk to us. and I think he now has a title, is that correct? Yeah, no, not really. And do you, do you use the blackboard? Yes, you use blackboard, okay, good. Would you like the microphone? I don't think I'll do it. Okay. So, my apologies for not having a title. did not plan to talk when I came here. I came here with no talk prepared. But Don and Vaughn particularly have been very welcoming and have urged me on several occasions to at least try to say some words because I've done some work in the past which is largely unpublished on the history's formalism, which is an approach that I'm going to try to explain a little bit later, that might be of some value at a conference like this. It seeks to address the problems raised by relativity theory and the inability to define simultaneity, the inability to separate space from time, in a way that is nonetheless compatible with quantum cannons. So all of this introduction is merely a semi-apology about my lack of preparation and therefore what you're all about to go through. All right. So in order to get started, I want to, first of all, address some of these relational issues that we've

15:00 seen in the course of this conference in order to try to make a point, which will lead me to this history's approach. The statement is that it's really not hard to come up with a relational theory. There's been a lot of work on this. Recently, there's a paper by Gambini, Porto, and Polin, that's on the net, as you can see, that I'm going to try to summarize a little bit, that has to do with the use of real clocks made of real stuff to do quantum mechanics. So their proposal is not at all controversial. They're not changing the theory of quantum mechanics in any particular way. In particular, they have, in the background of their calculation, an ordinary external Newtonian time. that I'm going to call tau. And they imagine that they have a system described by some variable x. So x is some observable having to do with the system. And in addition, they have a clock system t. over here. And each of these systems is described by some density matrix. And of course, the state of the universe, consisting of the clock and the system, lives in a tensor product of the two Hilbert spaces, and they make some mild assumptions about the dynamics in here. I guess the key of them is that certainly if we're looking at any reasonable clock that you'd actually want to use experimentally, it should be statistically independent of the state of the system over here. That's one of the features that a good clock should have. And therefore, they assume two things. First of all, that the density matrix can be written in this form, so there's no statistical correlations between the state of the clock and the state of the system, and also that the Hamiltonian can be written in a similar form, so that the clock

17:30 evolves independently of the system. And when I say evolves here, I mean it evolves in this external Newtonian time, whose origin is not of interest. what we want to do is start asking questions about what an observer who is using this clock to study the dynamics of this system will observe. So we want to ask questions like, what is the probability that the system is going to exhibit some particular behavior, so we could observe this variable x and find the result little x, given that the clock variable has some particular time. And this is just a conditional probability, which you can write as the probability that both x is equal to x and that t is equal to t, divided by the probability just that t is equal to t, independent of what x is doing. Incidentally, because we've assumed that these systems are independent, this is just a simple product of the probabilities that x is equal to x, and that t is equal to t. So, the key thing is that we want to answer this question that doesn't depend on this external Newtonian time. And it's quite easy to get rid of the external Newtonian time. If we assume, and this is an assumption, so I think this is a slight flaw in the paper, but it's a flaw of the sort that physicists overlook all the time. If we assume that these coincidences where x is equal to x and that t is equal to t are exclusive, that is, there's not two values of tau where this happens, then this probability can simply be written as a sum over all of the different possible values of tau. Or an integral. if that was continuous, but it doesn't.

20:00 So, this is the probability an observer using this clock to study this system would measure the system to be in the, you know, if they measured x when their clock reads t, this is the probability that they would get the result, little x. And so like I said at the beginning, there's absolutely nothing, there's no deviation here from ordinary quantum mechanics. We are saying that there is a Newtonian time. We don't know where it comes from. Maybe there's some collective excitations of other bits of the universe or something about the environment that really does tell us what that Newtonian time is. We don't care. We're approaching this like a statistical mechanic. These are things that we don't know and don't really care about. But in order to do our calculations, we assume that there's some highly organized behavior that the universe will exhibit. And what are we doing? We're integrating it out. We don't care at what value of the Newtonian time this happens. What we're interested in is measuring the probability of the coincidence between the clock variable and the system variable. So this, one could argue, is a relational theory of quantum mechanics. it's like Newton's notion of absolute space which he was fundamentally unable to reconcile with Galilean relativity so he said there is absolute space we just can't really figure out what the rest frame of the universe is it's out there but we can't measure it no experiment that you could ever do would measure that thing well here if all we ever measure are these probabilities there's no experiment that you could ever do that would detect this Newtonian time you only get to use your clock times And one could then define a good clock to be one that is good with respect to other clocks. That's, you know, very interesting. You call it a clock on a rock, and then all the clocks are good. Yeah, well, it's not a good clock. We thought it was on kind of a range of what's going to carry like that system.

22:30 It's not just any. We can talk about this offline, so I'm supposing that there's good clocks because we have this notion of Newtonian time. And we know what systems we can use to measure that Newtonian time in the regions where it's well defined and so forth. And those are the regions of the universe that we're trying to do this little game. And so we're saying, yeah, I can generate as many relational theories, if you want to call them that, as you like, using this very, very simple technique. So the question is, what are the observable consequences of this? In order to explain that, what Gambini and Pullen do is they define two things. And the first thing is the probability. So suppose that you measure, you look at your clock and you observe the time t. What we need is the probability that at the time of that observation, when you made that observation, the background Newtonian time, the unphysical time that we're trying to get rid of, was actually t. And you can calculate this, this is just the probability that T of tau is equal to T, divided by the sum of these probabilities, so I'm assuming that there is a time when you observe T, divided by the sum of these probabilities, overall, tau. So given that you do observe t, little t, this is the probability that you observed it at the time tau. And then using this thing, you can define what you might call a relational density matrix. which is not going to evolve, so to speak, in the fictitious Newtonian time. It's going to evolve in clock time, this quantum mechanical clock time. And it turns out that if you sum, again,

25:00 in order to get rid of the Newtonian time, you sum over all possible values that you take, the probability that that was the fictitious time when your clock variable read t times the density matrix of the system at that value of the Newtonian time. It's not clear that this is the right thing to do. However, if you follow through the statistics, you will see that this density matrix gives you these correlations if you assume that the evolution in the Newtonian time, the invisible time, is unitary. So there's no change to quantum mechanics whatsoever, but there's this sort of effective relational density matrix that you can use to calculate. And then the question is, well, what equation of motion does that relational density matrix satisfy? the answer So we assume, first of all, that we've got a really good clock. That is, I'm going to assume, I'm going to allow that there is some Newtonian time that we can make sense of. And that the clock that we're using actually does track it. But the thing that's unavoidable in quantum mechanics is that no matter how well your clock tracks this ideal clock, there's going to be some dispersion. And that's the thing that you have to build in. We're going to assume that this probability here, T sub T of tau, is given by some function. Remember, each of these is just a number. There's no capital T operator anymore. These are just numbers. And it's going to be sharply peaked so that T is about equal to tau. I'm also going to assume, well, they also assume, I beg your pardon, that it's symmetric. Yeah, this is the direct density. So I'm saying that we've got a really, really sharply peaked Gaussian. And you can expand it. And there's higher order terms here. This isn't necessarily the best expansion. But it's the one that most clearly emphasizes the idea that this thing is sharply peaked.

27:30 So, what's going on with, all right, you're not taking derivatives of the right delta course to get something that's going to do that integration by parts and wind up being Yeah, yeah, yeah, you integrate by parts. Yeah, so this is, oh, the fact that I've left out the first derivative. There could be a coefficient of the first derivative in here, but that wouldn't be symmetric forward and backward in time. In other words, the probability that your real clock was ahead of the fictitious clock would be higher than the probability that it was behind by the same amount. And that's, it's just a symmetry argument. It's not essential. It's not essential. You can do this with the first derivative in as well. So the question is, with this assumption what is the equation of motion for this relational density matrix in the lowest order term I should be clearer here this means d by t rho of t minus i the Hamiltonian that describes the system itself with respect to the Newtonian time commutator with rho of t and then there's minus the derivative of t which is expected to be times the second order commutator and then of course there's going to be higher order terms depending on exactly what's going on up here as your clock variable spreads out as you revolve forward in the Newtonian time For the sake of argument, Gambini and Pullen assume that this is a constant sigma, which describes the spread of the clock variable. And certainly this will be true if you look on short enough timescales. I mean, we know that dynamically it's going to continue to spread out, but on short timescales this will be okay. And the interesting thing is that this form of a correction to the Schrodinger equation is not new. But this is a standard form for a decoherence term in the equation. I'm sorry, quick. Can I ask, is there some special criteria for selecting the initial density matrix?

30:00 Is there a question of the ensemble then? Let me answer that question by doing this. And this is the last thing that I'm going to steal from Gambini and Poulin. not my work. This is just a way to motivate the thing. So the question is going to be, what implications does this have for studies that we might do in space-time? So what we could do is calculate the matrix elements of this system when the clock variable reads T. And this turns out to be quite straightforward to do. It's just the matrix elements at T to zero. And again, there's a little conceptual problem here that I haven't quite sorted out. So delta omega here is the difference in the frequencies. These are energy eigenstates with definite frequencies omega. Delta omega is the difference between them. So the fact that we have only delta omegas over here means that this thing preserves probability. That is, if I take the trace of the density matrix, it remains 1 at all times. The second thing is that if we look, if the density matrix represents an energy eigenstate, then the evolution is Schrodinger. I can't, I'm not getting some, you have a imaginary term, that'll be inventory, but the decay term is not the same. When delta omega, when delta omega is zero, there's no decay. Therefore, when we trace this thing, when we add up over all, you know, we set M equal to N and we add up over all possible energy daggons tables. I'm sorry. You know, these are all zero for the trace and therefore probability is concerned. The leading order behavior here in delta omega is the standard evolution that we would get from Schrodinger, and then you've got this decay term that hits all the off-diagonals. So if there's a difference between the energy eigenstates, then that component of the density matrix is suppressed,

32:30 which of course means that you don't preserve the square of the density matrix and pure states, if they are superpositions of energy eigenstates, evolve into mixed states. and there are other people who have come up with the same thing Milburn is probably the best known or at least the most recent best known there's a question what's going on here why is this happening well if we have a system that's in a state that's a superposition of energy eigenstate then basically the whole problem spread in the clock variable. When we measure the system, when our clock reads t, we don't exactly know where it is in its oscillation under Schrodinger evolution in terms of the background Newtonian time that we've integrated out of the problem. So it suggests that if we try to do a sort of cosmological problem where the clock variable is definitely part of the universe that we live in, there should be some fundamental fuzziness to the delta t's when we do our measurement. We must always account for that possibility that we can't do things at sharply defined times. Anyway, that's the prediction of this thing, and I think that if you really try to formulate a relational theory with a clock variable and the clock is quantum mechanical, this seems to me the simplest possible thing that it could be. It's a new term in the Schrodinger equation that accounts for the dispersion in the clock variable. Okay, so what does this mean when we try to do things in space-time? I'm sorry, but I put this behind a ways. I think fuzzes out of time. Rebelli will be involved with the clearance with this fuzzy stuff. I don't know Rebelli, but I know it's fuzzy. Maybe we should talk about this

35:00 after. Okay. what is time in general relativity? suggests that time is a Cauchy surface that you would draw in space-time. So if we had a sharply defined time, that would mean that you had a sharply defined Cauchy surface. There are all sorts of problems when you try to do quantum field theory with arbitrary Cauchy surfaces in space-time. We know about these. You can prove, for example, that even if you're doing free scalar field theory, which is the loveliest of all field theories, on a flat Minkowski background, and you want to calculate the evolution from the Cauchy data at, say, an inertial slice at t equals 0 to some slice that's not inertial later on, that, yes, there's a canonical transformation at the classical level, but no, it's not unitarily implementable, so the evolution is not unitary. We can still mess around with the algebraic quantum field theory approach gets the sensible results, but I think the problem only gets worse if these Cauchy surfaces become fuzzed out. And it's complicated more so by the fact that when we seek to calculate things like these matrix elements that I've just erased with the density matrix, we are the dynamics and the kinematics of the system. And this is the observation that led to the formulation of this history's approach. So in ordinary quantum field theory, you have a canonical connotation relation that says that when you commute the coordinate and the momentum, at the same time, you get a delta function in position. And then to calculate the commutator at other times, equation as operators. So time and space here are clearly being treated on a different footing.

37:30 So the history approach replaces this, this is the bio-opter, with the history in which everything is treated on the same basis. What is being said here? This algebra, this history algebra, cannot be represented on the same Hilbert space that carries a representation of this operator. There are, in some sense, too few degrees of freedom in this Hilbert space that you have here. Because here, T and T prime are allowed to be completely different. So this is a different algebra to write down. So the picture that one should have is that the Hilbert space of the history's approach is not in any sense square-integrable functions of the initial data, but rather square-integrable functions of entire histories, whether or not those histories satisfy the classical equations of motion. So when we look for a Hilbert space that carries a representation of this algebra, which we can do, Chris Isham and his students, Dina Sabadew particularly, have done a lot of work on this in particular cases. I'm going to say a brief word about that before I wrap up here. if I may. So you can find Hobert's basis that have these that carry representations of this in simple enough examples. But, you know, why? The advantage of this is that we can formulate sort of an algebra of temporal propositions. So four covariants with fuzzy time, which means that we can't be looking at little slices of the space-time. We have to be looking at little balls of the space-time. We need to be able to formulate the logic of a system of projection operators that correspond to measurements of what a system is doing in a certain region of space-time. That's what the history formalism lets you do. It would be really nice if I could put this stuff on the board, but I'm almost out of time.

40:00 So let me say one word about what I think possibly could be done with this. Because there's a big problem. The cases that Chris and Dina looked at largely were history formalisms for theories on Minkowski's base, basically, where one has a lot of structure. The Hilbert space that you end up representing this thing on, where these measurements of your system at different times are kinematically related. There is no dynamics that you use to entangle the measurements at one time with another. This is necessary in order to get the algebra represented correctly. So they use the following Hilbert space. So you basically end up with a Hilbert space that is a tensor product over all times of the instantaneous Hilbert space at time t. So you get this huge, in the case of a continuous time variant, you get this huge non-separable Hilbert space. There's two possible definitions of this object. There's one to von Neumann, and the two are distinguished by the definition of the inner product itself. So there's one due to Fomboimant, and there's another due to Streeter. Let me say that, let me do the Streeter definition first, because this is the one that Aichem and Sabadew actually used. It is that if I have two state functions in this Hilbert space here, their inner product is given by the exponential of the integral dt, where t is the variable that labels all of these instantaneous overspaces, of the log these components. So this idea of taking the log, adding up, and then exponentiating is quite a familiar one. so, and obviously you can't define this in all vectors, but you restrict to a sufficient quantity, and this screener tensor product is the completion in that inner product. The Feynoyman inner product

42:30 is simply the product over all times of these instantaneous inner products. The difference between these, so the reason that Aysen and Sabadell were using this Streeter inner product, is because it has this integral dt here. So basically speaking, that's why you can have a delta function here instead of a Kronecker delta. That's the logic. You need this continuous measure built in because this delta function is defined by integration against this menu. Down here, there is absolutely no notion of continuity in time. All you are left with in the Feynman inner product At most, what you're left with is the order of the moments of time. Now, at first, it seems that this is absolutely necessary to do. So it works really well. You can do your free-field theory on the costume space and formulate a history over its space and look at wave functions and so forth. But it doesn't work when you have a generally covariant system because then there's no dt. There's no preferred dt. You can try to do some tricks and make this a dynamical variable like that, but then you have the problem that your inner product depends on a dynamical variable, like the lapse function or something like that. So basically, this has to go away is what I'm suggesting. So for a diffeomorphism invariant theory, it seems to me that the Feynman inner product is much preferable. What's interesting is that as people have developed the loop quantum gravity approach, they've developed representations of ordinary quantum mechanics that are defined on overspaces that are very much like this one. Tensor products of the complex plane at each point of space. On that, they have represented not the vial algebra, not this thing here, canonical commutation relations, not the generators of translations, the momentum operators, but rather the exponentiated forms of those, the translation operators. In a similar way, one could hope to make some progress here. in formulating some diffeomorphism invariant theory that does not depend on having a preferred notion of dt to define this inner plot. I was going to make one more point, which was...

45:00 Okay, I'm sorry. If anybody has a question, I will likely think of what I was about to say just as you're asking the question. Thank you. If you reduce things, take out structure, make it primitive, just the trichotomy, you've got something to get the wrong, quick, and very, in the construction of the theory. Trichotomy meaning an ordering? Yeah, ordering. I just reset what you said to warm you up, so remember what you're going to say. That part of your talk about the probabilistic approach to measure time is actually, I believe, it is used in time methodology when people do this time average, for example, atomic time, which is reported by one ray. You mean in order to make measurement, error estimates of the... Ensemble of clocks. They have to know how that ensemble of clocks behaves in order to extract from that ensemble. It doesn't show over the whole time. And so you take second flow, second flow, and so on, and everything. So you have to have some . Yeah. I think that they use quite a lot of probability theory and . Yeah, so I think the probability theory is likely to be quite similar. But what's being said here is that if we could get, as Jorge would put it, we could get a large enough system of bosons in a condensed state where we could actually look at energy states that are separated by a large enough delta omega and leave it sit there, and we could keep it in that state for an hour, we would be able to see this extra term. We would be able to see this experiment. So I remembered the thing that I was going to say. And so everything I've told you here is nice. It looks like we can represent the dipheomorphism, blah, blah, blah, nothing in this product here that knows much about the geometry of spacetime. The problem

47:30 is dynamics. In the classical theory, the dynamical equations of motion can completely disappear when you look at the history phase space, the classical analog of this Hilbert space. But then you can recover them quite easily. The equations of motion are simply that you write down the action function on this history phase space and you're looking for stationary points. In the quantum mechanics, there are no Hamiltonians anymore, there's no, we have deliberately taken the time evolution, the unitary time evolution, out of the equation and we have reduced the relationships between observations that you would make in space-time from being entangled via dynamics which of course depends on having an external time parameter and we have made them purely kinematical. We have to restore the dependence on some time parameter, whether it's internal or external. And one can also question whether we've really gotten rid of the notion of external time here. We've gotten it down to the point where we're just looking at a collection of points here, with no topology, no geometry. But that doesn't mean that we've eliminated Newtonian time. Can you say a fact? Maybe one quick question. It's really very quick, but I'm just wondering that we get a relation behind the extra function, or a medical function. As you know, we can derive the type of principle from the particular relationship. I'm wondering if that would go by what we would obtain from the inequalities, from the inequalities. It could, but we'd have to have the dynamics. We have to know what the dynamics are doing, because I would imagine that it would show up in commutators between time displaced. up in commutators between time-displaced observations. And so, of course, we'd have to define carefully what we mean by that. I got an NSF grant to work on astrophysics, so I discovered the real problem of time and haven't worked on this in a couple years. So I'm trying to remember what the drawbacks to doing this procedure are.

50:00 Recovering the commutation relations of the ordinary theory is going to be difficult, and it's not even clear exactly what form they should take, especially if the time that you want to do unequal time commutation relations and get some propagator over here, or some Green's function. you have to think about what you're actually trying to do in this theory that's what I'm proposing I'm just suggesting this as a framework where one could try to approach this problem posed by the quantum fuzziness of time which is an experimental well, fact, almost It's close to being testable. Into a theory where time is defined in a covariant way, as some sort of fuzzy slicing of space time. Well, thank you very much for doing this. Sorry, I took a break. I am really grateful because I really did get, I was instructed and I got some insights into how the history's approach works. Thanks very much for doing that. we have reached the close of our activities over these two days, and I guess nothing remains but to thank you once again for coming, and we have not yet decided if we will be able to do anything with the proceedings of this symposium, possibly we may be able to make some things available. Yes, we might be able to do that at the very least, but we'll be deliberating. And if you yourselves have some suggestions, then what we might do, we'd welcome them. So, thank you very much for coming. Hope you've enjoyed your stay here in Sherman, Texas. And hope we'll see you all again sometime soon. So, farewell. Thank you. Thank you.