General Covariance in Quantum GR

0:00 This is a famous 1940 paper on the CPT theorem, on the spin statistics, I don't know. All right, this is the entirety of what he said as a justification for the principle of locality. This is the entirety. The justification for a positive lies in the fact that measurements at two space points, meaning space nine points, with a space-like distance can never disturb each other. Since no signals can be transmitted with velocities greater than that of light. Now, as a methodic physicist, I'm not convinced that this is a justification for the principle of locality. I mean, in order to justify this, one would have to actually seriously examine what it means to send signals in quantum field theory, and then to actually prove that whatever you come up with your formulation of what it means to send these signals, that Einstein's causality would entail that in fact those operators must commute. Right? Well, there is no such investigation in literature of the relationship between signals in quantum field theory, about which we have no correlation whatsoever, and the commensurability of space-like separated observables. There is none whatsoever. Now, 40 years of alternating quantum field theory has shown that the principle of locality is a very powerful one to be. So, whatever extent is the level of justification, if you've done it, you can do a lot with it. It's very, very useful. And concrete models have been constructed. There are many, many, many models of physical interest in which the principle of locality poses about. I'd like to also point out there are also models of physical interest in which it does not hold. Okay, so this is an open, totally open question. What is really the connection between locality

2:30 and Einstein causality? I open it up to anybody's future investigation. It's completely open. I have nothing to say about that. So let me pose a weaker question. Okay, what really is the connection between the commensurability condition, this locality condition? I'm reading not specifically commensurability, that observables, localized space-like separated regions must commute at the commensurability center. Because Harvey suggests that because you can't send these signals, these measurements cannot disturb each other. In some sense, these measurements must be independent and therefore must commute. So it's this last thing, independent therefore is much smaller than all the lines, because Pauly's chain of argument goes, signals imply independence, implies commutativity. So I'm just looking at this last question, this last step. What is the relation between commutability and independence? Well, Klaus has already used the word independence. And as I mentioned at the beginning, he told you the strongest possible formulation of the independence of two subsystems. So to make a clear to you that this question is not empty, because if one hands Klaus's independence, then it certainly follows the other's community, a tensor problem. But to make a clear to you that the matter is not a curiosity, let me consider this other notion of independence. So, let's say we've got two algebras and they're representing the algebras as observables associated with two different cell systems. So one says that, at least I do, that these two algebras are C star independent if for any state of A and In any state of B, you can find a state of the entire algebra whose restriction to A is the original prescribed state and its restriction to B is the original prescribed state.

5:00 Now that may look a little bit like a model, but what it means operationally is, it says that no matter how you prepare this one subsystem, you can prepare the other subsystem in any other state you like. You can prepare this subsystem and this subsystem completely independently of how the other subsystem was prepared. There is a very strong independence property, ladies and gentlemen. Well, unfortunately, there are C star-independent periods of cultures which do not compute, which are not commensurable, and there are commuting, i.e. commensurable, algebras which do not satisfy this independence property, which has complete operational meaning and significant operational meaning. So, the connection between commensurability and independence This is not clear. So let me briefly mention, when did we start? More than 10 minutes. Well, that's not enough. Just to tell you the theorem, I don't be able to tell you the operational relation board. So let me tell this theorem to you, which was recently I'm going to assume I get the Tuchel to myself. We define a state of the full algebra to be AB uncorrelated if for any two projections, E and A and F and B, when you form a beat, this is the maximal projection contained in both B and F, then the expectation in the state by, the special state by, of this week is just the product of the expectations of the factors. So I don't have time to explain to you what the operational meaning, what the operational motivation for this year is, because if you exist, maybe some will be time enough for that to be in the discussion section. But what we dreamed was that if there exists such an A, B, unquadded state, it's not to have one, then in fact the algebras commute, they are commensurable.

7:30 And I'll just mention on the side, they're not only commensurable, but they satisfy Klaus's very strong independence condition. Okay, but still, even with this, there's still a question of, under what general physical principles does it follow that there exists an AD uncorriated state? That is no good question. Now, the third aspect of causality, I'm going to discuss, is that of common causes. Alright, so let's say we've got two regions, O1, O2, and consider the union, the backward-backward light cones of all the points in O1 and O2, but then I take out the sets O1, O2, and I'm going to just consider the green part here, yes? Let me call that O1, O2-tilde. Alright, so let's say I've got two projections, localized in each of these regions, and a state for which the expectation of this meat is strictly greater than this. Let me mention, for this third part, I'm going to assume a gravity hole, so the algebras here do commute, and therefore the meat is nothing other than the product AB. All right? In fact, the meet A of which B is A times B if and only if A and B commute. That is a theorem. Okay, so what this is saying in this context is in fact that phi of A times B is greater than phi of A times phi of B. There is a correlation, a positive correlation in the state phi of A and B. So, following Reichenbach's suggestion, Miklos and I made an extension of Reichenbach's considerations from classical correlative theory to this setting by declaring that a projection is a common cause of this positive correlation if the following four conditions are satisfied. And if you recall that these conditional expectations are obtained this way, at least as long as the numerator of the denominator is not zero,

10:00 and we are sure, and we see to it that the denominator is not zero in this theorem, then you will recognize these four conditions as precisely Reichenbach's four conditions. All right, so there's an additional condition which is required because we have non-commuting algebras, and that is we require in addition that this common cause commutes with both A and B. All right, so A, B, and C all commute with each other. So that's a common cause. So, Mitlos and I proved the following theory, under general conditions which include the first two expressions of causality, local period causality and locality. Under general conditions, including those conditions, for any space-like separated regions O1 and O2, and any locally faithful fine, I'll come back to it in a moment, and for any A, B, and Phi satisfying those two conditions. There does indeed exist a common cause, and where is it localized? It's localized here in this region. or in the past of both, all right? In the past of both. Now, I'm sure everybody knows that Reichenbach didn't propose his conditions, his study of common causes and the study of causality, he studied as a cause study in the direction of time. So it's a bit of an abuse to take his considerations and think of them in terms of causality. But, in fact, the common cause that we prove and obtains in quantum field theory satisfies this condition. C is a projection which is strictly less than both A and B. That has its consequence that in any state whatsoever, not just in the original state psi, in any state omega, the expectation of the common cause is bounded by both these expectations of A and B. In other words, if you have a preparation in which C responds,

12:30 A and B must respond, all right? And in addition, C, the support of C, the localization of C, is not simply anywhere in this huge green region. You can find one, for example, in this place, and you can find one in this place, et cetera, okay? So in particular, are strictly in the path of both, all right? Localized in a bounded region, and therefore, there are many locally made states. Under the general assumptions that we have there, one knows that every state of bounded energy is faithful on these local algebras. And since no preparation that man can perform will involve infinite energy, yes, that means that all the states of physical interest in this given representation satisfy the conditions that it be locally faithful. And therefore, the expectation of 5c is not zero, all right, and those conditional expectations can be written. Alright, so in fact, although writing box original petitions are only absolutely minimal petitions for real cause, we've shown that there is in fact a very strong cause for every colonization in quantum field theory. And I guess I'm done. Thank you. On this last transparency, why this service was, I would say that the whole problem, something should be in the intersection of this life. Well, okay, so I know you would say that, but my assertion is that in light of the considerations of classical theory I just made, your rejection is not thought of it.

15:00 In the classical theory? That's right, it's not thought of the classical field theory, because the data in the intersection are not sufficient to determine the values of the classical field solution in these two regions. All right? It's not determining, but to screen off, you see. Right. So it is a separate question, where there is a cause, but not in our strong sense. Right? But there's some cause which is more like writing talks original sense, that you have just this weak, restrictions on the cause that you're extremely, and that you have the of the one is at least increased by the presence of the other, all right? But that is a separate question. It does not obviate the significance of the existence of these strong classes, and you will not find a strong class in the assumption. Now I can tell you, you can't even find it for a classical field theory, and therefore you can't find it for a free field theory, and therefore you can't find it for a general field theory. Intuitively, because there could be stuff coming in along the way. Of course, there's more to stop the sufficiency. Absolutely, absolutely. But those things cannot contribute to the correlation. But they do contribute to the correlation. The question is how much do they contribute? The question is, how much do they contribute? Anyhow, it's very difficult to take this solution as a generalization of the . Well, as I have indicated, I think that Breichenbach's conditions are a minimum set of conditions. What we showed here was this very strong contribution is to be found, as I indicated, and I've also indicated that it find such a strong cause in the intersection. All right? Now, as I said, it still needs to open the question of whether that weak, right-and-box weak idea of common

17:30 cause can be found in the intersection. That is separate to the independent question. Oh, sorry. So I was intrigued by your worry about signaling cases that have been analyzed. The notion of signals when analyzed in the QF2 cases I was wondering what you might see was wrong with the falling sort of, perhaps a little bit hand-waving of running. So you might say it's necessarily a sufficient condition to have the possibility of signaling for there to be a conditional probability for the outcomes of measurements over here depending on something you do over here, and that there's an ability to change what one does over here. So in that case, if you have that sort of conditional probability, we can understand that channel would be a non-zero-meter information um yeah it's one way to this would be one way to interpret and so would you worry me that we don't have a sufficiently clear notion of what freedom to do something over here it wants to well i don't know because the conditional for quality stuff i don't know what to say though is i mean what you're talking about these conditional expectations right this would be a very abstract way of trying to get a large set of things which would maybe include also signals, right? But signals, I think, are rather more specific than that. And then how specific? I don't know. The matter is uninvestigated. But if you have that scenario, that would be a communication channel in Shannon's sense. If you agree that what Shannon's talking about is the general, most general setting, what a signal wants, then at least you have a signal in Shannon's sense. very general way of including things which might also include signals. Well, this is related, in a way, to a two-part question. I suppose that if one was determined to investigate it with a sort of concrete meaning of signal, you would think about excitations of the field. That's right. Roughly speaking, you point a laser from Alice to Bob and you try to think of an AQ&A description of an excitation. But my historical question would be, when they first talked about commutation and anti-commutation in the late 20s,

20:00 didn't they... That was not an altruistic. Right. But didn't they then... They presumably talked about how the statistics at the distance would not be ordered by a non-selective measurement locally performed. I mean, that's the normal way of talking. Maybe you have the notion of localizability then. So what would locally performed operations mean? It wouldn't mean what we mean in our field theory. Not in this sort of Exactly. The principle of locality is not simply commensurability, it is space-like separated, observables are commensurables. Alright? That is the principle of locality. And its justification the hand-waving Einstein causality. It's like a mantra. it comes with, but it's not investigated. If you model, you're setting on the signal just by applying the unitary of the local algebra, then I think this will give the morality, so what is that? It gives it immediately, because the unitaries generate the algebra. Yeah. Right. What is wrong with this argument? Well, because the argument, it just moves the question further. This is what he's talking about. So, as I ridiculed yesterday, it is my feeling that no physical insight or philosophical of insight is obtained by appealing to a mathematical triviality. It is mathematically trivial that if this holds, that the algorithms commute. That's a triviality. Alright? So, it simply pushes the question to what is the justification for this? Alright? So, fine.

22:30 Right. Let's work on that question then. But it just, okay, the question is just pushed somewhere else. The essential question is still not meant. In my opinion. You said that you hoped somebody would ask you a question about the operational motivation for the condition on the meat of the algebras. What is the operational motivation for the condition on the meat of the algebras? Okay, so we came to this condition by considering what it means to talk about coincidence experiments of non-commensurable observables, which is not entirely trivial. And so let's consider the simplest case. two projections, yes-no experiments, and the coincidence experiments for such. So in a coincident experiment of these guys, there are these four coincident results, and we asked ourselves the question, what would be the correct observable, The quick mathematical representative of a yes-no experiment in a coincidence experiment like this, whereby yes means plus plus counts. So it would be a yes-no observable for such a coincidence experiment. So what are the constraints? Well, we wanted a yes-no, so the operator is going to be the quantum projection. The other constraint was we have a plus-plus, so that means when I thereafter test for going on e, then I better get yes again. And if I test for f later, I better get yes again, right? So it must be the case that whenever the apparatus yields yes, in this yes-no equivalence experiment, in any sort of measurement of E and F plus also yield yes, and that means that this orthogonal projection is bounded by both E and F. So it's

25:00 an orthogonal projection which is dominated by both. And then there gives an additional practical operational consideration, and that is when you design a coincidence experiment, that is a good experiment, designs it so that the acceptance rate of his device is as large as possible. He doesn't want to sit there for hours and hours and hours and get no coincidence because his stupid apparatus lets everything through without registering. He tries to maximum. So that means we want, in line of these two conditions, we want a orthogonal projection followed by both DLF. So if we want to maximize the acceptance rate, that means that we want the largest projection followed by both. And that means we're slightly meat. Okay, so now we know what metaphorical object, what observable, should be used to model that yes-no-wizard experiment, all right? Okay, now as I mentioned, the meat of two projections is just their product if and only if they commute, all right? Okay, so remember, what we're trying to do, we're trying to find sufficient conditions which imply that they commute, right? So commutation is going to be a necessary condition for the truth of whatever condition would ever be coming up, right? So if we're going after it's something that is going to commute, and that means the meat is going to end up to the actual product. And that means that in the situation that's of interest when you're trying to divide locality, that you should be thinking in the back of your mind that that's what you're talking about. And for a number of reasons I will go into, this is an actual independence condition.

27:30 Okay, so, once we found what we thought was the right observable, we said, okay, let's just pause this condition. If there exists a state which satisfies this condition, all right? Well, this is something which can be measured, all right? This can be measured. You can perform the promises experiment. You can perform these experiments. You can compare the numbers. All right, so, and we prove that this, the existence of one such normal state satisfying this condition, where all projections E and A, or projections F and B, implies that the algorithms commute, and indeed it implies Claus's strongest form of independence. This spatial tensor product structure. Thank you.