11th UK Foundations of Physics Conference — Time, Clocks & Causality

0:00 So, metrology is the science of measurement. Very precise measurement standards are made available by such authorities as the National Physical Department of this country. it's an accepted norm that everyone should use the same measure length of time etc enshrined within this notion the measurement process must not affect the standard applied but let us look at the case of relativity theory we should take what might be called a received opinion this is Science of Measurement, based on the general theory of relativity, which says that physical events take longer, meaning the clocks run more slowly in an intense gravitational field than in one less intense. J.D. Jackson has recently updated his standard text on the dynamics. A moving clock runs more slowly than a stationary clock. He quotes the long mean-free path for ion decays due to time dilation. Now, there does seem to be something different about relativity theory, yet we are quite accustomed to the views expressed above, so I take it that most physicists and philosophers of physics would more or less go along with these views. Now, I would like to take an alternative view, one which is certainly not physically equivalent, yet is closer to the tenets of classical metrology. This is a short talk and so it contains a few ideas. The first arises because I believe there is a distinction to be drawn, which is not normally drawn, normally overlooked. We want to consider two clocks, A and B, standing side by side and starting at 12 o'clock.

2:30 We're only concerned here with the minute hands. The clock's run for one hour. After one hour, the hand on A is again at 12, while that at B is five minutes behind. If clock A is known to be accurate, so we say it's a standard clock, then we say that B is slow. So that B is slow or has run slowly. Now this is what we ordinarily mean when we say a clock is slow. Now we consider a similar situation in which clocks A and C stand on that side and after After one hour, the hand on A is at 12, while that of C is five minutes behind. An identical situation to the one we had before. Let us say that we have a Peta Trace printout showing the beats of A and C. We find that the beats of C coincide precisely with the beats of A, but that C stopped five minutes before 12. Now we do not say that clock C is slow or ran slowly, merely that it recorded less time than A. If we were to compare the paper trace printout for A and B, B was the earlier clock set against A as you remember, we find that the beats of B are always behind those already. Finally, we count the number of beats produced by B and the number produced by C and find that they are identical. So our first conclusion, we must therefore distinguish between clocks which run slow and clocks which run at the correct rate but record less time.

5:00 Now, we next consider the relationship of clock readings to physical events. We take the previous examples, but this time run a single test with the three clocks side side by side. Clocks are started at a physical event which occurs at their precise location. This is indicated by E1. At E2, clock C is stopped. At E3, clocks A and B are stopped. We say that A has recorded the correct time between E1 and E3, B has recorded the wrong time between the two events and C has recorded the correct time interval between E1 and E2. Now it is only the presence of a standard clock A which enables to conclude that B ran slowly between E1 and E3. Further, had C been in a different location and again recorded a smaller time interval between a different should be saying E1 and E2 with an underline, then no conclusion could have been drawn about its rate. In general, when clocks record time intervals between different event pairs, no conclusion can immediately be drawn about their rates. Procedures for comparing the rates of distant clocks and uniformly translation clocks will be discussed later. You'll find that so-called good clocks at one level, we might say those sustaining the highest density of precision, run at the same rate following the chromometric condition. Clocks record time intervals between event pairs, their rates being unaffected by any particular pair of things.

7:30 I'll turn to general relativity. and the case that we're interested in has been expressed thus the time dilation of a clock in a gravitational field or gravitational redshift now this is just some standard book work so I'll go through it very quickly I'm sure you're all familiar with it we consider a spark shield space-time the central mass n with a spherically symmetric universe the light element is there standard notation with the proper time on the left, the coordinate times and coordinates on the right with the metrical coefficients and the standard substitution of small n. So in the line it will also give us infinitesimal proper time intervals so for a clock at a fixed point in space with R35 constant we have dTor is 1 minus 2 over R to the half dt. The situation is as I've there the large mass on the left here the radius of the emitter and the radius of the receiver with emitted signals and received signals At the emitter we have that relationship following on the hand before and at the receiver the same thing. I've just missed an E off the small bar at the top and an R off the small bar at the bottom which you will see obviously. So we then take the passage times between the emitter and the receiver to both be the same in the standard way. so we finally get to dT at the emitter equals dT at the receiver that's the standard bit of book work that we do having done that those two terms cancel when we take the ratio so we get d tau r by dT is 1 minus 2 ohm over r the big R sub x to the half, A over 1 minus 12 over R, E sub x to the half. Then with R at the receiver

10:00 larger than R at the emitter, we get d to R by d2 is greater than 1, T in one time. Now the emitter is close to the central mass as we just pointed out and so in a stronger gravitational field. We are well acquainted with the idea of time dilation in special relativity. So we might accept that here we have another case of it. Time elapses more slowly in a gravitational field. This is the familiar redshift or slowing of clocks. But why should we give up the accepted interpretation that the clock in the stronger field has run slow, to be correct, run slowly. Well, first, we should note that the time intervals measured were between different event pairs, so we cannot reach conclusions about the rates of the two clocks. Secondly, in the accepted interpretation, quite naturally, it might be said, the gravitational gravitational field is invoked to explain clock slaying. We now consider the effective removal of the gravitational field. Returning to our earlier redshift considerations, we're going to take victorius of 1 minus termo rather than hard dt, and note that for a given coordinate interval, the metric coefficient is only dependent upon M and R. Thus, rather than thinking of the clocks at the emitter and the receiver as being fixed at their respective radii, we put them into free fall. Now, the condition we require is that during the interval of transmission, the associated clock is substantially at rest in free form. During the interval of reception, the associated

12:30 clock is substantially at rest in free form. The results of such an experiment... by that? We start with the clocks of fixed radii going out from the source and no doubt they could have been on the tower for example. In this case, instead of being on the tower, we think of instantaneously dropping clocks from the tower. So the first clock is 20 minutes, we let it go at the point of emission and similarly when the second clock receives its pulse it must also be similarly okay the results of such an experimental course conform to be tall and thus be identical with those who are experimented with the clocks for so-called experiencing the gravitational field. Importantly the clocks in free form will satisfy special relativity and when instantaneously stationary will run at the same rate. Thus we establish that we get the classical rate shift result when it is known that the clocks of emitter and receiver are running at the same rate. We conclude also that the clocks in the more usual experiment on fixed radii run at the same rate. The correct logic is that we first establish the equality of plot rates at different locations. Only then are we in a position to make time interval comparisons. Between the event pair at the receiver, the associated clock records a longer time interval than the time interval recorded at the transmitter between a different pair of events, both clocks running at the same rate. These are some self-tests. A few examples may be used for the diagram.

15:00 In the diagram, the observer views a laser under varying conditions. At A, the laser and observer are fixed radii in a Schwarzschild space-time. B is the same as A, but with the laser and observer virtually at rest in frame 4. At C, the observer is in a geostationary satellite, and at D, the laser is in a geostationary satellite. For A, B and C, the same redshift, and of course a blueshift is observed in D. Clocks have become ever more reliable survey instruments than in the Berser et al. gravitational time dilation experiments. Hydrogen laser clocks are subject to accelerations in 20 G without being effective. I want to move on now to the case of special relativity to pick up one of the term data points that we made over. We now turn to special relativity, the question of time dilation, length contraction, twin paradox, etc. of occupying physicists and philosophers almost half a century. While in GOR there is a natural asymmetry because of the ever-increasing radiance away from the central mass, in SR there appears to be a certain symmetry between observers considered. This has engendered much discussion and is central to the question of reciprocity between inertial frames of reference. Recall on our earlier note where we said we would return to the question of comparing the rates of distant clocks and uniform mean translated clocks. The first difficulty to be overcome in cloth comparison is that of spatial separation, while the second is that of different inertial frames. We now consider spatial separation. Light signals are originated at the midpoint between two mirrors, the reflected light oscillates

17:30 between the mirrors of the formula light clock that we're familiar with at Poppe Marder, 1971, at the spatial positions indicated by one to two reputed reflections set up events sequences, which we call ticks, and form identically spatially separated clocks in the inertial frame F. Each mirror both reflects and allows transmission, so that signals pass through the first pair of mirrors to be reflected and transmitted at three and four and so on. The temporal extension of mirrors has been meted out for clarity. Three and four, of course, are again identical clocks and the lower pulse recurrence rates of the distant clocks are easily allowed for. Thus, a standard frame time is established. In the dark way, we have viewed the light signals as elements of a light clock structure. But ordinary clocks can be positioned at the various mirror locations. The signals now enable comparison between distant clocks and we use Einstein in synchrony and then for any symmetric event pair the two times are the same. Thus we have addressed the first difficulty in clock comparison by making a standard time available to spatially separated clocks in a single inertial frame. We now come to the second difficulty, that of uniformly translating inertial frames. We retain our original reference frame F and introduce inertial frames A and B, translating left and right, each with uniform speed U in F. exactly the same events that is used. A passes through one and three, etc. and B passes through the even number of events.

20:00 Consideration can now be given to relating time in A and B to time in F. Light signals originating in frame F and an event such as P arrive in A and B at events 1 and 2 respectively. Signal transit times depend on frame speed and the speed of light, and we apply factor K to a higher interval in F and a related interval in A or B. We get that the time at event 1 in A and the time at event 2 in B is A times the time at event P in F. That is, record time from the 0 event. So the time in frame A at event 0 is equal to time in frame B at event 2. Whether or not physical clocks are carried by frames A and B is not important, since following our discussion for the fixed frame, the light signals form clocks in the moving frames. If it is the case that good clocks are synchronised in A and B at event zero, then they will be synchronised at one and two. So clocks in A and B run at the same rate, such as one and two that I call reciprocal dents. Having established the equality of clock rates, we can turn our attention to the question of measuring rods in each frame. We continue the established light signal structure. For frame A, the light path from event 1 via q back to 3 gives the time of event q. The heli line depicts a measuring rod extending from q to the origin of A. Finding the time of q is equivalent to a radar measurement, giving the distance of q, which is the length of the measuring rod. I've just said, of course,

22:30 there that the time in R at A, which is the time at Q at A. So there they're just the two ends of it. Frame B employs the light path 2Q4 and of course gets the same sort of results. So we've shown that both frames can be equipped with measuring rods of equal length, or our use of light signals to define the length of measuring rods conforms to the international metrological standards where length is defined in terms of time. Measuring what is this application of the camera document. Just coming to a conclusion. All of our considerations are based on time, whether clocks are in close proximity, spatially separated or uniformly translating, we have seen that rate comparisons can be made using the same light network. The frame F introduced at the start can be expensive, in fact you can actually start with it, and of course nothing is changed by using the more familiar representation shown here this is exactly the same diagram to the ones we've been looking at except that I've just drawn it in the fluid form. The k calculus which we very briefly refer to gives k equals that thing where v is relative velocity of a and v and as you would expect this is the relativistic doctor shift the first bracket gives the classical doctor factor and the second sector gamma. If we take a classical example, since 1 and 2 are reciprocal events, the time of 2 in A equals the time of 1 in B. Those other two relations that I put in now are simply the ends of the rules. So we get that the time of event 2 in A is gamma times the time at event 2 in B, and the time at event 1 in B is gathered time to time at event 1 in A. The clocks in both frames have the same rate, so between 0 and 2, the B clock records a smaller elapsed time in A,

25:00 between 0 and 1, A records a smaller elapsed time in B. We see the complete reciprocity between the frames, and we avoid the term rest frame, rather preferring fiducial frame where we can switch our allegiance at any time during any experiment we may care to before. A brief summary. All of the standard results in general relativity and special relativity remain the same but their interpretation is fundamentally changed. Good clocks are not affected by gravitational fields or by wind formation in relativity physics all clocks run at the same rate recording the last time between the length pairs according to the relevant metric curve. If all the old questions in S I will find a straightforward resolution, for example that of reciprocity, why don't both clocks go slow, neither clock goes slow, there is complete symmetry between the respective clock readings on symmetrical pairs of events. Most significantly, we want to distinguish between clocks which run slowly and slow in the traditional sets and clocks which record a smaller time interval between a given pair of events while maintaining a standard rate. Our aim is to draw a distinction and show how it is possible to revise There was something important early on that I was not able to follow or else you could explain it again. Yeah. You had three clocks A, B, and C. Yeah. And you said that the paper trace of A and C had, let's see, somehow clock C had registered, showed five fewer minutes having passed, but had put the beats on the paper trace at the same time. Okay, yeah. Let me make sure I understand what that means.

27:30 Sure, fine. My watch has a second hand that is jerky while the minute hand is the hour hand. some little pencil that comes down and makes a mark on a piece of paper that's kind of going by I guess I don't see how it's possible how the situation with AAC is possible ANC you want first I was going to say with A&B A&B, right, A&B okay, let's do A&B A and B are fine. A and B are fine. But A and C aren't, so good. Okay. This is A, isn't it? And this is C. Now, that's the, they're going to start at 12. And they go around this way and they finish up there. A gets back to exactly 12. clock, but C doesn't get back to exactly 12. It stops 5 minutes before. Now, what we want to say is that C is a clock which is exactly the same as A, and it's running at exactly the same rate as a clock, but we, if you like, stop it at 5 minutes to 12. So... Oh, we stopped it early. We stopped it, yeah, we stopped it. Oh, I thought it ran and just happened somehow. No, sorry. Okay, I wasn't. What we're saying, really, is that this just happens to be a different event. If they started, if it was E1, E2, the whole elapsed time or the two events that we ran between now this one here just we stopped it at a different event in a fridge okay and that's that's what I'm saying happens with all clocks that's the actual situation so what you do is simply record a time interval which is relevant to the metric coefficient in that frame between that pair of events Are you also saying that the two clocks have the same number of beats, one in 60 minutes and the other in 55 minutes?

30:00 No, no, no. The first clock will have, I should say, 60 beats in the 60 minutes. And the second clock will have 55 beats in 55 minutes. It's a perfectly good clock. It stays precisely with A. It's simply that we come along and stop it, or there is an event at which it stops. that's the sort of crunch point really but it must always be keeping the same rate and so the point is before we can start drawing conclusions about clock rates you know we must make sure the clocks are going at the same rate if we can get the clocks going at the same rate then we can start saying what is it of course is the difference. The difference occurs because one clock is stopped and it's not given as much time to run, if you like. I'm going to put it that way. I'll do one quick one. I'm just a bit confused by something you're very... In special relativity, for example, it's absolutely essential to give any operational meaning to the court that assume that if you have ideal clocks that are at rest if those same rods and clocks are boosted so that now it rests with respect to a second frame they remain ideal in that sense all clocks in relative motion run at the same rate that's something that's absolutely accepted by in fact sense in the sense i just mentioned i mean i can't see anything in your argument that isn't perfectly consistent with the standard interpretation of time violation so what i let me ask you a specific let me let me just answer that little bit you get the same oh thank you you get the same conclusion but you get there by a different route the clocks, you see, whether the clock actually runs slowly that's why I made this point earlier on whether it runs slowly or runs for less time

32:30 gives you the same finishing result if you come along later and look at the clocks you can't tell which clock you're looking at so you will, as you say, get exactly The outcome is the same if you look afterwards, but if you were to compare paper-traced pinnaps, as I said, you would find that there was a difference. Another way to look at it is that these clocks that I've talked about are running at a slightly higher power rating than the clocks in special relativity. They all use the same amount of energy that these clocks are running at a slightly higher rate. They run at the same power as a standard clock sitting in a natural physical environment. Okay, I think we'll have to call that. Next we have, I apologise for the pronunciation, Henry Sinkin-Nagel. Very good, very good. He's going to speak on time clocks and causality, could Hume's problem of induction give what is the basis for the philosophy of physics? Yes, thank you. That's the title, and I should start by saying, for those of you who are familiar with Hume's problems, most of you, I guess, that I will not be talking about the problem of induction in a general sense. That is, I will not try to argue that when you've seen a finite number of what to one, what to one to what. I think Hume was right on that score. But I think when Hume is using his problem of induction with respect to causality, it goes wrong. And I think this problem really lies at the very heart of the philosophy of physics, or foundations of physics, if you like. And I try to suggest that Hume is simply wrong on this score, with quite a number of consequences like that. So, what is Hume's problem of induction with respect to causality? Well, very straightforward He said in the inquiry, we cannot be sure that the sun will rise again tomorrow, since we cannot infer from past experiences of causality to experiences in the future. So just 50 years after Newton came along and told us how the universe worked more or less, Hume came and said, well, you thought you had the laws there, but in fact, you don't

35:00 really have anything because it might be that tomorrow none of these things will work. Just because you've seen the Earth rotating a lot of times doesn't mean that it will happen in the future. It was quite a shock, I think, when Hume did this. Now, the basic assumption behind Hume's reasoning is that only formal logic and mathematics, on general relation of ideas express certain knowledge. The rest of our scientific knowledge is contingent. That means it could have been different, and for Hume in particular it might possibly change. And since causality does not fall under formalizing and mathematics, well then it is contingent. This is what Hume was saying. Another way of putting Hume's point is that only formal logic and mathematics express necessarily sufficient criteria for philosophical or scientific assertions to be meaningful. That is, as long as you don't contradict yourself, as long as you're consistent with your use of symbols, well then you can say anything you like. This is just that that Hume is actually using. Now this has quite serious consequences, I think, for our very way of thinking of physics, It effectively means there is no rational justification for our belief in the future validity of physical theories, or for our belief that the future will be similar to the past. This has, for instance, the consequence of our belief in the experimental method. This is where we have the idea that we should reproduce an experiment. Comes on a sort of a not very good grounding, because we have no rational belief. We have no rational justification for this belief that a future experiment will sort of reproduce experiment in the sense when we do an experiment we would like to have it reproducible so that the conditions of the future are similar to the conditions of the past. Otherwise repeating an experiment does not make a whole lot of sense. And what Kuhn is saying is that this belief we will have to just take on as a matter of faith and not as we cannot give a rational justification. I think there are a number And a couple of other consequences that I'll come back to. Now what I'll do is I'll try to show, or at least sketch, why I think that Hume's argument is flawed. And I think what I'll try to show, or at least sketch, is that the concepts of time, clocks,

37:30 and causality are intimately related in such a sense that Hume cannot meaningfully refer to the future without presupposing causality. So, come back to what I mean by me. Okay. So, just to give an idea, I think it's very fair to ask about the meaning of time in Hume's argument, because I'm not going to tell you what time is or anything like that. I just think when Hume is claiming something so extraordinary, it's fair to ask him what is exactly the meaning of tomorrow in his argument. Now, in Hume's thinking, as you will know, Hume took the concept that he was using. He was always trying to relate that to the impressions, the sense impressions that he had. And obviously we don't really experience direct to the future, so it's not very clear what tomorrow actually means in his thinking. And obviously, if by tomorrow we mean when the sun rises again, then clearly it's contradictory and meaningless to assert that the sun might not last tomorrow. That is pretty straightforward. Now, of course, the first response to this is, well, of course, you can just think of something else as being tomorrow. You might say that, well, let's try with tomorrow's when something X occurs. For instance, when I read Midnight on my watch, I want that much. But I would try to, I would suggest at least, that this will not work for him. There's no way he can specify what he means by tomorrow and still maintain his argument. And there are two steps in that, so I'll go through briefly. First I'll claim that there's a logical relation between time and clock in the following sense. But we cannot meaningfully refer to time without referring to a physical system which can be used as a clock, and vice versa. Of course, there's nothing new in saying that time and clocks are related. What I'm saying here is, I don't really see how to mean a sense of time without . I'll come back to that one. It's not a definition I'm giving here of time or of clocks. It's simply a minimal criterion, I'd say, to discuss either of these kinds of things. I cannot give here a full theory of meaning, but I should just say that I'm thinking, when I say meaningful, I say, well, to be meaningful is roughly to be useful in descriptions and

40:00 arguments. That sort of has a bit of fun and a flavor for it. And also that knowing the meaning of a concept involves knowing its relations to other concepts. That was a short meaning theory there. That was the first step, the logical relation between time and clocks. The second step is that Hume cannot, I think, speak meaningful, speak about physical systems which can be used as clocks without presupposing causality. So if it's so that you cannot talk about time without clocks and clocks without causality, then there's no point in saying that maybe causality will not be valid in the future. Excuse me, what do you call causality? I will come to that in just a second. Before I do that, let me say a little bit about this thing about the logical relation between time and clocks, or physical system which can be used as clocks, clocks for sure. Because the first thing one might think of when I say that there's something logical or something necessary, a necessary connection between time and clocks would be, well, the concept of time has surely changed in history. So how can there be something intrinsically related? How can there be such a logical relation, you might say? But of course, that the concept has changed in the course of history presupposes that there is something about the time concept which has not changed. Otherwise, we cannot refer to history. When you make a reference to the historical change of the time concept, you're using the time concept. So, I cannot go through the history of time here by any means, I'll just say that a good starting point might be Aristotle, who said something like, time is a mission of motion and time and movement define each other. Now that's fine from the point of view of my logical relation, those moving like physical systems and time and movement define each other, they are related, that's all well and good. What's not well and good is when we come to Newton. Because as you will recall, Newton had this famous distinction between absolute and relative time, saying that absolute, true and mathematical time, of itself and from his own nature, flows equally without relation to anything external. Relative time is some measure of duration, by the means measure. And the second part here is more or less what could fit with what Aristotle is saying, and that certainly fits with my logical relation there. But the first thing, to say that absolute time is something which of its own nature flows without relation to anything external, is obviously not, or not directly at least, not very clearly related to any cloud.

42:30 Lots of people have been thinking about what Newton could possibly mean by this. But I think the basic fact of the matter is that it makes no sense what he's saying. And it makes no sense because when you think about it, what does it mean to flow equally? Well, if something flows, you would think about something like a river. And in a river, the water passes the riverbed, so it flows in relation to the riverbed. And if you say that something flows equally, steadily or uniformly, you'd like to know equally with respect to what? So I think, in effect, that it does not make sense what Newton's saying. Now, Newton can get away with this, because it doesn't really matter to the whole of his physics, because I think that every time he's using a time concept, apart from discussions about what God is and things like that, he actually does specify what he means by equal flow, because he will connect equal flow with notions like uniform movement. And, of course, he has a notion of a uniform movement, and that notion is specified in his first law of motion, where a particle traces out intervals in equal times. And this, of course, directly refers to the motion of an idealized physical system. Also, I should say that the word here, mathematical, may suggest that, well, it's some kind of a mathematical notion, the time notion that Neusen is using, such as the T we have in our equations, Of course, the little t we have in our equations do not flow equally from its own nature. That's a similar feature of parameters in mathematical equations. So I really think that in order to give any meaning to this concept, he must, as he does, be related to our emotion of business. Now this in effect means that I don't think this notion of absolute time in Newton makes sense without relation to anything. external. On the other hand, I do think that there is something about Newton that we really cannot get around. And that is what I would call the causality concept of Newton's loss. And I think that that is necessary for me and for most of the time in the future. So

45:00 now I'll come on to the second step, meaning about causality. So what I'm claiming is that the absolute time part of Newton's argument does not make sense. Nevertheless, there's a core of the eternal mechanics that we cannot get around. So on to causality and physical systems which can be used as well. That was the second step, remember, the argument. Now, a very general notion of causality, as you've seen, every event has a cause, not very specific, can be used in a lot of different settings. But I think when we talk about causality with respect, when we're talking about the motion of physical bodies, there is a precise version of causality which actually Newton provides, or it's provided by Newtonian mechanics. I think that might be called the causal core of Newton's laws. It says, roughly, a body in uniform motion continues its motion unless the body is caused to change its motion. And that the same core is acting in the same circumstances and at the same stance. So that's just an attempt to extract the causal concept of first and second law basically and without specifying exactly the principal form of the second law. So that's, I'll call that the causal core of Newton's laws and you have a perfectly well-known example of that. You have three other ones coming in, it hits the white one, the red ball stops until you get to zero and the white one goes off. Now, I didn't specify that it must be linear in here. We can also, in fact, think of uniform rotation. That's not important now. Now, back to Hume, because if this is sort of a causality concept of Newton's laws, what is Hume exactly claiming? Well, Hume is claiming something like the following. that a ball may stop without a course. That is, contrary to this one, a ball may stop without a course. You do an idea of frictionless circumstances. I'll come back to that in a second. And more generally, that the course of cognitive mechanics might cease to be valid in the future. Since this is not formality, it's not mathematics, well, then it might cease to be valid. Now, it's important, this thing about the frictionless circumstances, because the situation that Hume has in mind that he describes is this one. A red billiard ball comes in, white and nothing else happens. Now everybody here who has tried to play billiards know

47:30 that this is perfectly possible on a real billiard table. You just give the red ball exactly the right speed and everything is fine. But Hume's point is that even in the idealism, disregarding friction, which is effectively what you're using on a billiard table, even without friction, this could happen. This is what Hume is claiming. Or equivalently, if you just think of the red ball without the white ball, so just a red and suddenly it just stops. This is what Hume claim is possible, and that is it. We're going to say that maybe the sun will not rise tomorrow, maybe the rotation of the earth will simply just stop. Now, what I will try to show is that if this causal part here for returning mechanics is no longer necessarily valid, at t equals 0, then there's a similar physically meaningful concept of t larger than t is 0. So if that's true, then it does not make sense this imagination of humans. In order to give you at least a feeling for what I had in mind, I'll just sketch the argument. Roughly, I I think we can agree that a clock is a system undergoing a more or less regular constant or repetitive process. This determines more or less equal time intervals. I include here more or less because it has been known since the beginning of clock-making history that some clocks are more regular than others. But I think one can fairly easily convince oneself that in order to speak about more or less regular clocks, you'd have to also speak about and refer to completely regular. For instance, if you want to say that the earth and the sun system is not a perfectly good clock, that's because we are referring to frictional forces and we are sort of assuming that if there were no frictional forces and no other disturbing things, no tidal forces, I should say, frictional tidal forces, then it would be an ideal clock. So I think we are carrying along with us the concept of an ideal clock, although they are not, of course, realized anymore. So, this means that for an assertion about time to be meaningful, I think we must presuppose the reference if we made to an ideal clock. This is exactly what I think that Hume cannot possibly do. Because if we assume with Hume that the causal core of Newton's laws is not necessarily valid

50:00 in the future, then it is possible for an ideal clock indeed for any clock, any ideal clock, to stop at any given moment of time, say it is zero, without a course. But if this is possible, then it is possible to speak about a future moment of time, that is T larger than T naught, independently of an ideal clock. Since every clock, every ideal clock, we start at T zero. And of course, the stop clock is not completely over the clock. And I even think that it can, that Hume cannot jump, so to speak, from clock to clock. That's what we do in ordinary life, when our old clock doesn't work anymore, we throw it out and get a new one. It's not available for Hume since he's re-under the assumption that the course of core of using laws is not any longer valid. So nothing guarantees that there might be a future point where there's a plot. So I think this means that future is simply not a meaningful concept under Hume's assumption and so it cannot be assumed that the course of core of using laws might be valid in the future. Now, unfortunately, returning physics is not the whole story, as you know. Anyway, I think that the case can be made that although relativity and quantum theory demonstrated the limitations of classical physics, it's also called Newton's law in order to have a meaningful concept of time, even in these theories. And I'll put that really detailed argument here, but I'll just sketch it. Well, I think in relativity, both in special and general relativity, we rely on standard clerks in inertial systems. But in general relativity, in fact, just on time-like water lines, but anyway. And also that the standard clerks in relativity theory, they are actually just like the ideal clerks in internal mechanics. that they are ideal clients. They respect both the course of court, which is lost. Now, apparently, well, Einstein was not very happy with this, which I've learned from a paper by Harmi and Oliver, because Einstein apparently thought that sooner or later it must be able to give a more fundamental description of rocks and rocks as far as I've learned. But I don't think that would do. I don't think Einstein really could get around these clocks, simply because also in quantum theory I would say that, well, we need classical clocks.

52:30 The T in our wave function and our time-dependent tracking equation, hence the reference to the revolution of a quantum system, refers to an ordinary classical conception of time. not only because t is not quantized, but simply because what we mean by an evolution of a quantum system turns back to a classical clock. Now, this is in conformity with what Bohr was saying about point mechanics, but I don't think you have to buy all the boring stuff if you don't lie in order to accept this point. And I think this holds true even if you consider atomic clocks. Because even the usefulness of atomic clocks rest on the validity of people's involvement of material physics. In that both in the surrounding system of an atomic clock, and in fact also in the micro-physical system, in the heart of the clock, there's a certain amount of nosality allowing you to draw a very quick atomic clock. Using microwave cavity, cesium atoms are sprayed in here, and you try to induce hyperfine transition effect flips of these atoms, and you have a detector up here, and the detector goes back to a microwave emitter, which sort of controls the microwave cavity there. And so the object is to tune in the electromagnetic frequency here, so as to induce as many atomic transitions as possible. Now this of course, this whole device assumes of course that causality is valid in the surrounding system. For instance, when you send out a signal here, it better come down here. And also, on average, you would have causality in the micro-system, in fact, because what you're utilizing is that you send in a frequency with the right frequency, and in average that will induce a flip of the atoms. That is, because you have many season atoms in here, you have, on average, causality. So, just a brief summary and some conclusions. I tried to, if not sure, then at least sketch how time is a meaningful concept only due to the logical relation to the concept of a physical system that can be used as a clock. And also that the causality content, or the call of users' laws, of notion, constitutes a necessary condition for meaningful discussions

55:00 And that therefore Hume's problem of induction with respect to causality cannot be formulated in a meaningful way. So if the sun should have not to rise tomorrow, which could of course happen if there's an explosion in the sun or a comet comes by, but it will certainly not be because the cause of the core ceases to be valid. It's simply not possible. Then we lose what we mean by the future, by tomorrow. This has some implications. I think, first of all, that this gives at least the basis for rational justification for the future will be similar to the past, which is quite nice for our belief in the experimental method. Also I think it induces a rethinking of the relations between physical theories, so returning mechanics is not just a little increase of relativity and quantum physics, it's right there in the core of these theories, in order to give meanings of the concepts we're using. And finally, I haven't talked about this, but I think also this implies an invalidation of the container view of space-time, but I haven't talked about space, but I'm assuming something similar could be said, and an invalidation of the container view of the absolute view of space-time simply because not a time of physical clocks, physical systems can be used as clocks, are not a lot of compliance with the other. It's not that time is the most fundamental thing, or that the clocks are the most fundamental, And I think this might be of some interest for the absolutist relativist debate in physics. Okay, thank you. I think that was a very clever argument. but I think it's fairly common to distinguish a kind of hierarchy in the properties of time where at a very fundamental level say you have ordering relations relations like earlier later between and then at a much higher level in the hierarchy you have metricized properties which allow you to say something like the time-lapse between these two events is exactly equal or twice as much as the time-lapse between the other events now I agree completely with you that in order to have these

57:30 the metrical properties of time, you have to have recourse to the color that you have to follow. However, I think one could very well challenge the idea, whether also these very bare ordinary properties of time are only meaningful if you have a clock. And you will know that that's all that you need. Absolutely. My claim is, I have no idea what you mean by before, after, and now, unless you refer to it. I mean, I don't really know this thing about separating these two things. I wouldn't really know what you mean by before, after, and now, without that. You're not wearing a watch, but you know the difference between now and the middle of it. You don't know that it's exactly a minute ago, you know that it was a minute ago. Sure, because I am essentially referring to my biological clock or my notion, my experiences of clocks. That's what I mean when I say it's not now a minute ago. That's exactly what I mean. Otherwise I wouldn't know what it means. But one could say, perhaps some people would say it's immediately given to the consciousness. What's that? That it's immediately given to the consciousness. You mean to be given to the consciousness? Yeah, of course there is... That's a first perception. It's a first perception, isn't it? We see the things that are in our consciousness and things that we believe could be there. Of course, this whole thing about consciousness and what we see is, of course, central to the whole human program about everything that we see, everything that we have, all the content we have are due to our sense impressions. But I think if we are to discuss sort of explicate what we mean when we say time, there is really no choice. I mean, it doesn't help to say it's a first impression. First impression of what? May I make a little comment in defense of you? In my belief, what he said about steadily flowing time is quite meaningful. Stretching your analogy with a river And you cannot say whether a river flows at what speed exactly, unless you see the riverbed.

1:00:00 But you can say whether or not it flows with acceleration. The same is using this. Maybe I cannot say without close whether or not it flows at what speed does it mean. But I can say whether or not it flows steadily. I can be reformulated in realistic terms. If all possible bodies suddenly get acceleration, I can see for some reasons for this or I can say that times don't flow steadily anymore. Okay, but can you define acceleration without referring to velocity? without referring to any i don't need any reference uh reference system i don't need clocks exactly but yeah i will experience it as a gravitational force i don't need clocks or i don't need rocks to feel a gravitational force and this will indicate some acceleration Okay, we can have two more questions. Chris? No, I think I didn't. First, a comment. In reply to your point, Henry, you might say that you still need a clock, but it's just a nonmetrical clock, which just gives you an order, and that would be one way of replying to that. Secondly, your question. I'm sympathetic to the idea that there's something intrinsically self-refusing in trying to argue against induction. But I wonder what happens with your particular argument if you state Hume's conclusion in terms of denying as you can infer from the observed to the undeserved, rather from the past and the future. Right, I mean, certainly that's sometimes what people say, that the central thing about human is not future-past, but observed-unobserved. But I think there at least some kind of relation, because now everything which is in the future is certainly unobserved, although some of the things in the past are observed and some are not observed. So I still think it makes perfectly sense to ask, particularly in the context of this argument about the sun, what he means by the future.

1:02:30 Yeah, I can see what you mean, but I still think that you could say that I don't need a clock to talk about the unobserved, but it's quite possible to talk about the unobserved in the same way you can talk about things that don't exist. Although usually when you say something is observed, you say something has not been observed, or something is going to be observed but you needn't it's simply a fact that it hasn't been observed it hasn't, but what means it hasn't we're already using past tense that's why yeah, there's a way to then turn on the concept of a person perhaps a few times persons are important too, I think yeah, in fact, still quite a certain place okay, well, can we thank Ashley again? Okay, the final speaker of this session is Juan Leon, the MyMath and Spay, who's going to talk about dynamic profound variables and plots. First of all, thank you very much to the organiser for giving me the opportunity to present this. What I want to present here is the result of something that started years ago with cosmology, with the Wiener-de-Witt equation. I'd like to understand what is called the problem of time and then I realised that many things I didn't understand at the time. I don't understand the same kind of things, so I used my field of inquiry to simpler and simpler and simpler things. At the end, I was asking the very simple question, when can I predict the time when something will occur? So, this is the topic of my talk, and I will do that in classical and in quantum systems. At the same time, before that, I will just make some distinction between time and dynamical time,

1:05:00 that for me is the distinction between absolute time and relative time, if you want. But I'm not very strong about that one. I don't think it's necessary. So first of all, time is an external parameter, as the previous speakers said before, in dynamics, Newtonian dynamics, and also in quantum mechanics. a lot of attempts to include time among the variables, among the operators of the system if we are talking about quantum mechanics. And there has been, from the very beginning of quantum mechanics, a lot of troubles with that just because Pauli, in his famous book about quantum mechanics in one of the good notes, he showed that it was impossible to have time as a canonical conjugate operator to the Hamiltonian because then there are a lot of troubles but one of these is that then we will run into non-bounded energies or negative energies and things of the like okay this is due to the stone from Neumann theory theorem for representation of conjugate variables or something like that so i'm not going to enter into the mathematics so from very beginning it was formally said time cannot be unbreak okay then the question of time research i mean search again with several some people in the 50s and 60s in particular haranokambon and other russian people that began to think about time and then it has been continue with people like Paul in analysis of physics and people like Alok, that he did a long work showing that it was very difficult to give physical meaning to this time operator and so on and so forth. The President Pioski in Poland, he made also a lot of work showing very clearly how can and we understand time from some point of view that you see in the operational terms. So, first, what I want to show you is something that is not mine. This belongs to your colleague, Hildor.

1:07:30 He's a man that for me understand better this, your time. And he produced very nice papers in the American Journal of Physics. And his papers contain this. So, momenta and energy are the dynamical variables of the system. I am an elementary particle physicist. So, for me, the systems are made of particles. And I start thinking about the particles. The properties of particles are given by their momentum, the momentum, the energy, and the spin. Okay? In the boncari group. Because you can think about that in a group or whatever you want. So, these are the dynamical parallels. However, coordinates and time are labels of space-time. So, for instance, if you are doing kinematics, the time when you are doing kinematics for me is absolute time. Okay? No relation to physics at all. No relation to any clock. At most, relation to the length of the trajectory, if you want to find, that is fixed, independent of matter. Good. Now, positions are dynamic variables. In fact, one of the first exercises that one makes in modern quantum mechanics in the courses of modern quantum mechanics is just to get the positions for a physical system in terms of the momentum and the energy. And then you have this, that is dynamics. If you want, you can put here Poisson brackets or computator, you have the dynamics in classic Darwin quantum pieces. But remember, x is a parameter, and then it makes no sense to do that in classic Darwin quantum theory. and p is a parameter. I mean, it's 25 past 3. So, I thank you Henry, because you prepared the ground for this before. So, let me go now, how can I tell time in classical mechanics? The goal is the following, we have the phase classical, the phase space. I am thinking always on an autonomous system, Hamiltonian system autonomous,

1:10:00 because I don't want to have a god outside, outdoors, that tells me something that I am not prepared to, and things like that, okay? So, the idea is that I should be able to extend the phase space including time and the conjugate momentum of time, so time will become a variable, a dynamic variable. have this kind of constraint and the idea is that you can trade time by some position. That's the goal, okay? So assume that you say, well, I take the value of q1 of this position, the coordinate x as a parameter, and then I take p1 from here, I get p1 in terms of this, and then what I do is this theory in which I have What was time is now position, okay? And the Hamiltonian I had before is now this Hamiltonian P1, okay? This is the idea of having time as a dynamical variable. This is the relative time, okay? I am putting time relative to something physical, that is the position of the value of some dynamical variable, okay? The theorem will go to clocks. This is not possible. This is not possible because of integrability. Okay? Remember, you mean integrability. You need to have... What I mean is the following. I only know how to do this in this case. Okay? Maybe you have better answers and then please tell me. The following thing you need. You need a number of conservation laws that equals the number of degrees of freedom and then you need that these define isolated intervals so that you can just extract the momenta from this. This comes from this, excuse me, from this immediately. And then the point is that in that case, time is something that is independent of the path to follow. So you have, say in another way, you have horonomic coordinates as a solution of the equations of motion, okay? I'm going to construct explicitly the time in the classical situation. This is very simple because this is the standard... How do you call that? Jacobi? Okay, this is the standard in Jacobi?

1:12:30 So, this is the standard transformation to ignore our coordinates. I choose this because it's not what you used to see. In general, one puts here, i, that is the action times ohm, that are the frequencies. And these are the action angle variables, but you don't need to have angle variables. So I am thinking about unscattering problem here, just to depart from the usual way of reading things. It gives you a flavor that is going on, that I think is better. Then the solution of the Hamilton-Yakob equation is this one and this is path independent. And then you have the momentum in terms of this phi i and the new coordinates given here, okay? So, if you ask the question that is in red there, okay, what is the... I prepare a mistake, a particle here, a QP, the initial instant. And here I have my detector. When the detector clicks? This is the answer. The point here, which is a very important point, is that this is a mess. Because I can have Hamiltonian, which is very involved, etc. But if the system is integrable, which is a very, very heavy, very strong restriction, then what I have is just a linear flow. Okay? In the phase space of the generally called action angle variables. Okay. So, let me be a little bit more explicit. So, integrability, in that sense, what it means is that I have the simultaneous existence of independent planes, or those, if you want, one for each degree of freedom, okay? And this independence is integrability, right? It's the incompatibility condition I've had before, and the isolated integrals I was asking for. So, then you see that one unique time permits correlation, one-to-one correlation, I put one-to-one in the sense that one-to-one, I mean, it's nothing formal, I don't want to be formal here.

1:15:00 So, in some sense, you can do, okay, now I introduce the clock and then I go to the, in some sense, the clock. I close one of the coordinates, okay, and produce a torus, right, and then the Hamiltonian in these variables will be i omega, or j omega. Okay, so I can write for that variable this equation, that is the equivalent to the equation for the rest of the dynamical variable, and then what you have is the clock, so I know what is marking the hand of the clock when I arrive to the position x, x is a vector in general or whatever, it corresponds to many positions. The point is that you have one equation of this for each degree of freedom. This is the point of integrability. It's amazing, okay? So, the point is the following. You can dispense t, and t is a gauge variable. Never mind if in a plane you decide to take t squared, like the momenta and the other some kind of function of t, like the time and so on and so forth, because at the end But what is important here is that for each position here, you know where the other positions in the respective phases are. So, time is like a gauge value. For integrable systems, relative time is like the absolute time in your words. I'm not trying to be serious about that, but more or less I think this is the answer. In general, what you do is just choose one of the axes as the direction of the throw in this linear space. This is for the classical system. If I go to the quantum system, I'm putting here some kind of summary of what I am going to do. What I do is the following, I take for granted that I know how to tell the time, how to predict the time for free particles, okay?

1:17:30 For linear flows, the equivalent in killer space to the linear flows, or the logoidal flows of space space. I perform a canonical transformation, here everything, Q, P and H are operators, okay? Then I realized that the operator performing the canonical transformation is the molar wave operator. And then I realized that the solution, I mean the operator that gives me the time of arrival is the same. I prepare here the state, I put here the detector, but in between there is an interaction. An electric field, a magnetic field, the gravitational field of Earth, I can have here, how do you call that, a cavity, and then send atoms with a dipole, electric moment, and then, you know, all these things, you can do that. So the answer, the solution is this one, okay? And then what comes later on is to just explore the consequences. I have to explore all this, but I have no time to show you that. So, I will show you how to produce the dynamical time for the free particle. This is not mine. This is what Aranov and Bohm started to do, I don't know, 40 years ago. So they didn't give any physical meaning to that. It was Paul and then Kiyoski who started to understand the physical meaning to that. People like Roveli and some other people that are working in quantum cosmology and the problem of time and quantum gravity use this in order to start to understand the problem of time in quantum cosmology. I think they understood nothing but they did a very nice work that helped me a lot. So, this is the formalism. The operator that you have to produce is the quantum version of this. This is the distance divided by velocity. This is the quantum question you have to to symmetrize here

1:20:00 that is more or less this and then you see that the time of arrival at some point x is the time of arrival at the origin times some kind of translation of the situation, this is covariance there is also covariance with respect to time here, okay, so this is a kind of covariance in primitive system as somebody said the other day okay, the eigenstates are these, where where this pi, this projector, is the projector over the states that are moving to the right or moving to the left. Remember that the Hamiltonian for free particles has this degeneracy, okay? So I have to take this into account. Then you have a problem with the mathematical properties, and then what you have is a positive operator value measure. So, in terms of the interpretation of this formula is the following. I can give you the probability of arriving at the point sex in the interval t1, t2, is this one. The fact momentum of this probability gives you the value of the operator. It's a kind of a spectral decomposition, not in terms of a projector, but a positive operator. And then you have the probability that if I cut the Psi state here, it will arrive at some interval of time in the vector, and this is the average value, the expectation value of time. Now, this is 43 particles. Now, I'm going to show you the canonical transformation. Remember, I told you, I take for granted this one, but I have shown you that I know how to work in the free case. Okay? Is what I have done right now. Now, I perform the canonical transformation, and then I look at the results that are the answer to the problem. The canonical transformation, I have the executive summary of the canonical transformation here. You don't need to look at all the equations because it's impossible. But here is the following. If H0 is the free Hamiltonian, H is the current Hamiltonian, the one you are interested in,

1:22:30 Omera will be the Meller wave operator, the one that is used in the scientific theory. and then I'm considering a case without bound states in order to be simple so omega is unitary in other cases it is isometric and the formula has become more complex I prefer to be simple here now assume that T0 is the time operator for the free case the time operator for the case with interaction is this one this is the result And there is a cool way here, so that it started, that's Dirac, and Dirac was the one who invented this, when he in some way, in the transformation theory, and started some, well, his book is hidden there, okay, you're going to do that, the first is, you know, the book. And then it turns out that the transformation that performed the canonical transformation, the operator, is the molecule operator. So, let me go now to the case that I have arrived at, to the interacting case. So I assume that we all have done all the operations that are of non-interred because we cannot follow them. But at the end we have something that is very similar. These are the eigenstates and this is the D operator. Look at this. This is time, this is X. So time is the eigenvalue. X is a parameter. X plays the role of time in the usual formulation. And X corresponds to the right movers or left movers that were at the free description and that are now corresponding. These are related to the scattering states in one dimension. I am putting only one dimension here because of simplicity. If you go to three dimensions, there are still, you have to introduce some constraints. I am not very involved. This is the formalism. The interpretation is quite similar to the interpretation I had before. I prepared the state of psi here, and then this is the average of the expected value of psi.

1:25:00 The probability of arrival to x in some intervals is this one, and this p of x is this, that is the probability of eventually arriving to x, okay? Look that even for classically forbidden regions, the p of x is different from 0, you can reach the other side of the barrier and so on and so forth. Okay? So, as I said before, deformities account for a lot of things that are not in classical dynamics, so this is completely quantum theory. The first thing to look at is the Wiener time delay. You remember that Wiener introduced in the 50-something in a scattering theory that was the time delay, and he realized that time delay was related to the derivative with respect to the momentum of the phase shift. So, what I'm going to get here, what I'm getting here is very similar to that. It's a quantum version of that, because the Wiener time delay is just a parameter. Now, I am going to give you some examples, two examples, okay? Assume that I choose a Gaussian as an i-initialist type. It's the simplest thing I can choose, okay? A Gaussian that displays a Q0 with mean momentum P0, okay? That means possible. I am in the face of a picture all the time, I realize that, so... Then, assume that at some place, to the right, the interaction starts. Then, the probability of arrival at some points, x to the left or to the right of this point, is given by this. If you are at the left, the probability is just one. If you are at the right, the probability is given by this measure. This is the same measure that when you can go to the Bohm book, or the Bohm book on quantum theory, where he trades this, the old one in the 50's, the original. Now, the time of arrival at the position is, of the interaction, what you get is, this is x minus q0, I mean the distance, divided by the velocity, and this is average.

1:27:30 Okay? Is the average of the classical thing. If you are derived, what you get is something that is more or less the same. You have here, this part is the same part, okay? Is the velocity, I mean, is the distance from q0 to 0 divided by the velocity. And then here, what you have is the same, but instead of having a constant momentum to define the velocity, you have this, the P of Q, is the P that you have at some position, where the potential is V of Q. Okay, it's hard to get, this is the, I'm using WK here, because it's the simplest way to give a physical meaning to that, but let me go now to something that has no classical contact part, and I've said that this question, this problem of the step potential, didn't have classical contact part, well, anyhow, what is clear here is that you have a situation in which you have reflexion, I forgot about reflection before. So, what do you have here? It depends, but you have the probability. I'm not looking at X positive. I'm not looking at this part. So I'm looking, I put the detector here and look to the probability of arrival, of ever arrival at the position X. What you see is the following. If you have a momentum that is above PV, associated to this potential, then you have something that is constant. And if you have a momentum that is below PV, I mean, you enter into a forbidden region, you see that there is an exponential decrease here, something that was suspected. If you look at the average time of arrival at some point, x to the right, You realize the same thing. I mean you have something here. If you have a momentum that is above the barrier, classically you can reach that point. And then you have an exponential decreasing contribution if you have a momentum that is below the barrier. okay so this is the t's that are here that corresponds to the classical parts plus something

1:30:00 that is precisely the Wigner tiny legs okay this is zero here in the case that you are above the barrier because then this t is is real but if you are below the barrier the prime is imaginary then this is not zero, because it becomes complex. So, these two examples give you a bit of the idea of what is going on. Look at this, the important thing is that you see this different behavior, very different behavior. When you have here, if you have momenta, if this weight packet that you prepared here in this initial state has low momenta in comparison with the state of the barrier, it will only arrive here at the border and will not go any longer because of this curve. However, if you have a momenta that are above the barrier, you have mainly this and then you will go, you will travel along the state without any problem. This is something that is not the complete history of that, I have many examples that I have been working on, like tunneling, and then you realize that tunneling is up to some point instantaneous, but that means the following, the time employed in crossing a barrier is slow, okay, this is the Harman effect, but this is coming here along with the probabilities, okay, which is something that is new. in this formula, it was not done before, okay? Also, for instance, if you have photons and you have a photon band gap, I'm talking now about the experiments at Berkeley. You remember the experiments, you send a laser pin, you have here a photon, excuse me, nonlinear crystal, and then you produce two photons, two equal photons, and if you make both photon travel equivalent optical paths, Then you have here one of these interferometers, and then you detect no coincidence at all, okay, when both paths are equal. If you put in one of the paths, you put in mirror, a photon, okay, then what you get is that one of the photons is accelerated or delayed, okay, so you have to move some, to shift some screws in order to get the interference.

1:32:30 And then doing that experiment, they realize that the photon crosses the band gap at the speeds that are superluminal. OK, this can be done. I have done the calculation here and get this. And these are things that I am just looking at this move, OK? So thank you very much. This is all I want to do. OK, any questions or comments? back to this issue of absolute time in some formulations of dynamics there are two T's, namely the observation time and the other one usually the proper time is the canonical conjugate to the Hamiltonian whereas the observation time is usually conjugate to the mass in which case For the time conjugate to the Hamiltonian, that's a non-metric kind of thing. So it advances absolutely, or not absolutely, or it just advances. And then when it advances, the whole system moves forward. It's like an iteration parameter in a calculation. When the calculation moves one step forward, then the whole machinery does one more step. so the advancing of the iteration of course is an arbitrary thing or outside the dynamics in that sense is absolute and what I didn't see in your formalism was a distinction between the proper time and observation time or frame time or whatever it's called by different people. I have been working in the non-relativistic situation You can do that, I did this in the free particle case in the relativistic situation, and then it turns out that what you have is that you can produce time of arrival, let me say, in terms of the Newton-Bigner position operator. I have done that for the scalar particle, but I also have done that for the photon, which is usually said it has no position operator, but time ago, maybe two years ago,

1:35:00 Margaret Thornton's construct. Self-action position operator for the photon, that is the generalization of the Newton-Bringer operator, I didn't use that and this calculation is what I perform here for me but this is not what you are asking me, what you are asking me is a different thing what you are telling me is that you have a whole line and then a mass but then you have a Hamiltonian, Hamiltonian gives you a description of the system That is a relative time. Not absolutely. Or line. Relative to what? To the system. To itself. It grows the properties of the system. To itself only. I mean, it's not referential. Well, yeah, yeah, but the system, remember, I put the clock here. Okay, if you have a, you can add the clock as an ID, okay? And then the clock is just running. The point here, and this is what for me, I still don't understand well why it's going on like this. In general, when people put clocks in quantum mechanics, I think that it starts with Von Neumann in his book, they may put an interaction between the clock hand and the momentum of the particle that is moving or so, so that he produces a correlation between the position of the hand and the position of the particle. Here, is that there is no that kind of interruption, because this, when you have this integrable system and the quantum versions of the integrable system, okay? I did ask again and again to people, do you know the clock? That is like the clock that you put in the quantum hamiltonian to tell the time. This clock in which the position of the system is multiplied by the, I don't know, or the momentum of the system is multiplied by the position of the hand or something like an interaction of Hamiltonian in order to make correlations between the physical system and the clock. and nobody has answered me yeah, this is the clock

1:37:30 always the clocks are something that are independent I can't tell the time but if I start to interact with the time probably I break the time and I cannot get the right time so better don't interact with the clock is what I am getting here classical quantum mechanics I was intrigued by your mention position operator for the photon. Could you say again, who came up with this photon You are asking about the position operator of the photon? Yes. The question is the following. The photon Those are just a reference. The name. No, I give you that. Margaret Houghton. H-A-W-D-O-N Margaret Since I don't know who Listen, maybe there are three physical reviews in the last few years about this point, but let me tell you how it works. I mean, the point is the following. The photon is not professionally invariant because it only has two degrees of freedom. it has plus and minus polarization, it must be zero. So you need to put together both the partners of the photon to produce the photon and then you have a kind of link that's telling you that if you perform a rotation you will never get zero spin. So that makes you that makes it possible to produce a reasonable position operator but this woman D was just to produce a matrix position operator that is self-adjoint and that gives you the right spectrum. It's very interesting. I think that her work is a kind of generalization to the foldable cushion position operator. Okay, well let's thank our speaker again. Thank you.