The quantum theory of measurement; subsequent discussion

0:00 I think the answer to some degree is, first of all, the problem is about why does the universe look the way it does when we look at it, in spite of the fact that there is all these other branches all over the place. Why are we generally only perceiving only one branch? Why do you say one? Why do I say why? One. Why one? When you look at, when you find it. Discriminate among several alternatives for something. Yes. You're narrowing the class. No one, fortunately, has ever succeeded in narrowing it to one. That doesn't change the problem. Yes, that doesn't change the problem very much. I mean, so you have a projection onto a subspace rather than onto a ray. I don't think that changes this problem. Let me put it this way. Can I toss a quantum coin and depending on the outcome of the tossing, can I go with a lot of money and someone to Las Vegas in one branch? Thank you very much for your time, and I hope to see you again in the next lecture. No, it's not so much fuel, but I think it's why does the consciousness choose to perceive all the single separate branches? I don't want to go into some discussion for people who don't see that as a problem, but let me suggest an answer for those who do. And that is, in the way you're speaking of this, you talk about making observations at a particular time, in the standard way of speaking of everything. But I think if instead you take the point of view that one should be talking about a path integral interpretation where you talk about amplitudes for a particular history, then you don't have this snapshot problem of first asking the probability and then you have a whole choice of different branches and the next time you ask a question you may be located on a different branch and asking the probability in a different context.

2:30 So one could talk about a sequence of conditional observations and break it up into a sequence that way in time. Or one could simply talk about the amplitudes of some fixed history, and then you don't have these half-dead problems. This has to be the end. But if one of you could really wrap it up, it would be good. No, I open another problem. The problem is this. You have a system which you couple to the apparatus and then the apparatus is coupled to the environment and that defines the different ways. And so the apparatus cannot be in a superposition of two distinct position eigenfunctions. Now, I understood how the apparatus functions, but I do not understand now why the systems are quantum and chemical, because after all, the system can be in a superposition of two distinct position eigenstates, but on the other hand, it's not coupled to the environment only via the apparatus, it may be coupled to the environment directly. Why is it? Those superpositions are not destroyed as fast as you discussed it for the apparatus. I'm asking now, why do we observe quantum behavior at all? Thank you for bringing it up, yes. I tend to put back the transparency, but this will make it five minutes and I don't want that. The point that's important is a comparison between two timescales. One of them is what I would call quantum Poincaré timescale. In a hydrogen atom, that's the time it takes electrons to go around. That's one time-scale. The other time-scale that's relevant is the time-scale on which you leak a substantial amount of information to the environment. That's the time-scale that's defined by the coupling component.

5:00 The point is, the quantum systems are such that the leakage time-scale is very long compared to the dynamical time-scale. You see the Schrodinger equation is valid for periods much larger than the relaxation time period, the correlation, the correlation. Classicals, it's the opposite. In other words, you would have to have temperature of 10,000 degrees Celsius to make a real event, okay? The ratio of the masses that shows up in the body weight, that's, you know, really good. I think also, to a large degree, the theory that I've written down here is not really the correct theory for the electors, because we know that this theory, for instance, will tell you that you will slow down to the velocity to zero, and we know that there is no velocity to zero, because we're going to divide it. So, you know, but I don't think that's, that's an objection against having a gas of potency, and against that being the case. This will help people ask questions anyway. Historical comment here. This is extremely good stuff. These problems were here. The point is forth and fifth. Discussion is omnium in the universe, which is minimal and admirably circulated. I don't have an open mind, but basically I do. One of them had an open mind. There was a great consensus among my mind. Why has there been a difference? Why is there so much more readiness to accept it? Isn't it really because we have been concerned in the intervening years with such things as quantum cosmology, and the interface between particle physics and cosmology? The other startling change, the same particle the other day, which simply said flatly, the reason this land unified field theory is definitely a fact.

7:30 Fifteen years ago it was quite the other way around. Why bother with gravity at all? It's been a real change. Well, thank you very much. To present Murray Gelman, who is going to talk on the quantum mechanics of this specific universe. It's a pleasure because he's made profound contributions to our knowledge of nature. It's a personal pleasure for me because we were... Schoolmates as undergraduates many, many years ago. And I also think it's something else. I feel that Murray is a link in a very important relation that I have. He is on speaking terms with the logos of the universe, primarily in Greek, and I know him. And therefore, via him, I feel I'm related to the logos of the universe, and I think many of the rest of us could feel the same way. Well, thank you, Edmund. I don't know what I can say to that. You're the oracle. I have probably eight hours worth of material to select rapidly on my feet, those little bits that I think might be... And it's all written out on these transparencies which I will rush through so fast you won't be able to see them and it'll be kind of a blur but every once in a while I will stop and it's very nice that Wojciech Żurek was able to speak first because he emphasized a number of the things that are important in this area of the quantum mechanics applied to the whole universe and to all of science. It's not invariably a sign of, it's not, there's not universal agreement that interests in this subject as a sign of synology. Ah, another old school man. I started thinking about the relationship between quantum mechanics and science as a whole around 1963 in discussions with Felix Villar and occasionally Richard Feynman.

10:00 And we discussed the point of view, which I never wrote up. It turned out later to be very closely related to the many worlds picture of Everett, who was encouraged by Wheeler. I never heard of Everett's interpretation until sometime in the 70s, but apparently he wrote it up in 1957. The main thing is that he explained this interpretation in ways that bother a lot of people because of connotations of some of the words. And if one gets rid of those particular connotations, then I don't see any problem with it. But there's a severe problem of it. I emphasize, as then as now, density matrix for the universe and the description of the environmental sciences as well as fundamental physics. All of them, after all, have to be covered by quantum mechanics. As Bryce pointed out, nowadays there's quantum cosmology, the work of Hartle and Hawking and a number of others, and the climate is different for this kind of thing. It's much easier to talk about it now than it was 25 years ago. I started thinking about it again a couple of years ago, two and a half years ago, with Valentine Pelegny, who encouraged me. I had always avoided the idea of any professional commitment to the hobby, but Valentine suggested that it was important to work with it. It would have been better if he told me that in 1963! And then I got together with my old friend Jim Hartle in Santa Barbara a couple of years ago, and since then he and I have been working on this together, and what I'd like to present today is a joint effort, which we already know. The mistakes I make, however, in the presentation are my own. I'm not very happy with murky philosophical questions, and I have on my wall a prescription from a physician forbidding me to debate people on philosophical questions. I had a colleague, unfortunately recently deceased, who claimed falsely to have such a prescription, but I actually do have one. I always had trouble in my research with people telling me at first that it was crazy, and then later in retrospect that it was trivia.

12:30 In this field I discover you can have the pleasure of being told both, right? Now, let me go quickly into this. I don't discuss here any possible modifications of quantum mechanics like anything else in science is subject to modification. We talk about questions like this, how can quantum mechanics permit quantum cosmology, already partially addressed by Witt here. How can the universe appear to have a history when quantum mechanics appears to be... Contradiction with the idea of history. What is the status of the information about this specific universe or this class of, we should always say, this class of universes, plays such an important role in the environmental sciences? And what's the source of the regularity that's exhibited by the environment? What constitutes an observation? And how do we characterize observers? We won't use the word observer so much as the term information gathering and utilizing system. It's a new term that we introduced simply to avoid some anthropomorphic connotations. And we know, as in the previous talk, that the old Copenhagen interpretation had these faults, that the probabilities, the statistical probabilities requiring an identical number of identical situations actually available in practice. Second, the observer is considered apart from the system studied. And third, it's assumed that classical objects not only exist, they obviously do, but that they obey some separate set of classical laws outside of quantum mechanics. None of these can be preserved unmodified if we're really going to have quantum mechanics. I think if it hadn't been for the strong personality of Niels Bohr brainwashing people in the 20s, no adult would have swallowed these things anyway. So there's a density matrix for the whole universe road of a Schrodinger equation. I will use the Schrodinger and Heisenberg pictures interchangeably, unfortunately, in the talk. I haven't gone through the transparencies to edit them for Schrodinger versus Heisenberg, but in a proper version of it. Yes, yes, I'm going to go on with that. Don't ask any questions. Be patient.

15:00 There may not be, I say here, a well-defined time variable. I'm not going to get into it. You can argue with my colleagues about it. Some over past formalisms, some people allege, might get over this problem, but I'm not going to get into it at all. I'm assuming that after a certain time, after a long time or whatever, that we can have, essentially, a well-defined time variable, a well-defined energy. And I'm not going to worry about those other questions. To you. And for the moment, I adopt the solipsist point of view that I am the observer, you're all figments of my imagination, there's only one observer, and I make observations and so on. I'll have to correct this later, otherwise what's the point of my wasting my brain on all these figments of my imagination? You have a rich fantasy life. So, yeah, well I'm going to fix that later, if I get around to it. Not too many interruptions I may get to. I assume talk about projection operators, summing to one, orthogonal, usually. And also I'm going to make a ridiculous idealization that measurements or observations are carried out at a definite time and in a small region of space. I can't be bothered to get into the very complicated questions that are raised by you, maybe later, but I doubt it will be time. Projections P1 with alternatives alpha 1 and later time T2, I have a bunch of projections P2 for alternatives alpha 2, and this can be for one or more commuting variables, call them all one variable. Then there's a density matrix for the whole universe. Some people take it to be a pure density matrix, it doesn't have to be. Its only function is to yield probabilities, but these are a priori probabilities, they're not statistical, not necessarily statistical.

17:30 And when this a priori probability for some correlation in the universe is near one, then you can make a real prediction. Pardon? Pure means a pure state estimate. For example, Hartle and Hawking assume a pure state estimate. The statistical probability means that you have available to you a coin that you can toss a large number of times or some array of identical things to play with, protons that you can scatter one after another and so on. A priori means you may have just one. Calculating whether your marriage is going to be successful. The kind of thing you do all the time, right? I mean, you don't have the statistical elements. Some of us in the course of a lifetime have them. Now, it's easy to show, many people, including Jim, have shown that if the universe contains an ensemble of protons or something like that that are identical that you can play with in large numbers, then the a priori probabilities for the universe yield statistical probabilities for the subsystem that you're producing. Now, in the new inflationary cosmology, we are in one Robertson-Walker domain, a bubble of reheating or something like that, and the others we may not come into causal contact with for a very, very long time, so you can apply, if you want, statistics to those and think of the a priori probabilities as being statistical probabilities as an aid to thinking in those other Robertson-Walker domains, but you can do even better than that. As you know, in the new inflationary cosmology, the visible universe is just a tiny corner, very, very tiny corner. Some people think 10 to the minus 30,000 or something like that. A very tiny corner of a huge bubble, Robertson-Walker domain, a bubble of reheating after inflation. But that's only a part of the universe. The universe is much bigger than that. And how the rest of the universe works with the space-time foam and so on, I'm certainly not going to get into and especially not here. But I'll talk about that bubble as the sub-universe. It's part of a much bigger thing, and that whole universe, the entire universe, is considered by some cosmologists to be possibly a quantum fluctuation, among others, with no causal connection whatsoever among these, so that treating those as a statistical ensemble has zero practical consequence, exactly zero practical consequence.

20:00 The same is true if the universe is one topologically disconnected part of some multiverse, again with no possibility of any contact ever among them in the future, again no practical utility whatsoever, but you can, if you like, visualize the many universes interpretation of quantum mechanics as being displayed over these as a statistical ensemble. If you wish, it's harmless. It's a nice conceit, though. It gets you away from a lot of the annoying connotations of the kind of stuff that Everett wrote, distinguishing his prose here from his ideas. If your ideas are correct, the prose is misleading. I didn't want to say that. You're actually taking the sins of the world on your own. First to fourth declension, which is much worse. This is a mess, but all it says is that the probability formulas trace, and here are the probabilities for the last thing, for the next to the last, for the next to the next to the last, and so on, in between somewhere is rho. And in the Schrodinger representation, this rho can be evaluated at some very early time, t0, in the universe, just after time became a good thing. Now an important feature of this formula is that everything has to be time-ordered. If we allowed zigzags, then we would get a non-zero joint probability for two mutually exclusive alternatives at the same time, which we obviously better not have. So zigzags are absolutely forbidden. Everything has to be time-ordered as part of the rule. In the sum over paths formalism, that comes out very naturally because you sum over all paths e to the is for the path times, for example, two field operators, phi of a spacetime point x and phi of a spacetime point y, divided by the normalization sum, and that's equivalent in Heisenberg language of operator quantum mechanics to psi the ordered product of phi of x, phi of y. So the ordered products come out perfectly natural. If you wanted the reverse order, you'd take e to the minus i. In the Feynman diagrams, there are zigzags, but those zigzags are purely mathematical zigzags having to do with the perturbation theory, the Striegel-Barrack-Feynman perturbation theory. Nothing real.

22:30 Can you tell us what we do in quantum cosmology when we're using a factor on time? No, no, I'm not going to do that. If we want to reverse things, we change e to the i s, e to the minus i s. We shall discuss later, when we talk about the arrows of time, how the main time arrow comes from the fact that the universe was in a particularly simple condition near T-nau, some 10 to the 10th years away from now, and that direction we call ago. If we wish to put that in the future, we change e to the i-s, e to the minus i-s, and run things backwards to form a CPT transformation of the whole theory. In the early universe there are some theories according to which you have e to the i-s and e to the minus i-s. And those have to be justified, because as soon as you get out of this early time, you better have mainly the IS. But I'm not going to get into that. Now, the joint probabilities can be used to construct conditional probabilities, like this, by summing over all these things that have happened to the last bunch of alternatives. And then these are the useful probabilities. If I want to say that certain things have happened, that the universe is restricted by alpha 1 choice, alpha 2 choice, and alpha 3 choice, up to alpha n minus 1 choice, then I would use these renormalized probabilities. And you can see from the formula that if I'm talking about the last one, the next to the last one, the next to the next to the last one, and so on, and I sum over them, then that sum over those probabilities is the same as the probability of the first few up to there. It's important to say that. It sounds trivial, but it's important. So after a number of measurements have been made, then I'm dealing effectively with the renormalized row, with this trace here renormalizing it in the denominator, which covers just these things that have been. It's a row of t naught, and then all these measurements that have been made, observations that have been made. They don't have to be active. Astronomers, after all, learn things, too, and they just go out and look. And other scientists learn things, too. We're talking about all of science, not just physics. So the probability of future observations can then be expressed with this row embodying the first few measurements and then the projections onto the next ones.

25:00 Now a lot has been made of this. People call it reducing the wave packet, but it's only those variables that are affected by the projection operators under discussion that are being in any way reduced. Most variables in the universe aren't being, having anything done to them, so that's an extremely poor name, reduction of the wave packet. Sometimes it's called selection among many worlds, and as long as those many worlds are thought of as being really separate in the way we discuss, many universes if you like, then it's fine. I've just gotten some information about which class of universes I'm in, restricting the class by P1, by P2, by P3 as the alphas are selected. It's similar, although not identical, to what happens in a horse race when you originally have a joint probability for the eight races and after the first three races have been won and the winner is announced, you renormalize the probabilities for the remaining five races. The only difference is in quantum mechanics you're dealing with non-commuting operators, so that in the EPRB situation, for example, you may learn, say, the right polarization, in which case you know the other photon is right polarized, or you might learn the linear polarization. In which case you learn the other photon is linearly polarized. Pardon me, it's Bohm who reformulated last time's situation with David Bohm. And mentioning him it doesn't mean I endorse a lot of the things he did. I was with him in Princeton during the days when he was concocting all these things. First as a Marxist he felt he had to reject quantum mechanics. And he was under great strain then. Personal strain because of all kinds of personal problems. Political strain because he was a witch and the witch hunts were going on in Washington. But anyway, he wrote a quantum mechanics book in order to strengthen his faith in quantum mechanics. And having written it, he decided that he really believed in the theory of measurement now that he had explained it himself in his own book. And he was waiting eagerly for an opportunity to talk with Einstein. He asked me if I would arrange an appointment with Einstein. As a new arrival as a postdoc at the Institute for Advanced Study, I was scarcely the person to do this.

27:30 And the next day, Bohm said, wasn't necessary. Einstein called me. As soon as the book came out, he bought it and read it. He immediately phoned me and said it was the best presentation of the case against me, against him, that he had ever seen. And tomorrow I have to talk with him. Fine. They met the next day. The following evening I saw David and I said, how did it come out? And he said, I don't believe it anymore. Now, it's very important to emphasize again that when I learn something, I'm learning only about an infinitesimal number of all the variables in the unit. And lots of variables, of course, are normally ignored. Radiation escaping to infinity, things in the deep interior of macroscopic objects, things that are too far away to distinguish. Stephen Hawking and other people made a big fuss about black holes, gravitationally collapsing stars. Whether or not Stephen is right in saying that there's something in principle about the information that's falling into a black hole, certainly in practice it's not terribly available to us, so again, it's the kind of thing you would normally sum over, and that of course makes a density matrix into an even more mixed density matrix when you have to sum over all these things. So the fact that measurements or observations ignore most of the variables in the universe makes possible two pervasive features. These are key terms of our experience of the world. Decoherence and irreversibility. Both of these are essential to the notion of observation and measurement. And I'll oscillate between them. I'll talk a little bit about decoherence and a little bit about irreversibility, back and forth. Decoherence, you remember we said that we had this normalization formula for the probabilities because the denominator can be expressed as the sum over the probabilities of all the events. If you sum over the last one, the next to the last one, and so on and so forth. But if you sum over one in the middle, or at the beginning, or scattered ones through the formula, it's false. It's not true anymore in quantum mechanics, as you can easily see from looking at the formula. So you don't have classical sum rules like this or this, which hold classically. Only if you sum over the last one, the next to the last one, because then you can move the projection operators through, sum over them, get one, move the next pair through, sum over them, get one.

30:00 But you can't do that if they're in the middle. And this is like the famous two-slit experiment. This is as much as I will do to draw a Gedanken experiment. This tapeworm is my idea of a detector, a set of detectors. And here's a set of photons and two slits in the screen. And when you sum over the upper and lower in this position next to the row, with the detectors out on the end, you don't get the answer. The answer, because there are cross-matrix elements that are not zero, where you have upper here and lower here. Those are the interference terms. But because of the existence of the interference terms, you can't simply sum these two and get that. You can't sum over the inner projections, the probabilities for the inner projections and get this, because the cross-terms contribute also. So the only way you ever get the classical formula to hold is if the cross-terms essentially vanish, like here. But as soon as alpha n is different from alpha n prime, this thing is zero. Then you would get the classical rule when you sum. Because then the sum over both of them, which does give one, is equal to the sum over just the equal values of the alpha n. So you get the classical sum rule to the extent that the cross terms are zero. If all the cross terms are zero, so you have a string of decoherencies, a complete string of decoherencies, then you get all the classical probability formulas, and you're dealing with a classical. This classical situation is necessary if you're going to talk about measurement, observation, learning, acquiring information, storing information, any of the things that you've done in measurement, in what's called a valid measurement. You've got to restore this classical formula by getting rid of all the off-diagonal terms. Now, no one is allowed, there's a strict rule that no one is allowed to discuss this subject without mentioning Stern-Gerlach, so I will say Stern-Gerlach. If you can bring the two beams back coherently, as in one of Wojciech's beautifully drawn pictures, if you can bring them back together, then you have made a valid measurement. It's only if you can't practically bring them back together, if the phases have been dispersed someplace where you can't get them back practically, that a valid measurement.

32:30 So you have to have this irreversible escalation to a persistent macroscopic mark. Amplification, if you like, by a signal above the ambient noise. The information is then typically stored redundantly, all of this described beautifully by Wojciech, in such a way that restoration of coherence is not practical. But we have to say what we mean by irreversibility. Irreversibility is not a self-defining thing. In fact, the equations are essentially reversible, and irreversibility has to be defined with respect to a particular way of talking. But it's crucial that the results of sequences of valid measurements can be stored in records that are describable by operators that approximately commute equal times. These records, which we call present data at any given time slice, these data are the things that you use to predict the future and to reconstruct probable histories of the past. In an observer or information-gathering and utilizing system, we will talk about these as memory slots, but of course they also exist just as material things. Now, with this persistence of the mark, then I can share my results with you, and you no longer have to be figments of my imagination. We can all share things, and we can have a collective experience, and we can become a collective igus, and we and the apparatus can function in the same way. Various other parts of nature can function in the same way. So you can get barring, error, erasure, and so on and so on. You can get agreement on results and escape solipsism. You have to show these results in the same measurement. Yeah, of course. The point is the correlations are very high once you get through a certain situation. And I'll be describing, trying to describe that situation. And we will talk about an information gathering and utilizing system, as I said, in place of using the word. Most of the examples are living things or devices constructed by living things, but that doesn't necessarily have to be so. The functions of the IGUS include utilizing the probability formula in some approximation or other, maybe a classical approximation, maybe an exceedingly crude approximation of some kind,

35:00 but using it somehow to predict or to reconstruct history, and renormalizing the probabilities to take account of the alternatives that are being checked off. As the class of universes is somewhat narrow. And then it's only the renormalized or conditional probabilities that are relevant for further prediction, the other classes of universes are then irrelevant. Now, there are other functions of an igus, but those are in no way require anything as fancy as an igus. Actually performing the measurement or observation and so on and so forth. This part we will call the measurement situation. And that occurs over and over again in nature, in inorganic nature, in perfectly ordinary circumstances. And so the functions of the igus are actually very, the functions that require the igus are actually very narrow ones. The simpler function can be performed not only by artificial pieces of apparatus, which physicists like to talk about, photographic grains, supercooled bubbles, things like that. But also inorganic mechanisms of widespread occurrence in the universe. We will see that they are all related to the remoteness from equilibrium in terms of suitable variables of the density matrix rho of t naught of the early universe. And might as well say now, because I will probably say it several more times, but I might as well begin now to say that the main thing is to emphasize rho of t naught. Or rho of all times, if you like, just related by the Schrodinger equation for the Hamiltonian of all the elementary particles. The main sin of the people of sixty years ago and also many more recently and even some today is trying to answer puzzles about quantum mechanics within the formalism of quantum mechanics rather than realizing that the answer lies in rho of t naught. Rho of t naught is the solution to most of the old puzzling questions. Just staring at the formalism of quantum mechanics without invoking the special character of Rho of T naught, which is what permits us to exist, any igus to exist, permits all these circumstances that allow the notion of measurement and observation to exist, is foolish. It doesn't lead anywhere. I'll use one instructive example. Ancient fission tracks in a mineral such as mica. These are actually used, as many of you know, in science.

37:30 Uranium nucleus and naturally occurring impurity decays and leaves behind in the mica tracks of defects in the crystal. You can assume the crystal to be perfect initially if you like, even zero temperature initially. Of course it isn't that way, but those are permitted idealizations. Say there are two fission fragments, you get two tracks in opposite directions left behind in the mica. They are left there for thousands and thousands of years, can be detected later by the scattering of light. This is an example, I would say, of a natural measurement situation. The amplification and persistence are obvious. The correlations that lead to the formation of a track are the same ones that were discussed by Mott 60 years ago. What interests us is the matter of the irreversibility of the situation and the decoherence. So we've got everything here but the igus. Then Mr. Fleischer comes along and the picture is complete. Now, decoherence is often talked about in terms of a reduced density matrix, where we divide the variables of the universe into two classes and sum over one class as being comparatively uninteresting, keep our attention fixed on the other. We'll call that the spore, and we'll call the trace over the remaining variables a little trace, TR. Then you write the density matrix for the whole thing, and then you sum over all the variables that are being ignored and construct a much more complicated equation than the Schrodinger equation. For Spur of Rho, it permits all kinds of things that the closed system wouldn't have. Spur Rho has increase of entropy, it has canon, it has dissipation, it has all sorts of things that an open system typically has, and decoherence. Decoherence cannot occur, as we shall see, for a closed system with perfectly fine-grained entropy, but it can occur as soon as you have coarse-graining or as soon as you sum over something variable. Now, the separation of variables into two classes like this is often awkward. It's not general enough, and we'll generalize the definition later, suitably, to include all situations. But for the moment, let's look at this case of dividing the world into two kinds of variables and summing over one kind.

40:00 Here are some of the people who have studied the problem. Through carelessness on the part of the publishers, this name was radically misspelled, leading me not to understand how it was pronounced. Anyway, Ugo Fano, who has turned it mysteriously into V. Fano in the literature, in the references, but he's actually Ugo. Of course, in Latin, the two letters were not distinguished. Particularly, I will be talking about the brilliant contribution of Wojciech Żurek. But anyway, these are people who have looked at it. So you have Hamiltonian, H1 plus H2 plus V12, and you assume for simplicity that rho is a product initially. Not in the early moments of the universe but at some time t equals zero. It's a product of a row for the one set of variables times the row for the other and then you construct the equation for say the interaction representation version of the reduced row and you get an equation for it here and this second term is usually the first one that contributes. You notice the spore is taken here and row two appears here so you have expected values of all these. The usual problem is an oscillator coupled to a lot of other oscillators with a bilinear coupling, so that you can solve it exactly, and the information gets distributed from the oscillator of interest to the others, which are assumed to be very numerous. There's this interesting insight of Jurek, which he described here, which is that in a ridiculously short time, The position of the interesting oscillator decoheres with itself as a result of the summation over the influences of all the other oscillators and this ratio lambda bar of a delta x squared is 10 to the minus so these huge numbers always occur. So Spur-Roh effectively becomes operator decoherent. It becomes a sum over P alpha Spur-Roh P alpha. The diagonal terms simply vanish from the operator. Now that cannot happen to Rho.

42:30 As we'll see later, there's an easy proof from entropy that rho can never undergo this, rho can never go into sum over alpha, p alpha, rho p alpha, unless it starts that way. But Spur-Roh can. Just as Spur-Roh can do all sorts of other special things. Now, it's useful to look at the formalism that Feynman and Vernon introduced in 1963, sum over histories formalism. Actually adopted by Caldera and Leggett. It doesn't do Caldera and Leggett much good. In fact, I think they would have done better not to use it, but for us it's useful here. So we have the Spur row with a time tau, referred to as initial time zero, and I take x and x prime on the two sides of the reduced density matrix. These are the coordinates of the selected oscillator. We integrate over the initial positions of the oscillator on the left and on the right, and then we integrate over the paths that it follows, x from zero to tau and x prime from zero to tau, and here is the amplitude, which is e to the is of x of t, e to the minus is of x prime of t, and then e to the iw of x and x prime, which expresses, this is a functional that expresses the integral over all the other junk. And this e to the iw was called the influence function by Feynman and Vernon, and it tells you what all the other stuff does. Now the approximate decoherence, the habitual repeated decoherence of x over time, expressed in terms of projections p, when we narrow those projections onto narrow intervals of x, can be related directly to the sum over histories. It becomes a decoherence of the histories. This decoherence of the histories then means that contributions come mainly when the trajectory on the two sides of the density matrix is approximately the same, x of t and x prime of t follow similar trajectories. In the heat bath example that we've been considering where the oscillators are assigned some definite t, the uninteresting oscillators are assigned a heat bath temperature t, if you go to the Fokker-Planck approximation, which was actually mentioned by Jurek in his talk, Then the imaginary part of W is proportional to the integral of x minus x prime squared, dt. A beautiful example of squeezing, because e to the minus a constant times this is what you get there. I times I times a constant in the exponential gives you e to the minus a constant times this. And that squeezes the x and x prime together. So the decoherence results from that squeezing in an obvious way.

45:00 That's the general phenomenon that occupied almost the whole of Wojcik's talk. Let me say a word about reversibility and irreversibility, by the way. If you go to that approximation, Fokker-Planck approximation, you get something called the Decker equation for spore rho of t, which is a simple differential equation for spore rho with a single time derivative. Then the friction, radiation, and other irreversible processes occur in that equation multiplied And so the equation remains perfectly symmetric, but all these irreversible things turn out to be multiplied by this, so that the irreversibility is introduced by the fact that we use the equation only for positive time, the reason being that we are operating forwards, and we are setting the thing up at t equals zero, and considering as physical only the things that happen in the equation after t equals zero, that's what's introducing the asymmetry. The equation itself is perfectly symmetric. Pardon? The distinction is that t equals zero is uncorrelated. And that the future is the time one way from that. We run the equation one way from that. We do not apply it the other way. If we applied it the other way, all the friction would go backwards and so on. It would be all opposite. So the important thing is that we're not using here the density matrix of the sub-universe or the universe. We're using the effective density matrix that reflects the results of measurements or observations that have been made up to t equals zero. Now, the initial density matrix rho of t naught is such that the mechanism of decoherence that we've been discussing applies to many operators in nature. Contributes to the nearly classical behavior that's observed for so many macroscopic objects. Center of mass of a planet is a sort of typical example that people argue about. Yosensei, already referred to, argued that the scattering by the planet of low density material, even just the three degree photons, Decoheres the planet. In other words, the x and x prime on the two sides of the density matrix is squeezed together with a huge coefficient just by the encountering of those little...