Bohr, Penrose & Hawking / Limits of QM from GR — A Clash of Principles

0:00 Thank you. So just keep it running until you go to dinner. All right, I'll see you tomorrow. The pass always coming. I'll just stick it in the car. Well, I've just been saying to be able to carry it down to London back in here. I'm just thinking nothing to do with it. What? Where are you staying tonight? Yeah, I can just stick it in the car. Yeah, are you going back to London tonight? No, but I'm going out with loads. Okay, I'd better take the winter. It's all right, I can stick it in the car, it doesn't matter. Mike! No problem, I'll just put it in the car. I'll just stick that in the car. Thank you. Thank you.

2:30 Thank you. Hello, it is my pleasure to introduce Vin Onruf from the University of British Columbia. and he will give a talk that is not a talk until it is not heard. Okay. What I want to talk about is still definitely work in progress. I thought I would be slightly farther along, but I'm not. So what I want to do is put out some ideas and I guess get some feedback to them. In some sense, the theme of the talk is, I guess from Niels Bohr originally, at least transmitted to me through John Wheeler, that no phenomenon is a phenomenon until it is an observed phenomenon and in particular I want to discuss a few issues about observations in gravity and quantum mechanics and I will have a caveat in a few minutes and of course all started way back when with more with Heisenberg than before I guess or in fact from what I recall was objected to the Heisenberg microscope experiment where Heisenberg, trying to think about the uncertainty principle that he had derived for his matrix mechanics, thought about this

5:00 little microscope and observing a particle by bouncing photons off of it and showed that by using these photons you got an uncertainty principle that looked like this for these for this electron in agreement with the arrived. There's not enough room here. Is this thing needed? What is it? There's a gale force wind blowing on that side. And we also recall Bohr's involvement in the argument by Parles and someone else that the electric field, that there were fundamental limitations on the measurement of the electric field. They thought about placing a little electron into the electric field and trying to measure the value of the electromagnetic field, and by using the uncertainty principle for the position of that electron in trying to measure the force came up with an uncertainty principle which said that you couldn't measure the electromagnetic field beyond a certain accuracy. Bohr and Rosenfeld came back and said no this was nonsense. They introduced the idea of compensating fields, compensating forces on this electron in order to show that you could measure the electric field to arbitrary accuracy. In fact, all of those compensating forces and so forth are really irrelevant. One can sort of do everything almost that they did in one page. Basically, if you look at this thing from the Heisenberg representation rather than the Schrodinger representation, if you try and measure the electric field, you stick in a little electron there, you want to, for example, see how far the electron moves, and what you find is that the position of the electron afterwards is related to the position

7:30 before by an equation like this. In other words, the uncertain momentum that came in by specifying the position of the electron messes up the measurement of the force. On the other hand, that only happens because you're trying to measure the wrong variable. and measure the variable x afterwards instead of the variable which cancels out that initial value of the momentum. For example, if you measure the field x minus pt, where t is supposed to represent the full measuring time, then in that dynamical variable, the initial momentum cancels out, and you simply have something that expresses the, if you make a measurement of this final guy, that's simply expressed in terms of that initial position uncertainty and the field that you want to measure. So by making the initial position as accurately, as accurate as you want, you can measure the strength of this field to arbitrary accuracy, as long as at the end of the period you simply measure this variable instead of trying to measure, let's say, x or p on its own. They introduced this whole compensatory mechanism in order to automatically have the dynamical system of the electron more or less compensate for this initial uncertainty in the momentum, but that's simply unneeded as that one makes the appropriate measurements afterwards. However, there's a salutary lesson in this example, namely that arguments from measurement theory can be very dangerous, that all that arguments made from measurement theory may demonstrate is the lack of imagination of the arguer rather than anything about the physical world. And I want to put this up here just to let you know that I'm cognizant of the danger and that I may be falling into this trap just as Pauls and whoever else it was fell into it with the electromagnetic field. Well, the story sort of starts...

10:00 I don't know why there's a gale-force wind at that time, at that end. In some work that I did way back in about 1980, where I was thinking about the application of measurements to the gravitational field. And in particular, I'm going to review this just very quickly. Let's say that you want to try and measure the metric, g mu nu. Now, the first reaction of a general relativist is, what in the world do you mean by measure the metric? After all, the metric is something that relies on a coordinate system. And what I'm going to mean is that we've got our coordinate system basically supplied by some sort of physics, some kind of physical variables. So, for example, I may call my electron at position x equals zero, and the collision when these two electrons collide, I'm going to call that t equals zero, etc. So I simply place onto the space-time a coordinate system, which is determined by some physics going on within that space-time. I don't necessarily imply anything about the distance between these objects, etc. So what I want to do now is I want to measure g mu nu, little g mu nu, the metric variables, in this coordinate system. That means I have to try and make measurements between those various coordinate values. Let's say between t equals 0 and t equals 1. g mu and u is just the thing that tells you what distances are between various things. And the argument that I made in that was that this also implies uncertainties in the curvature of the space-time. Namely, that one cannot simply do these measurements without also at the same time upsetting the space-time around one. And the arguments I gave were something like, for example, if we want to measure the distance between two points, then we've got to put something like a ruler in there. If we're going to measure that distance to a certain accuracy, then that's going to imply momentum uncertainty. What is the momentum uncertainty? The momentum uncertainty comes about basically because the forces which this ruler that you stick in there apply on the two

12:30 things at the end that you're trying to measure the distance between i.e. there's an uncertainty relationship between the the distance that you're trying to measure and these forces that exist within this ruler the stresses that exist within the ruler using einstein's equation those stresses are, of course, related to the curvature of the space-time. And as a result, going through that argument, you get something like the uncertainty in the metric components, xx, and the uncertainty in the curvature tensor, gxx, times the volume of the space-time that you're, this is the four volume of the space-time that you're trying to do this measurement over, must be greater than or equal to something like h-bar. Similarly, if you're trying to measure times using clocks, you use the same uncertainty relationship, namely delta T times delta capital E greater than or equal to h-bar. This, of course, is a very, very dangerous uncertainty relationship because numerous people throughout the history of physics have badly misinterpreted it. For example, this says nothing about one's ability to measure the energy in a certain time interval, as many, many people have argued that that's what this actually does. What this uncertainty relationship rather refers to is the ability of a physical system to act as a clock, the accuracy with which a physical system acts as a clock, versus the uncertainty and the energy of that internal clock. In other words, if you want a clock to have a certain accuracy, delta T, then that clock has to have an internal uncertainty and energy, delta E, in order to act as a clock of that kind of an accuracy. That's what this uncertainty relationship means. It doesn't mean anything about mutual measurements. Anyway, using this uncertainty relationship, again, I argue that one got an uncertainty relationship on the metric component G0-0. simply related to the Einstein metric the capital G naught naught component of the Einstein metric again averaged over a space-time volume. So that just leads to this kind of thing. Well this leads me to the first of or I guess the second of

15:00 the names that I had in my title, that I took in vain in my title, namely that of Roger Penrose. Roger K. has come for a number of years, has come up with a very interesting idea, and has argued that somehow or other quantum gravity should automatically and inherently lead to decoherence. And the argument as I understand it is the following. Let's say that you split up into two pieces and then you send it through an interferometer. So this mass could occupy either this position or this position in some sort of coherent way. Then the problem is that that mass in this position will affect the running of time in a different way than it would affect the running of time at this position. ability to synchronize your clocks and the ability to know what time it was, and therefore the ability to know what kind of a phase shift you got, because after all phase shifts are determined by the time development, is determined by the location of these masses. And this uncertainty in exactly how much time had elapsed on these two paths then led to a decoherence between these two masses going along these two paths. This is, this argument I've always been of very much two minds about. it. One of the things, I'm not sure why that first instance bothered me, but I guess it was a comment. Any theory of quantum gravity that has, that relates energies to time should have this kind of stuff. But one of the things that bothered me is that we know that there's a sort of idea of false decoherence. Things can seem to be decohering without actually happening. And the example that I have is, for example, if we take an electron with its

17:30 electromagnetic field. For example, an electron used in an electron microscope. Now, that electron is in a coherent superposition. It goes along two different paths. If we look at the electromagnetic field surrounding that electron, we find that those two electromagnetic fields due to the electron sitting in this position versus the electron sitting at this position, the overlap between those two electromagnetic fields is very, very small. Because the electron and its own electromagnetic field are very tightly correlated, that means that if we look at the electron itself and ask for the density matrix, if you will, for that electron traveling along those two paths, one finds that when you separate the electron, the density matrix basically gets reduced down to its diagonal because the electromagnetic field itself has this very small overlap. One finds this sort of as a very generic principle when you have any mass or anything coupled to an outside set of fields. The thing about this, however, is that if we bring those two electrons back together again, we find that they interfere perfectly well. In fact, experimentally, we know that they interfere perfectly well. And so this is a sort of false decoherence, because what's happening is that one's tracing out over an environment, if you will, But that environment of the electromagnetic field is very tightly coupled to the electron itself. So when you separate the two, that environment cancels out any coherence between those two electrons, but that coherence just completely reappears when you get to bring those two electrons back together again. And I've done a few calculations in various models like that. And this worried me that this is the same kind of decoherence going on here. So the question that came up in my mind is, is the Penrose intuition the same thing? On the other hand, Diossi has argued that one can use this kind of uncertainty relationship that I derived, in fact, to come up with exactly the same kind of arguments that Penrose did,

20:00 the same kind of decoherence arguments that Penrose came up with. This bothered me, I mean, it gave me a little bit of the same kind of feeling as perhaps Einstein felt when Bohr walked into the room after a nightless sleep and said, well, the reason that the time-energy uncertainty relationship and this weighing of energy debate, you all know the weighing of energy debate, I assume, where Einstein tried to disprove the delta T, delta E uncertainty relationship, the correct uncertainty relationship by the way by weighing the box, determining the energy by weighing the box in the gravitational field. And Bohr came back the next morning and said, ah, the reason that your little example doesn't work is because of general relativity. Incidentally, that problem was asked in the general's exam in Princeton the year after I took it, fortunately, which I always felt was an incredibly unfair thing to do. It took Bohr a whole night to come up with the right answer. If you happened to have seen it, then you would have been okay. If you didn't, there was no way that you were going to come up with that answer overnight. besides there must be something wrong with the answer because surely the consistency of quantum mechanics does not require the truth of general relativity. It turns out it doesn't. Any theory which allows you to weigh energy and apply Einstein's argument for how one could determine the energy of the box also would apply, would also apply, imply that you get this redshift relationship, which I managed to show a few years later. But anyway, that's a complete sidetrack. So part of the question in looking at this was to ask, what I want to look at is to ask whether or not conventional general relativity and quantum mechanics can say anything about this question. Now, the real problems in quantum gravity, of course, have to do with things like non-renormalizability, etc. They really have to do with the independent dynamical degrees of the gravitational field. Or, to put it another way, they have to do with gravitational radiation.

22:30 The fact that you can get gravitational radiation fluctuations of arbitrarily small dimensions the problems in quantum gravity. But we know that there's one system in which this isn't a problem, namely the spherically symmetric system. Because in a spherically symmetric system, you have no independent degrees of freedom. Just as with an electron and its electromagnetic field, the degrees of freedom of the gravitational field are completely tied to the dynamics of the system, to the dynamics of the matter field. So the question is, if I just use that system and try and quantize it, a spherically symmetric system, that should have all of the features that Roger's example has in it, and should allow me to see whether or not this kind of decoherence, whether there's some room for within the system. Let me get a little bit technical here. We can quantize the system in the following way. I warn you, those who do, those of you in computer science or whatever can go to sleep for a few minutes. So if this is the metric, where's that green pointer? If this is the metric, so we write the metric in the standard so-called ADM form in a way in which we can easily split it up into a Hamiltonian kind of metric, then as usual one gets an action which is the Hamiltonian type of action together with these so-called constraints within the gravitational field, if we have as our dynamic variables this factor u here, which corresponds to the metric in the radial direction, distances in the radial direction, and this factor r, which corresponds to the surface area of the spheres of symmetry of this space-time, then the Hamiltonian looks, turns out to look like this. The matter Hamiltonian, what I'm going to do is to take from a matter Hamiltonian, basically, something like dust,

25:00 i.e. it has the action, the Lagrangian, if you will, of a free particle, but I'm going to allow there to be some radial, some angular tensions in these dust shells, which I can used to control the motion of the dust. I'm not going to be using that very much. All right. Ralph has left. So if this is the action for the particles, the dust shells, then it turns out if I Hamiltonianize it, it turns out to have exactly that very nice ADM form, which really surprised me. I mean, this looks like a real mess. It's got N's and NR's under the square root, the M on the outside of the square root, but when you just go ahead and do the standard Hamiltonian transformations, it drops into exactly the nice kind of N times something which just depends on the position and the momentum of the particle, and NR times the momentum of the particle. really surprised me. Well, one can now use the equations of motion and also choose one's coordinate system. In particular, I'm going to choose my radial coordinate to just be R. So, sorry, the radial coordinate to just be this aerial coordinate. And I'm also going to choose the momentum conjugate to U to be equal to zero That implies by the equation of motion for r dot, r dot is of course equal to zero, that implies that this guy has got to be equal to zero. And the equation of motion which says that the Hamiltonian has got to be equal to zero, the constraint that that total Hamiltonian has to be zero, just becomes this equation, the radial derivative of this quantity here, has got to be equal to the source term of the matter. Similarly, the equation of motion which says that p mu dot has to be zero, if p mu is always zero, this had better be equal to zero, leads to an equation of motion for n, that vertical bar is just a slip of the pen. n times e to the mu is again proportional to something that's equal to the matter terms, and that thing we expect to go to zero at infinity

27:30 if these shells are at specific locations. So this is out beyond the last shell. That tells me that n is going to be proportional to e to the minus mu as we get out to infinity. If we substitute all of this into the action, into that original action I had there, so this is the original action up here, you see that together with this term down here, you see that my constraints said that all of the terms multiply n and all of the terms multiply nr are equal to zero. So that just tells me that the reduced action, once I've stuck in all of these coordinate conditions, is identically zero. Well, not quite. because although these guys are equal to zero, it turns out that that does not give us the right equations of motion, obviously, for the fields because of, primarily because of this term, where is it, this term right here, this u prime term. When I take the variation of that original action any of the variables, I've solved for u in terms of the matter variables, but when I take the variation of this action with respect to this variable, I have to do an integration by parts. And in doing that integration by parts, I have to get a contribution at infinity. And it's that contribution at infinity that becomes critical. So if I take the variation of the action, the reduced action, with respect to either of the matter variables, what I get is only this contribution from infinity that came from that integration by parts. Because near infinity, n is just proportional to e to the minus mu, this whole action can be now rewritten as just 4r times e to the minus u, e to the minus 2u, but that turns out to just be writable in terms of the mass that one sees at infinity.

30:00 So the reduced action just turns out to be proportional to the mass that you see at infinity. And that mass function obeys the differential equation that the radial derivative of the mass is just proportional to this term here, these matter terms. So every time you cross a matter shell, you pick up an additional contribution to this global mass that you see at infinity. Well, what all of this was an aid of was to try and derive Hamiltonian for the matter terms. And in particular, if one imagines that, for example, there's some inner shell over here with mass m, and we've got an outer shell here with mass little m, then the change in mass as we go across the shell just goes as we add to the Hamiltonian, the mass of this shell, but then we have to subtract from the Hamiltonian the mass of this shell times the mass of the inner shell times the radius at which this shell happens to be sitting. This is just a term, this comes from the energy of the shell, and this is just basically the gravitational potential energy between the inner shell and the outer shell. Now nothing that I've done here has been particularly classical. And I can quantize this Hamiltonian system simply by replacing all of those X's and P's by Hamiltonian operators. So for example, if I had two layers, I could take the inner shell to have Hamiltonian H1, the outer shell to have Hamiltonian H2, and the energy then, and I've dropped a minus sign here, that's supposed to be a minus sign there, the energy, the Hamiltonian for the whole system looks like this. For example, I can take H1 to be E divided by 2 times sigma 3. Sigma 3 is just a Pauli spin matrix. H2 is the same kind of thing, and I can now calculate in this Hamiltonian what's going to happen to my system.

32:30 And I can start off, for example, with this initial state, which is a combination of both particles, both shells being in the sigma-x eigenstate, or the sigma-1 eigenstate of these operators. Typically what one expects to happen is that they're going to rotate around. This extra interaction term is going to mess things up a little bit. This is just the interaction term down here. But the question is, does one lose coherence, and the answer is basically no. So if one looks at the expectation value, for example, if one looks at the expectation value, for example, of sigma 1 plus sigma 2, these two vectors, no, they don't completely rotate with respect to each other because of this interaction term, but, you know, so the, what it looks like is that the coherence between them disappears as cos epsilon t goes to 0, but then after a while it just builds back up again. Furthermore, if one looks at the another variable, namely the difference between these two sigma ones, then that again starts off as zero, but then builds up to be its maximum value when the other one goes to zero. So the two systems stay completely coherent. I.e., there's no evidence in this system, which is a quantum system, I've quantized the system where I've quantized the interaction, the gravitational interaction between them, there's no evidence here that one actually loses coherence. So the uncertainty in the space-time does not in and of itself lead to a loss of coherence. Now this is admittedly a very simple model, but it's a model that I think does correspond to Penrose's intuition. But this isn't the whole story. Ah. Thank you. I just need something to point at. This isn't the whole story. Because remember

35:00 what I said before was that it was the attempt to make measurements that led to the decoherence, or that led to that uncertainty relationship between x's, between g mu, little g mu, and me u. So what I have to do is to try and start to make measurements in this system. For example, if I stick a clock into this system. So my clock Hamiltonian is just going to be a simple harmonic oscillator, and the way in which I'm going to run this clock is to just use x or p as like being the hands of the clock. In order to get good accuracy, what I have one of these minimum uncertainty wave packets as far away from the origin as I can. The farther away from the origin it gets replaced, the smaller the delta t is that I'm going to be able to measure with this clock. But of course, the larger delta e is going to be corresponding to that system. So let me stick in a clock, for example, far away from, so these are my two shells, which represent H1 and H2, and here, far away from them, I stick a clock. What I find in the Hamiltonian is that there's this extra term corresponding to the energy of these guys inside here. These energies, remember, are uncertain because they're just quantum operators, but that uncertainty and that energy changes the frequency of this clock, changes the Hamiltonian of the clock, and in particular it causes the clock hands to go faster if this is a large number or go slower if this is a small number. So after a while, the clock hands, if this energy is small, the clock hands are going to be retarded, or if that goes fast, those clock hands are going to be advanced, and these two bits of the wave packet won't overlap with each other anymore, i.e. one is going to get a decoherence, an incoherence caused in this internal system, i.e. because in the

37:30 Hamiltonians for these two internal guys, simply caused by the fact that we're trying to use a clock to measure, let us say something about the time that's gone by in that internal system. So it's not the quantum gravity itself that causes the incoherence, but it's the attempt to measure it by using clocks and so forth that is causing decoherence. If I place a clock inside, I get the same kind of thing. In the previous case, what happened was that it was the total energy of that internal system that was driven incoherent. In this case, it's some other combination of the internal system that gets driven incoherent by the behavior of this clock. I.e., this is the term that multiplies the clock, and it's the various possible values of this total energy, well, it's not really an energy anymore, but this total combination of the Hamiltonians of the two internal systems that are driven incoherent by the use of the clock. Well, let me now try and get to an even more dicey example. And that is the whole black hole entropy puzzle. Let me review again. We've already heard from Ralph about this. We know that we have a black hole. We've got matter that collapses into a black hole. This blue line is supposed to represent the horizon. Up here, this green stuff is supposed to be matter, which collapses down to form this black hole. If one looks at the light rays very near this horizon, one finds this exponential squeezing. Well, squeezing, if we go into the past, it's exponential expansion, I guess, as I go into the future. And it's really this exponential expansion right near the horizon, which in a sense tears apart the wave packets inside the horizon from the wave packets outside the horizon that leads to this thermal emission with a temperature proportional to 1 over the mass of the black hole. The thing about that temperature well, that temperature then by very elementary thermodynamic arguments, since we have an energy proportional to M, we have a temperature proportional to 1 over M

40:00 we get an entropy, just using TDS equals DE going as M squared, i.e. the horizon of the black hole This means that it relies critically on the behavior of these modes, of the various fluctuations right near the horizon. Now, one of the things about this is that if it really is, if everything coming out of the black hole really is in this thermal state, that has to imply a non-unitary evolution. What do I mean by that? It means that no matter what I throw into the black hole, the same thing is going to come out i.e. this thermal state comes in no matter what gets thrown comes out no matter what gets thrown in originally so for example you know famous diagram like this you throw refrigerators or cats or whatever in to form your black hole and what comes out is just this thermal state no matter what got thrown in. While a many-to-one transformation is not a unitary transformation. A unitary transformation said that everything that comes out has to be directly, you know, if different things go in, different things have to come out. Now this non-unitarian-ness happens to be hated by most particle physicists. Most relativists I can't see what the problem is. I certainly can't. Banks and Suskin, for example, argued a number of years ago that if this process of stuff coming out of a black hole is not only unitary, but is Markovian, i.e. what gets thrown out just doesn't depend at all on what came out earlier, then you get violations of energy. But of course black hole evaporation doesn't have to be a completely Markovian process. And in fact we know that the black hole can remember how much energy was emitted because that energy is simply contained in the gravitational field surrounding the black hole. So the black hole has a way of remembering its energy it doesn't have a way of remembering its baryon number or its cap number

42:30 simply because there's nothing in the gravitational field outside the black hole. On the other hand, particle physicists want to have coherence, want to have this unitary evolution. And so the question is, how could you have it? You have all of these photons and gravitons and everything being emitted over some time scale like 10 to the 53rd or 10 to the 63rd years for a solar mass black hole. Now, I said most relativists before, and until a few months ago, that included Stephen. However, in his old age, Stephen has recanted. I would remind you that so did Galileo. And my feeling is that history will judge them about equally. however in this talk I don't want to worry about that let us accept that the radiation given off in some sense is unitary so even though I don't believe it let's take that as a given the word you did with Wall about 10 years ago you showed that a conservation of energy was independent upon unitarity And you think that's the more fundamental principle? Pardon? The conservation of energy is the only one you worry about. Well, energy, anger, momentum, the things that a black hole remembers in its external gravitational field. So it has a place to, I mean, the thing about unitarity is that the black hole has to remember what it emitted so that it can adjust its emission later on to comply with what it emitted earlier. So black holes need somewhere to maintain that memory. If one believes that the only thing one has is basically the gravitational field outside the black hole, then the only place it can store that memory is in the gravitational field outside the black hole. That gravitational field, the only thing it can remember is the energy and the angular momentum and the charge and maybe some of these non-abelian gauge theories, those sort of vague kinds of unstable hair that they could have. You know, but there are very, very few things that a black hole can actually remember about the way in which it was formed.

45:00 Those things it can remember, of course, it can use in determining what the subsequent radiation is. If it can't remember something, then it cannot use that information in order to determine the subsequent evolution. now Don Page worried about Don Page incidentally is another one of the people that has always felt that black holes are not that black hole evaporation is unitary anyway he looked at the system and tried to ask the question what happens with this decoherence. So what he thought about was an ordinary system, let's say a lump of coal that you heat red hot, and that emits its radiation. And he tried to estimate where is the coherence in that. And the coherence, he argued, was basically that he would expect this thing to keep emitting sort of what looks like completely incoherent radiation until about half of the energy in the system had been radiated. It's only, it's what really happens is that the radiation emitted in the second half of its life has to be correlated with the radiation that's emitted in the first half of its life. And it's in those correlations between those two bits of radiation that the information is contained. Now, much of the arguments that have gone on in this have always assumed that you could always hide these correlations in the phases. After all, the energy given off, the radiation given off by the black hole in early stages is supposed to be thermal. It's supposed to have a very particular form, just from the kinds of general arguments that Stephen came up with 30 years ago. Bronstein and Patti have recently argued differently that it has to be contained both in the phases and in the amplitudes. And the argument is actually very simple, that let us say that you've got an incoming state A, which emits some early radiation plus some late radiation.

47:30 And there's some other incoming state, B, which emits early radiation minus the late radiation. Then there exists another incoming state, namely A plus B. And if I send in a coherent sum of A plus B, then the light stuff just cancels out. And all we get is the early radiation. if I take A minus B, then the early stuff cancels out and all I get is the late radiation. I.e., if one is really going to have unitary evolution, one should be able to have the state where I throw in an appropriate combination of refrigerators and cats and so forth, in just such a way that one cancels out the early radiation for almost as long as one wants. So one could throw stuff into a black hole, wait for 10 to the 5th years, The black hole sits there emitting no radiation, and then suddenly it starts emitting its radiation. There should be some state that looks like that, which is of course a highly non-thermal state. However, all I'm going to use is that there at least has to be some phase information, that at least some of the information is encoded in the phases of the stuff that comes out of the black hole early and the stuff that comes out of the black hole late, 10 to the 53 years later. And the question is, can this be observed? If one is going to argue that the black hole really is, in some sense, producing some unitary stuff, then I believe that one should also be able to observe it. go ahead and collect all of the radiation given off by the black hole and recombine it in just such a way that one could actually see the existence of this phase information or the existence of these correlations between late times and early times. What's first reaction, of course, as a particle physicist? Of course, that's easy. Until you realize that in order to do that, you've got to build clocks. You have got to have some external system which allows you to collect the information in just such a way and at just such phases that you're going to be able to combine it together and produce some sort of interference

50:00 experiment which is going to demonstrate that the black hole evaporation actually is unitary, actually preserves this possibility of having interference between the And the problem is that you have to preserve these phases, you have to be able to control these phases accurately over an incredible time period like 10 to the 53rd years. Plus the masses that you have, and remember that the phase of a system goes basically as e to the i, e, or m, since c is equal to 1, e to the i, m, t. m is a large number. So therefore, in order to control the phases accurately, you've got to make sure that you can control your time to an accuracy of roughly 1 over m, where m is the mass of the black hole. However, many arguments have been made that there's no way in which you can measure time to an accuracy better than the Planck scale If this were a solar mass black hole that's something like 10 to the 38 Planck that means you've got to control time to better than one 10 to the minus 38 of Planck scale and there's simply no way of doing that There are no clocks that could allow you to do that So the suggestion that I have is that, in fact, it is impossible, even if, in some theoretical sense, one could argue that the evaporation from a black hole were, let us say, unitary, that that doesn't mean anything, and that ultimately our theory will have to be such that it demonstrates to us that it doesn't mean anything, that there's no way in which we can dynamically interfere the material coming out of a black hole in such a way as to demonstrate that the coherence is maintained. And our ultimate theory of gravity must give that same result. So even if a black hole is unitary, it doesn't matter. So even if the radiation coming out of a black hole is coherent, it is still incoherent. Because there's no way in which we can actually make measurements on that radiation to demonstrate its coherence. And if there's no way in which

52:30 we can make a measurement on a system, of an attribute of a system, then that attribute simply doesn't exist. So the coherence of this outgoing radiation doesn't exist. So I just end up with the same quote that I had before, no phenomenon is a phenomenon until phenomenon. Thank you. When you say your bias might be showing with the way you phrase it, because also no experiment can be done to show that the coherence doesn't exist. yes there's no experiment that can show that fairies don't exist if there's no possible way of doing an experiment to determine something then I don't believe it exists if even in principle it is not possible to determine or measure some attribute of a system then that attribute doesn't exist Heisenberg, et cetera, added you towards the physical world. But you can't measure that the thermalness of it exists either. All you can do is to tell that you can't tell the difference. Well, if you can't tell the difference, then that's what it is. What about a little black hole that only lasted for 10 minutes? let's say 10 to the minus 40 seconds much less, or 10 to the minus 50 seconds so it's less than the Planck scale yeah, for that kind of a black hole it's entirely possible that that could that that would just form and decay coherently there's nothing that gets in the way it's not clear what one would mean that one had a black hole of that size since the evaporation time scale is much, much, much shorter than the time scale it would take to make the black hole So it's just sort of a gravitationally induced scattering. But yes, those kinds of things. Rough arguments that I did in this indicated that the entropy, that although, that the entropy proportional to m squared certainly doesn't exist, suggests that perhaps there may be a type of entropy which is proportional to the mass. That this doesn't rule out that there is, you know,

55:00 that there is coherence of that order, much, much less, however, than the coherence, sorry, that there is still some residual coherence in the evaporation of a black hole. This doesn't completely rule that out. But that coherence is a very, very small fraction of the coherence that originally was there in forming the black hole. Roger wants to get in here. In your example of the Yeah, that's basically those extra terms in the Hamiltonian, those, those interaction terms in the Hamiltonian. Well, yes, I get it and I get a formula for that. That's at H1, H2 over X. It's just the gravitational potential of that system. If I make some clear from that question, when you were speaking about electrodes in an electron interchlorometer, Right. you said that the state of the electromagnetic field in one, on one arc and on the other, are almost orthogonal. Right. In the case of Roger's Tuchel, what about the state of the gravitational field? I don't know. I can't answer that off the top of my head. In this particular case. The mass of each shell is the same. I mean, it's one shell. No, no, no. So there are two. I mean, the problem I have here is you have two shells. But it's two shells. They're two shells. So you start off, let us say, you start off with two shells on top of each other. And each of those shells have some quantum uncertainty, okay? So the Hamiltonian of each of those shells is proportional to the quantum operator sigma-1. It's not a superposition of two different shells. That's what I misunderstood. No, it's a superposition of two shells with different energies, where you have a superposition of the energies in those two shells. So here you have one shell, and its energy is sigma 1, E times sigma 1.

57:30 Its energy operator is E times sigma 1, so its energy can be plus or minus E by 2. And you've got another shell out here, again, its energy can be plus or minus E by 2. But physically you've got two different shells. Physically, you've got two different shells, but I can arrange, I mean, I can arrange things so that, for example, one gets a superposition of the energy of this shell and the energy of this shell being correlated with each other. I'm a little confused why. Okay, so the energy, and this is the way in which I'm, for example, if I made the energy be this, then the energy could either be in this shell or in this shell out here. So that just corresponds to your superposition of these two shells. The shell is just a location. I'm a little confused. No, but this thing here is the energy at that location. Is that a mass? Is there a mass? Yes. My energy is the same as mass. A mass distribution on that shell is still opposed with another mass distribution on the other shell. I thought, yes. Yeah, so this just corresponds. So, for example, I could have, I could arrange the state to be such that there's zero mass on this shell, and there's mass, all the masses on this one, or there's zero mass on this one, and all the mass is there. Then if you take the difference mass distribution, and that's the thing. And then you work out the energy and gravitation's own energy. Yeah. But it's the same total mass, and is that taking into account? Yes, for example. its own vegetation. Actually, I think it is possible to preserve phases over 10 to 53 years by coding your photons in a quantum error correcting code. But the black hole doesn't coat the photons in a quantum error correcting code. As soon as they come out of the black hole, you can do that the right way and store them, accumulate them over time that way. I don't think so but I mean we should talk I will stop I'll stop worrying about another century or so do you finally the generalize second law and you're

1:00:00 evaporating black hole as well by pulling and Gambini they seem to do a very similar thing to you and they seem to have violated the generalize second law in their approach. Sorry, in which part? Well, if you're losing decoherence up in the future, then in effect you need to reduce entropy over large timescales. Why do you need to reduce entropy? If when the black hole evaporates again, you're not getting as much entropy before as what went in. No, no, no, there's just as much entropy afterwards as went in earlier in fact there's more it's very simple there's always more entropy afterwards and no but the the state the thermal state that comes out is more or less the state with maximal entropy right so given given some configuration of energy the thermal state will be the one which maximizes the entropy in that for that amount of energy they know it's an information terms, it's very low information though because you find it by a single temperature. It's, what do I want to say? No, it's still got the maximal, because, well, my mind has just switched off. Okay, I'll start with that. So, I want to go back to your answer to some of the questions about very small but old. Right. So you said that in this case, yeah, so you may have coherence between late and early. And so from this it follows that what you showed was not that we simply should not bother about whether it is coherent or not. scale this appears. You will not observe this for big black holes. But the same we have with any continuous effect in quantum mechanics. For example, we can recipropose fuller ends, maybe bacteria, but we will not dare recipropose planets. And so this doesn't catch really that. No, no, the question is, yeah, he's saying that in quantum mechanics, one often has this thing, you know, like Schrodinger's cat, that

1:02:30 to do an experiment with Schrodinger's cat to measure the interference there. But this is, I mean, I think what you're implying is that this does not say that there's anything in principle wrong with believing that this cat could be in the superposition. Or that we know that bacterias can be in the superposition. Or that, yeah, whatever. But the question, no, the question I'm asking here is, is in principle, not in practice, but in principle, it possible to do an experiment could see the coherence in this radiation given off by the black hole. And you said yes, with small black holes. And I suspect that it's yes for very small black holes, less than 10 to the minus 5 grams, which live much less than 10 to the minus 44 seconds. This is like quantum mechanics where we say... No, but at quantum mechanics, we believe that Trudinger's cat, in principle, that one could actually do an experiment with Trudinger's cat, which would be sensitive to the fact was in a superposition of being alive and dead. That in principle that experiment is possible. Okay, the question that I'm raising is, is it in principle possible to do an experiment which would see the coherence of the radiation given off by a black hole? Or do the limitations of our ability to measure time of controlling time and controlling phases place some ultimate limit on it. And the thing about it is that any clock and so forth that you use has gravitational effects. And those gravitational effects can mess up the ability of that clock to preserve the phases, to give you enough information about the phrases, to actually measure how much radiation is given off by that black hole or see the interference. And it may in fact tell you that you cannot ever see the interference caused by Schrodinger's cat. Okay, because Schrodinger's I assume he had big cats and that's much bigger than the Planck scale and so in order to see the phases there one has to be able to control time to much much smaller than the Planck scale so it may be possible that for any large object like this it is impossible to actually do any experiment to measure the coherence but in particular for these black holes now what I would like to do is still gotten that calculation or even quite know how to do it is to marry the stuff

1:05:00 where I was talking about using these clocks that gravitational effect in the spherically symmetric case and marry that directly to the black hole evaporation process to try and see whether I can get a remember I just sort of waved my hands and said that no clock could measure to better than the Planck scale which is sort of just using the standard everybody knows this kind of thing but trying to see whether or not that can come out of a more rigorous calculation you know in a in an example where one has gravity combined with quantum mechanics but doesn't have to worry about all of the issues of renormalize ability etc that one gets when everyone worries with one gravity When you talk about it being impossible to distinguish between two different, between the reality of the comparison for these long times or not, right, I mean, the real question is, is there a unitary theory that explains everything that you can observe? Now, the reality of the unitary is not what's important, but the reason you exclude things like fairies is because there are simpler explanations for them that seem to make more sense. But if you have two theories, one unitary and one non-unitary, and they both explain the same results, you might want to pick the simpler one, or you might want to pick the one you like better for some other reason, but it's completely symmetric from just the fact that you can't tell the difference whether it's unitary or not. The argument is not that. is that from these kinds of measurement calculations, it would suggest that there is no way in which one can determine the unitarity of the long-time evolution. Therefore, the quantum theory that one ultimately develops for quantum gravity should be of such a form that it also encodes that same impossibility in it. Just as Heisenberg's quantum encoded with it, in it, the information that it was impossible or that it was, you know, that you couldn't measure the position and the momentum at the same time to arbitrary accuracy, that got encoded into the theory, okay? Similarly, I would expect that ultimately the quantum theory of gravity would have encoded in it

1:07:30 the fact that you cannot measure the coherence in the long-term scale of the evaporation of a black hole, for example. It's not simply that, you know, oh, well, you can't measure it, and therefore that's the end of the game, but that this is a clue as to what the ultimate theory should look like. In particle physics, you use theory and experiment to generate a thing called a scattering matrix. Right. So a complete set of states of particles going in, a complete set of states of particles coming out, and there's a unitary matrix connecting the two. Right. Now, you just said that for the very tiniest black holes, they probably behave just like elementary particles. Right. respect. So, there's a question then of principle, of practice. I mean, the question of principle is, would there be any threshold saying that there's a black hole, there's a thing, if it's bigger than this, I call it a black hole, then it's no longer coherent. If it's smaller than this, I call it an elementary particle, then it is coherent. Why should there be a domain where you pass from one picture to the other? Planck scale. So that means, okay, that means any black hole bigger than the Planck scale would start to show deeper here. But could you not, in principle, also think of having a complete set of states for those figures? Of course I could imagine this in principle, okay? But what I'm arguing here is that from these measurement points of view, it seems that one could never actually determine or measure that decoherence. Therefore, what I would expect of the theory would be to tell us that it would be impossible, that that system doesn't have any input, that that, because it's unmeasurable, that it actually doesn't have any. If you say it's unmeasurable, if we look at the entropy record, take a series for a moment. Sure. Let's suggest how many states a thing can be in, which is very large, which is inconvenient, but it's finite. Of course. This isn't a fact argument that I'm making here. For all practical purposes, Bell introduced this phrase. You know, many discussions about quantum measurement theory are discussions about fact.

1:10:00 You know, would this be possible in practice? Can one actually, you know, build the thing, never mind in principle? That's in principle. So, yes, I could, could I imagine a theory which was unitary? The answer, of course, is yes. That's what I'm used to, that's what you're used to, etc. Of course we could imagine a unitary theory. That's not the question. The question is, does this kind of argument lead one to the conclusion that one's ultimate theory should be a non-unitary theory? Okay? That because it's unmeasurable in principle, I don't want to understand why it says it's unmeasurable in principle. In fact, in physics, we can measure everything in principle. Yes, precisely. So why should you think you're talking about not being measurable in principle? Well, that was the argument that in order to measure it in principle, one has to use things like clocks and so forth to keep track of the phases. one can make arguments from quantum gravity that one cannot have in principle cannot have clocks that maintain that kind of accuracy but in quantum physics we do have clocks which maintain the accuracy no we don't what do you mean you have clocks that maintain that accuracy you don't have any clocks that maintain accuracy to 10 minus how in quantum physics it works Yeah, that's the timescales of 10 to the minus 10 seconds, 10 to the minus 20 seconds. We're now down to 10 to the minus... Right, and almost all of those arguments lead to the conclusion that you cannot build a clock that's more accurate than the Planck scale. You take a billion blocks and you all deviate a lot with the average of 90 and very accurate. It creates a singularity if you try and make it more accurate. Yeah. You can't put them all in the same place either with the synchronization or not. But I suppose I'm interested in the asymptotic states, which are utterly far away. All these particles of vision are very far away from each other. It's all asymptotic states. I don't know, whereas it's not practical, but as a personal principle, I can't say any objections. Well, that's the question. That's the question I'm raising.

1:12:30 is there a matter of principle here which quantum gravity does to limit the accuracy with which one can measure it? I don't claim to have a final answer to this, only some suggestions of some directions to work in. Okay, well I think that on this note we can thank the speaker. We have a more formal talk here the Natalol and then there will be a reception the titles are all out there if you want to if you want to see them I'm also supposed to come down like a ton of bricks And our next speaker is Roger Penrose, who is going to speak on the limits to quantum accounts from general relativity. Thank you very much. I want to address the issue of quantum gravity, which, for most people, means the appropriate application of quantum field theoretic procedures to Einstein's general relativity, or possibly lead to some modification of Einstein's general relativity. But should we mean some more even-handed marriage between quantum field theory and general relativity with some give on both sides? That is to say, does quantum gravity mean standard quantum mechanics, or does this union

1:15:00 involve some change in the structure, or should the union involve some change in the structure of quantum mechanics? What's nature's idea about it as opposed to And I want to, the reason I'm bringing this up, of course, is that, in my view, there should be some change in the structure of quantum mechanics. And there are various reasons, I think, for believing this. I guess I could use all these other machines. I'll leave this here. Not quite there. I believe there are some reasons that there is nature's choice of quantum gravity should involve some non-standard quantum theory, and I want to give a list of these. I'm not going to address most of them in detail here, but I'll just mention them. Although I'm going to say something about this. Quantum theory does not provide a coherent picture, as it's a coherent ontology of the physical world, and this is connected with the measurement paradox. I'll say something about that shortly. And if a change in quantum theory is to be made, and some people, including myself, believe that this is a strong enough argument that at some level there should be a change in structure quantum mechanics, if the change is to be made, then general relativity is the most natural place It's a place where we have, in fact, different principles from other physical fields. And the very structure of space-time is involved. Well, that's not applied to other physical fields. That is to say the structure of space-time is what we're talking about. That's the field of gravity. And if quantum mechanics is to be applied to that, then it's to be applied to the structure of space-time, which is a serious thing. So, we do have to, which is a place where we expect big changes to take place. So, it's natural, if we are going to change quantum mechanics, that this should be the place where it is. Again, not a very strong argument, but I'm going to say again, I think this is a strong argument.

1:17:30 This is not such a strong argument. Then there's the forking black hole information loss paradox, which I personally think is a strong argument. I won't say a great deal about that, just mention it. Space-time singularities, we know there has been a standard argument, but one of the reasons one wants to do quantum gravity at all is that you have this problem that classical theory has these singularities, and this means that that's the place where we should actually look to quantum mechanics to see what we have to do about the classical theory. The classical theory evidently has to be changed in order that space-time singularities can and this seems to be a role for quantum gravity but there is something very strange about singularities that we see in the universe particularly the time asymmetry and its relation to the second law of thermodynamics and finally is this issue in which the title of this talk refers to is what I regard as a clash of basic principles and I put these principles here on the two sides On the left-hand side, we have the principle of general covariance and the principle of equivalence. People worry quite a bit in these many approaches to quantum gravity, particularly the loop variable approach. Worry about the issue of general covariance and try to make the quantum gravity theory, in some sense, generally covariant. The principle of equivalence, if you like, is the more basic thing to look at. And I regard these principles as important principles, with the principles of quantum mechanics. And when I say the quantum mechanics principles, I mean particularly the superposition principle. And so in this clash of principles, do we expect to see some give either on this side or on this side? And I'm really trying to argue that there should be give on the quantum mechanics side in order to fit in with the principles over here. Other people might find a mother of what's going on on this side. In any case, although the principle of superposition in quantum mechanics is a very beautiful principle, it certainly is, it's a linear principle, and certainly our experience in physics is that when you see linearity, you're very pleased, on the other hand, there's perhaps a suspicion

1:20:00 that that linearity is something which is an approximation to something which is at at a more profound level, or at a level which is more universal, one might find that it's an approximation to something non-linear. So, is the superposition principle of linear superposition principle of quantum mechanics something that will survive in this linear general relativity of quantum mechanics? And since I'm using these things, I'll put this on this slide, too. I don't know how to turn this one on. Okay. But just a remark or two about singularities. And the second, I won't say a great deal about it. I waffled on about these things endlessly in other places. But let me just mention it. This is the sort of standard pictures that one used to have of the various cosmological models. Nowadays, it doesn't have these pictures anymore, partly because there seems to be a cosmological constant, so they all look rather similar to each other now. You just go on expanding exponentially. But the other remark I want to make is that if you put irregularities in, then you have a singularity not only here, which is the Big Bang, but also all the way through where you have collapses to black holes and singularities in black holes. to deal with the Big Bang singularity should equally deal with these other singularities. Now, the fact is that what we expect to see in the universe is something extremely different in the detailed structure between the Big Bang and the others. The Big Bang seems to be extraordinarily uniform, corresponding to a very low entropy, which is the origin of the second law, and one can at least try and guess what the difference in the structure is. Here I've got a picture, this is more like a sort of conformal diagram, where the causal structure of the Big Bang is like a nice smooth space-like surface, whereas the causal structure of the singularities in black holes is some great mess, and one Think of that as the vile curvature as being either zero or something very small at the beginning, and wildly diverging

1:22:30 to infinity when you approach the singularities of black holes, or the big crunch if there had been one. But that seems out of date now, out of date in the sense that we have a cosmological constant which seems to be large enough to stop the collapse of the universe. Now, the vile curvature, which as I say seems to be pretty small at the beginning and wildly diverging at the end, describes the gravitational degrees of freedom. It's a bit like in Maxwell theory, you have two tensors which play a role. The Maxwell tensor, which describes the field degrees of freedom, and the charge current vector, which describes the source. In general relativity, unlike in electromagnetism, one has these two quantities at the same order of differentiation. In Maxwell theory, there's a slightly different order of differentiation, but that's just to do with the spin being 2 in the case of gravity and 1 in the case of electromagnetism. The two tensors that one's concerned with in general relativity are the Weill tensor, which is out of the curvature with the Ritchie part extracted from it. And that describes the gravitational degrees of freedom, as the Maxwell-Field tensor would. And the Ritchie tensor, which describes the sources in the same way that the charge current vector describes sources in electromagnetic theory. And it seems that the gravitational degrees of freedom are set to zero by, well, either it's an act of God, if you like that, where it should be about things, or it should be part of physics. If it's part of physics, it should be apparently the quantum gravity theory we're speaking, that's one of the jobs that quantum what quantum gravity is supposed to do for us, and if they're not sure that they have, it seems that there's something gross in time asymmetrical involved in this. Now, not everybody looks at it in the same way, but it does seem to me that there is at least a strong argument that there is something time asymmetrical involved in quantum gravity, or that nature's quantum gravity is. and it's hard to see how that can come about if we're just thinking in terms of classical general relativity in some sense quantized in a standard way

1:25:00 so that is an argument something should be different what the difference is of course isn't said by this but at least is an indication that there should be some difference let me make a quick remark about the Hawking black hole information loss paradox As I said last time, we call it a paradox if you come from particle physics and quantum field theory, you just say ho-hum, if you come from general relativity. Since I come from general relativity, that's more my reaction. Ho-hum perhaps isn't quite the right response, but at least it's an indication that maybe unitarity is not something which applies to all levels. there is some deviation from unitarity because things are extreme enough and things are there all the time. And if one believes that quantum mechanics has to be modified, this is to say that unitarity is going to be modified and the argument is that you're seeing it in a more extreme form of black hole galactic, that's there all the time. The problem is, at least one of the problems, is that you've got this information on parts of null infinity, which somehow is spread between the singularity And future non-infinity, in fact, this was the original Hawking argument of why we have black hole radiation in any case. That you only have partial information if you look at future non-infinity. But if you allow your black hole to evaporate and finally disappear, then the question is what happens to all the information that's producing the specific black hole that you have, which needs to get ironed out. there are many arguments that somehow it's got to come out again in this final explosion one problem with that as far as I see it is that the final explosion doesn't seem to know anything about how big the black hole was that we started with it could have been an enormous galactic scale black hole or it could have been an order of solar mass one or what have you and the amount of information that's been swallowed up by the black hole would be enormously different in those cases and if everything is happening at the last how is it that it's remembered that, of course, the big one or the little one, of course, it could have been coming out all the time, but then how does this part of the space-time know what's going on here? Because the destruction of the information doesn't occur

1:27:30 until you get near the singularity where you have flat-scale curvatures rising only in the neighborhood of here. So how is it that somehow out here that information is spewed out? So I always found that very hard to believe the information should come back out again. I don't want to dwell on that particular argument here, but if you believe that unitarity is going to go wrong at some level, well, then you're not so surprised. That's all I'm saying. So, okay, it's an interesting question, and how do we deal with it? But it's not such a puzzle because we aren't wedded to the unitarity at all levels. Okay, well now let's say something about the last point, which is up here, well not the last point, the principles, the last point up here, and I want to relate it also to the issues here of the measurement paradox, and then ultimately to this green one here. So let me say something about the superposition principle. Some of these transparencies I once said I would use in popular talks, I don't see that So that's any part of using it here. What is quantum linearity? Well, it manifests itself, for example, in this form. One might have a source of, say, a photon, which goes along and hits a green thing and produces a whole lot of junk. This can be in any kind of scale of things. Or you can have a mirror in between and it hits the brown thing and produces a whole lot of different junk. And because we have linearity, then you could have superposition of two by putting a green splitter here, and the green splitter puts the photon into a linear superposition of two derivatives. And then by quantum linearity, this must extend itself right up to all scales, and this means that you are left with the superposition of these two green-brown things. Well, that may be all right. Certainly, if those are still quantum-level activities, you wouldn't worry about it. But, of course, well, here I've just got a rather boring experiment. There's a source going along here and a detector, and it clicks if it sees the photon.

1:30:00 Too boring, so you'll pet it up a little bit. You'll attach a murderous weapon which kills the cat. Of course, in the middle of coming. You could save the cat by putting a little mirror in between instead. The cat's fine. And then of course, you end up by supposition, a dead cat. But then he will say, well, that's all very well, but you've forgotten all sorts of things like taking into account the environment. So then, as I say, I would do this in a popular talk, so let's say, you have the dead cat, Or you could have the slide map, and there's an environment on there. Sorry? It might be well on the floor. Has it gone on the floor? Do you like? It's hard to see environments, that's the whole point about it. There's its environment. So that's fine, and then of course you have the seat position. in the environments, and there they both are, and it's sort of hard to see what that's done for. Well, of course, the thing is that, I think it's because sometimes I use American between sizes and transparencies, and sometimes English ones in a little different shapes. It's a little dark, but never mind, that's just... The more of these you lay on, the more obscure things are. Now, let me just say a comment about environmental decoherence and how it tends to be used in ordinary, uh, arguments. Well, the point I'm trying to make is that the normal argument, uh, has a problem that it involves what I would call a double, a double, um, ontology

1:32:30 shift. And it's sort of, if you're trying to keep your mind on what you're actually doing, I find it very unclear. And the problem is that there's no unique interpretation of the density matrix as a probability mixture of states. And let me do it in a particular case of two-dimensional Hilbert space. So we have the Bloch sphere. So this is a northern sphere in three space. Point in the middle represents the unit density matrix. And points around the outside are what we And if you take any point in the Bloch sphere, let's say this point here, then there are lots of ways of representing that as a linear combination of, or a probability mixture, I should say, of two different states. You just draw any line through that point, and then the two pure states that you have on the outside, this thing can be right, written as a probability of those two. So there's no unique representation. Somebody might say, well, in this case, these are obvious ones here are not orthogonal. There's no reason why they should be orthogonal. It's very easy to produce examples using EPR examples where the probability mixture comes up as something where they're not orthogonal. So there's absolutely no reason why they should be orthogonal. There's also absolutely no reason why the number of them should be the same as the dimension of the Hilder space. Yes, you might have a whole lot of states which you're thinking of a probability mixture of and then, of course, there's not just a line to it here, you could represent it as a sort of weighted sum of three points on the boundary or any number you like. There are just endless different ways of reinterpreting the density matrix. So the problem I try to bring up here is what do you think is really going on? I mean, this sort of argument seems to be something like this, or easily not stated very specifically. But somehow you've got a state, which is supposed to be what the real thing is. That could be your cat, for example, and your superposition. And then you say, okay, you should bring in the environment, and since you have no control over that, you don't know, you can't get that information out, you say it's all lost in that environment. And then you say, well, let's trace over those states, and then you get a density matrix. And then somehow the density matrix is, you change your attitude as to what's real. And you think, oh, well, a density matrix is how you should be describing things in nature. And then you sort of surreptitiously kind of sneakily reproduced it,

1:35:00 re-express this in a different way, as a different probability mixture from the one you had before. Because originally it was dead or a large cat. But then you, no, so originally it wasn't that. It was some combination, and your photon was in some state originally and produced some linear combination. And then you reinterpret your density matrix. probability mixture of dead and a live cat. Well, you may be entitled to do that if you want to, together with some company environments. But then, you shift it back to a state. So you start with a state, and then you say, okay, this is real. And then you say, well, if that's real, then I'm allowed to reinterpret a different probability mixture. And then you say, let's choose one of them, because it's a probability mixture, and it might be one, it might be the other. So you see, there's at least two shifts in your ontological viewpoint So, as John Bell would say, it's a fact point of view, it's, if you like, it's telling you that the two principles that you use in quantum mechanics, unitary evolution on the one hand, and the reduction of the state on the other hand, although they're inconsistent with each other logically, they can sort of peacefully coexist. so it's a sort of peaceful coexistence principle if you really have no way of getting hold of the environment then, okay, it doesn't give you a consistent picture of the world but you're not going to run into trouble so long as you never actually can measure the environment I find this a very uncomfortable decision to take, it's alright for the moment perhaps, but you say well, maybe later technology will have ways of keeping track of environments and so on, and where are you going to push this It's a very unsatisfactory viewpoint, in my view, for an ultimate standpoint with regard to quantum mechanics. So, it'll do for a while, but can we survive with that viewpoint for a long time and what's going to happen in the future? If we want a better theory, it seems to me we've got to face up to this. Now, this is of relevance to what Bill Andrew was saying yesterday, because I think his point of view was rather different from mine, and I'll come to that shortly. He was basically, he kept saying that the, somehow, a form of deconference is supplied by the gravitational field.

1:37:30 Now, that might be a way of looking at it, but it's certainly not my point of view. my point of view is that we need to change quantum mechanics it's not that decoherence there's some form of decoherence we can't get rid of because decoherence you haven't got around this point here you still have this ontological problem which is unresolved I want just briefly to go back to caps and so on just because many I have a different view on this, which is not so much the environment. I can put the environment in there too, but it just makes the transgressions get very cloudy. So let me just remove it. You can put them in if you want to. On the other hand, I'll take the dead cat first. That was how I started it. And the other hand, he's going to look this way, they say, here's an observer comes along, looks at the dead cat, and you're led to some sort of many worlds point of view, and this really amounts to doing that, while doing this. So if you have a mirror there, then that's the other situation. You have a bean splitter, and you've got this. Of course, I put the mental state of the observer here. You might say that's how you describe that one mechanically. It doesn't matter so much, because you can look at the person's expression. And that's an expression for the live cat, and a Google expression for the other one. So, the question is, why is this superposition not committed? Now, if you're led, okay, you're somehow led into many worlds if you want to preserve unitarities or levels, but it's not resolved, doesn't resolve the problem, because you've got to say, why is a conscious being only allowed to see one or the other, rather than the superposition of two? And that needs a theory, consciousness, and so on and so forth. It seems to me to be taking a problem which is really a physics problem in a direction which is very hard to see how one's going to get a nice probability. laws and so on out of it, at least with our present level of understanding.

1:40:00 OK, well, bringing the cat in is a complicated thing to do. Schrodinger just did that because we wanted to make it dramatic. But there's no reason why we need a cat, just to make it a lot complicated. Here we have the two superposition of these two alternatives. one of them causes a lump of material to be moved from one location to the other and the other possibility leaves that lump of material in its original location now what I'm going to try and argue is that if we try to bring general relativity into this we are led to believe that this two position in fact that spontane linearity is somehow not reserved at all levels And that this superposition is unstable, reducing to one location or the other in a time scale of the order of h bar over eg, where eg is the gravitational self-energy of the difference between the two mass distributions. So you take this mass distribution and that one, subtract one from the other, and work out the gravitational self-energy of that, which is roughly the energy of displacement from one, if you imagine a lump sort of on top of each, two lumps on top of each other and you move one with respect to the other and calculate the energy that's involved in the gravitational field of separating one from the other, so there's a slight attraction between these two, and that energy is EG in the case of a rigid displacement. Of course it might not be a rigid displacement, but if it is, then the expressions come out the same. So this is the proposal, and I want to say something about why I think it's a reasonable thing to do, and why, how one can perhaps see it as true of nature. This is just a little qualitative picture, where this is the space-time diagram. You start off with the lump in one location, and then you've displaced one, or done, to this linear superposition of these two alternatives, and this is the argument that somehow this is unstable. The basic argument is that this thing, e.g., represents some fundamental uncertainty in the energy of the superposition, and that that is related by the Heisenberg time-energy uncertainty, as

1:42:30 as one would use this thing for an unstable nucleus. You say an unstable nucleus has a certain lifetime, and that lifetime is reciprocally related to a fundamental uncertainty in its energy. It's the same formula here that we all regard this as a legitimate use of time and energy uncertainty. So that's fine. Certainly it's part of standard physics. But I'm using it sort of in the reverse way here. Usually one says, you know the lifetime, energy by saying if we believe there's a fundamental uncertainty in the energy, this leads us to at least speculate that there might be a time scale in that superposition. And that's what I'm going to try and to persuade you that there, that it's reasonable that there should be such and time uncertainty. So I'll put this up here. I think I need all that for the moment. I'll give you the sort of argument that I've tended to give in the past. You see, I'm just considering a situation in which each of these individually is stationary, and I want to ask whether this superposition is stationary. ordinary quantum mechanics, that would be the case. Let's suppose that the energies of the two are the same, and you can just add them in each position like a state of energy, then so is the superposition. I would say for the same energy. And if you start taking into account general relativity, one thing you have to worry about is how do you write down the Schrodinger equation? Where you see normally you'd have a killing vector, and say that represents the d by dt in the Schrodinger equation. But then if you're considering something which is a superposition like this, how do you write down your Schrodinger equation? Because in some sense, you've got two different killing vectors, which are your d by dt's, and in what sense can you say these two killing vectors are the same? And I've drawn it in such a way that they don't look the same. Well, it's worse than that, because, now here's where the general principles of general relativity start coming in, we worry about the principle of general covariance. And the principle of general covariance will tell us if we've got two different space times, we shouldn't really be identifying

1:45:00 them at all. It's not just that the killing vectors are different, but the space times are different. And to identify the killing vectors, I would have to identify a point and one space within a point in the other space. And that's not fair according to the principle of general cobearance. There's no label on one space time that you're supposed to say is the same as the other one. Well, you might say, just give up at that point, or you go into some fancy things, you use loop variables and this and that, and it doesn't still quite seem to resolve this issue. But what I want to do is say, well, let's not be that Let's, first of all, consider the limit when the speed of light is taken infinitely. So, whereas problems without causality are removed, it's a Galilean limit, which means, in a sense, we're looking at Einstein-Cartan theory. You might think that the times are now the same. If you shouldn't... Well, the d by dt's are, of course, not the same, just because the times are the same. That's the standard, what Nick Woodhouse calls the second fundamental confusion of calculus. just because the t's are the same and of course the d by d t's are not the same because they depend on the spatial variables and so I have to worry about how you identify the spatial variables and that's the element here. So I'm going to say, well, I am going to cheat, I'm going to identify them but I'm going to take note of the fact that I'm cheating. And what I'm led to this is an argument I've given many times before a calculation rather like this know really how to identify these things, but I can say let's locally at least try and identify them. And what would you try and do? Well, you see, this is where the principle of equivalence comes in. I'm going to say a bit more about that in a minute. The argument is that I try to identify them and then estimate the error by looking at how the two falls in the two cases differ from each other and regard this as a measure of the error. Now, I'm going to give you a different argument shortly, which I think is much more powerful doing here. But if, let's say, the three falls are different, and I try to measure the error involved in making this identification, estimating that error by taking the square of that difference and integrating that over space, and then integrating by parts and so on, and that's where I end up by getting the gravitation of self-energy, the difference. So this is

1:47:30 why I'm saying this is an appropriate thing to look at. So let me bring that back here. In fact, you see, what I'm really doing is not quite what I've said here, because the problem is not so much with the killing vectors, the problem is with identifying the spaces before I find them, think about killing vectors. So, what's really the trouble? Now, the couple comes about, and I have a different picture here, and what I've drawn on this picture, not so much the Killingbeck's, but the notion of free-fall. So you see, that's what I'm saying, how the free-falls differ in the two pictures, and I'm regarding the difference between them, if you like, the gravitational force, that's what the emphasis is, the Newtonian gravitational force, and I think the difference, square it, integrate it over to space. So what I'm saying is, okay, now I just face one of these with respect to the other, and you see that these three folds, if you like the curvature that I have here, which is the representing the three folds, are different, and it's this difference which is the thing that I'm regarding as, I'm trying, shouldn't you see, really be trying to identify them? Nature would be saying, well, that's, these three folds should be identified, but I can't do that globally, so what I do is I do identify them and then try to estimate the error I've made, and the EG is calculated from that procedure. So I'm not trying to work out, if you like, some form of gravitational decoherence, because my view is that that doesn't help you anyway. That might be a way of looking at it, but it's not clear that it is. And in fact, I had a Let me look at Bill Unruh's example, because he was talking about that last time. Let me put this up here. This is the example with the different accelerations. I'll come back to this why I regard that as important in a moment, but let me look at Bill Unruh's example. here one has shells of matter time is going up the pictures always and inside you have no acceleration the field of

1:50:00 Newtonian field disappears whereas outside you have an acceleration field there's this other cylinder mass distributed further out and you're thinking of a superposition of those two I'm just putting them on top of each other, and you'll see that, indeed, in the middle or on the outside, the things agree. The only problem is that they don't agree between the two, and so there would, indeed, be a contribution, according to my calculation, to the different cases. And the thing is, it's okay, it's just what is nature doing? I mean, we get different answers, but okay, that's why it's useful to do experiments. If you want to know what the truth is, what does nature actually say? And my claim is that we would get different answers in this case. That's to say that this superposition would be unstable, and you could calculate the length of time it would take to decay by doing the integral that I just mentioned over this region here. And it's not the same as what Bill was doing. And again, he has his reasons for doing it his way. decoherence. In fact, that's good from my point of view, because I've always worried that if everything went through and the things I'm saying, suppose they were experimentally confirmed, I'll come to that in a moment, then some people would say, well, that's just another form of decoherence, this gravitation of decoherence. But if in fact they're different, that's very interesting. And so, thank you for your example. It certainly, as far as I can see, it's not an objection pointing out that there are differences. At this point of view, it's different from the one that Bill has been telling you about. Okay. Let me now try and provide a different reason from the one that I've been giving you, for thinking that one might have... You won't worry about the two positions when you have different three forms. And here, Bill took my name in vain in his talk, so I'm going to take his name in vain. Now, by thinking about what you might call the Galilean limit of the Anru effect. Now, I think this is really the way we're looking at it.

1:52:30 But let me first just think about the principle of equivalence in quantum mechanics, which is something people have worried about for a while. You could think of doing your quantum mechanics with a system just sitting on the table and taking the gravitational force as giving you a term in the Hamiltonian of the conventional type. So you've got an extra term which comes from the gravitational potential. That's fine. Or, you could think of your system in free fall, and then there isn't any gravitation for you. It's just falling freely, the gravitation will have disappeared. Do you get the same fun mechanics? Well, there's certainly experiments, this thing called the Cowl experiment, Colella, Overhase, and Werner just some years ago was done to show that in certain situations you get the same answer gravitational field can be followed one way or the other well, in fact this is also theoretically the case because there is a transformation from one system to the other I've used the capital letters for the 3-4 system and small letters for the system with reference to the fixed table there. And the capital of psi is the 3, 4, 1, and so on. I won't go through the calculations here, but it's a straightforward calculation. But what one finds is that the two, this is just for a single particle for the moment, the two are completely equivalent, provided that one psi, when you go from one psi to the other, you introduce a phase, which is this term here. Now, there are two terms in here. This term here is just a sort of energy effect. It's just the potential energy of the gravitational field. It's a straightforward thing. The other term is not so straightforward. It involves the time cubed. Now, when I say this is sort of, well, let's come to the Unruh effect in a minute. This is just a transformation. There's no general activity involved here. I'm just using Galilean physics. I'm just going from an acceneration frame to an accelerating frame. And in fact, you can go to a general system of many particles, and that's how it works.

1:55:00 The term in here is just the potential energy term again. But then you have this term with the t cubed, that's the time cubed, times the square of the gravitational acceleration. That's a little g. Now, let's imagine we're applying this now to the situation which I've been talking about. So this is, this is straight forward, you turn in, what do you call it, not you turn in, you turn in, you turn in, straight forward trading calculation. But now suppose we have a superposition of two gravitational fields. You have to imagine that you're an amoeba or something sitting here, and I've now displaced these bodies. Let's say it's a superposition of being here and being here. It's one body in a superposition of two locations. And I've now got this transformation that if I want to go from... Let me just rephrase it. The idea is to take Einstein seriously to say that in some sense, a gravitational field, in the sense of a gravitational Newtonian force is a secondary thing, and that free-fall is the natural thing. So this is a sort of philosophical standpoint. You say that you should be doing your quantum mechanics in a free-fall frame. You can transfer back to any other frame if you like, and then you have to introduce this funny old terminal. And then it looks as though you've got a force there. But the correct thing to do, according to the Einsteinian view, is to think of the 3-4 is the natural frame. Now, you can't do that here because you've got two different 3-4s. You've got this one and that one, the G-1 and the G-2. And the point that I'm trying to emphasize here, although I haven't done it yet, is that because we've got this T-cube in here, you actually have different vacuol. Now, I say this is a sort of Galilean limit of the Unruh effect. The Unruh effect, you think of, you can think of the Hawking temperature You can either, somebody sitting out at infinity or somebody using a frame which is fixed, and then you see a temperature, or you can fall freely into the hole and you don't feel the temperature. So, the Unruh, we're looking at it, is to think of it like a Rindler coordinates locally,

1:57:30 and I'm trying to do the same thing here. Now, it's not quite the same as the Unruh effect, because in the Unruh effect, if you take the Galilean limit, the temperature actually goes to zero. So, you'd think there wouldn't be a problem, if you like, because although the vacuum are different when the velocity of light is finite, you have a thermal vacuum in one case and not in the other case. Now, if we let the speed of light go to infinity, that temperature goes to zero, and so we don't have that particular problem. But the thing I'm pointing out is that you have a residual, which is this phase factor. It's not a thermal vacuum, but it's a different vacuum. So what I'm saying is that you, in fact, have two different vacuas here and here. And that in forming this superposition, you think of your amoeba, which is trying to make sense of what's going on here. And that poor amoeba thinks that it's got one vacuum because of this thing here and a slightly different vacuum because of that one. And it then says, well, when I learned about quantum field theory, I learned that you're not allowed to form superpositions between two different vacuas. So I'm cheating. So the claim here is that, yes, you're not allowed to. But you'll get away with it for a while. Now, the idea is it's a bit like what you do, say, in superconductors or something, where, in fact, you create superconductors, so you have a different vacuum. But then how do you ever get there? How do you build a superconductor in that? Well, you shouldn't be able to get from one vacuum to another just by building a superconductor. So the thing is that, OK, you have to look at it properly and say, well, no, there's some effect which should go up so far, and it's not exactly, and so on. So you have to do some estimator of the errors involved, and so on, and to what extent is it legitimate to treat this as a different value, and so on, when it's a trick, if you like. So I'm saying that the same thing is applying here. And I have to go off and do my homework, and I haven't had time to do it, I'm afraid, not for about a year or so, which is to find out what the right way people do that sort of thing is. You go and you actually try and do something that looks as though you're cheating by going from one vacuum to another, and you're not really cheating because you've seen how far up you go, how long you've got to wait, and so on and so forth. So the, this is just a guess, I don't know what I'm saying is right, but the guess is

2:00:00 that because the terms here, or this term here, which is the t cube, which is what's causing the problem, is exactly the same thing as that one has a neurons calculation here. it's the difference between these accelerations. Is it large to say that, okay, I can preserve this superposition for a while, but after a while, I'm going to have trouble? Because it's not really, I can pretend that they're not really different back to it for a while, but after a while, I'm going to run into trouble with that. And the suggestion is that I'm going to get a timescale which involves integrating this square. Thank you.