The collapse of the wave packet

0:00 Well, first of all, let's demonstrate this. Let's demonstrate that there is a way to do consistency, and then show that we can do without the collapse of the weight packet. So this is what I'm going to start with here, past B. And to do that, I'm going to consider the Stern-Gerlach experiment. Let's just call it Stern-Gerlach so I don't have to exercise my German. Stern-Gerlach experiment. Perhaps I should write that out in full because some people may be interested in testing that out. S.G. Stern-Geller and it goes as follows. We have, let's make our axes here, y. What we do is at some time, initial time t, we have a stream of silver ions moving along in the x direction like so. They pass between the poles of the magnet. There is a large positive Z deflection in the Z direction. If the ions are in the y-direction and the field does the opposite of t- there is a large negative set of collection with certainty if the ions are in the y-direction and the field does the opposite of t- there is a large negative set of collection with certainty if the ions are in the y-direction and the field does the opposite of t- there is a large negative set of collection with certainty

2:30 We can write this, and let's write this now formally, what this means. It means that if this is the Schrodinger propagator from time t dash to t, this is our initial state, f y positive, eigenstate of spin in the y direction, the value plus r, cross f z north. This is the eigenstate of z displacement, the value zero, and that is transformed by the interaction. All of these terms can be translated into Fy plus, conservative measurement, Fz positive, large z displacement, saying what this says here. Everybody happy with that? This transforms the initial state of t, this propagator transforms the initial state of t into the state of t dash. And the other side of the correlation is just a protruding propagator operating on Fy minus cross Fz north. It's this correlation here which allows us to call this a measure, the correlation between the spin and the large macroscopic and it's a conservative measurement in the sense that we require here because we get Fy plus, Fy plus, Fy minus, Fy minus that satisfies exactly the conditions of the conservative ideal measurement itself. What happens if we let the state, let the ions of T times be a superposition of S, Y, and I state, so a superposition of these two states here. So we let the ions of T be, let the ions of T be state Y positive, Z2 Fy minus, with zero Z displacement.

5:00 But they're now in a superposition of these two spin-y eigenstates. What happens then? Well, from these two equations here, let's call them 1 and 2. From 1 and 2, and the fact that the Schrodinger propagator is linear, it follows, therefore, this proposition S, the state at T' is capital Phi, where 2, as you would expect, C1... We can say in density operator language, what this means is that the density operator, the ions, is the projection operator onto the state of CWP, the collapse of the wave packet. CWP may be construed as saying the following, a construal of CWP, one way of interpreting CWP is as saying the following, each ion in some state, in particular it's either in Fy positive cross Fz positive or Fy negative.

7:30 That's one way of construing what the collapse of the wave is. It's a collapse down into an iron state here and here. One of those two there. But this clearly contradicts S. This clearly contradicts S if we construe S as saying the following. If we construe S here as saying iron state C dash is 5, The state of the ions at T' is phi. Notice the vague statement of the state of the ions. One way of construing that is that each ion is in the state phi. Now if we construe it that way, then you get a direct contradiction between CWP construed this way. So you get a straight contradiction. Now some people may object to this construal and they'll object to it on the grounds that I've assigned states to individual systems. If we do that, we have to construe CWP and S as if you can assign states to individual systems. Of course, a lot of physicists insist that you should assign states to ensembles of systems. And if we do that, we have to construe CWP and S differently. Do we still get a contradiction there? The answer is yes, and we can see this as follows. If we construe, assign states to ensembles of systems, then we have to rewrite... CWP in a more general form. It's then rewritten as follows, conservative measurements, conservative measurement, or Q, an ensemble, prepares them, prepares the whole ensemble in a mixture of eigenstates of Q, probability coefficients equal to the measured probabilities.

10:00 That's one general form of the collapse of the wave packet when you put it in terms of ensembles, rather than individual systems. For argument's sake, let's now accept this form of the collapse of the wave packet and see how we go. Well, what this means is, therefore, at T' we have a density operator, now ions plural, meaning the ensemble of ions, not each ion, but our ensemble of ions at T' have a density operator. There is a suitable mixture with suitable probability coefficients to match the measured probability and that will now just be mod C1 squared, projection operator onto Fy plus plus Fz plus, first element of the mixture, first component of the mixture corresponding to this state here, plus mod C2 squared, that's what we get if we use ensembles rather than individual systems. But this still contradicts S. This still contradicts S. Why? Because P-phi, now we'll of course have to use this form of S, because we're talking about density operators. Why? Because P-phi, the projection operator onto phi, is not equal to W. And indeed it's well known that you can find a constant Q such that the expectation value of Q in the state W is not equal to the expectation value of Q.

12:30 So these are definitively and experimentally differentiable, the state P5 and the state W. So, we still have a problem. So, however, we can see WP, the collapse of the wave packet. So, that's the first part of the program. She does realize, and I'll say what I'm going to, what she says about this, well, I must tell you now. What she does is she says, look, yes, I agree with you that it does contradict quantum mechanics, but unfortunately we need the collapse of the wave at the end of quantum mechanics. Therefore, we must give up realism in the context of quantum mechanics. So that's her move. So what I've found so far, Cartwright would agree with, absolutely, that she uses this to attack realism in the context of quantum mechanics. Now, I didn't want to say that until the end, because I think that's less interesting to physicists, but that's how Paltrow reacts to this. She's well aware of this derivation, but her reaction is, this is just the proof that the laws of physics in the context of quantum mechanics are inconsistent. If they're inconsistent, how do we react? We've got to take them non-realistically. The sentences of the theory are not to be interpreted as being either true or false. So, isn't it rather that she uses this argument to say, well, there will be some types of evolutions, some kind of measurements for which we need, let's say, collapse and weight packet, and so we should look for these types of measurements, we should try and look for these quantities of Q, and it's enough that the laws of quantum mechanics are wrong, but they're not universal, but they're true, maybe almost all the time. There are, I mean, the trouble depends which Nancy Carpenter you read. You read her earlier work, she does in fact take that sort of line, but if you read How the Laws of Physics Lie, this is very much an anti-realist book, and the whole crux of it is anti-realist. So there she tarts this argument up and uses it to argue against realism. That's the central claim, really, against realism. So she wants to say, look...

15:00 Take this very seriously. There is a formal contradiction between the collapse of the wave packet and quantum mechanics. It's not just that the laws are invalid at certain points. She says, look, this is a definite description of a measurement process that we have here. And there is a contradiction between what the Schrodinger propagator tells us and what the collapse of the wave packet tells us. It's not that it's incomplete somehow and collapses the way that it takes over where the Schrodinger propagator stops. The Schrodinger propagator applies right through this interaction. Of course the reason we can do that with the Stern-Gerlach experiment is very interesting because there's no observer in the measurement that I've described here. Right? We have a measurement just in virtue of a macroscopic effect here being correlated with a micro-phenomenon. No observer in here. There's no room for where condiments connect stops. And the collapse of the wave packet is usually brought in, mainly when you bring in the observer, who can't describe the object theoretically. Here, we've got a total description of the measurement of direction. Yes it does, but I mean some physicists would say this is not a bona fide measurement until you get an observer on to register this macroscopic effect. Well that's why I think it's a measurement, but what I'm saying is that von Neumann and London and Bari and a lot of people took the line, this is not a measurement until you get a registration in consciousness. Now, what I'm saying here is that I think that's a wrong concept of measurement. I'm agreeing with you absolutely. And I'm following here a group of people who made a very compelling argument to say, well, as long as you've got an interaction which correlates between a micro effect spin value and macroscopically distinct ways of registering that, you've got a measurement. And it's irrelevant whether you then go ahead and have consciousness interacting with the system. And if you take that line, then you've got a total description of this measurement process. And that's what enables us to say this is a contradiction. Not just that the laws of quantum mechanics take you so far but no further and then consciousness comes in.

17:30 You've got a total description within the laws of quantum mechanics and then you get a contradiction, a formal contradiction with the collapse of the wave. Oh yes, I think that's why they were prompted to refer to consciousness, because that's where they could locate us. That was very much the deus ex machina in the context of their theory of measurement. They were aware of this contradiction, so therefore they said, okay, this is where consciousness is needed. But the whole point is you accept this concept of measurement, then you have this known room there for the, well, you don't need the observer to perform a measurement of that, the interaction is actually finished. And this is what's nice about the Stern-Gerhardt experiment. You get a macroscopic effect totally described within the ambit of quantum mechanics, within the Schrödinger mechanics, and that's what enables you to say that you've got a contradiction here, a formal contradiction, and not just, as you were saying, that you get somehow a limitation of the laws of quantum mechanics, and then you can let in the observer. So that's why I think you've got a formal contradiction. I don't know, I don't know, I don't know, I don't know, I don't know, I don't know, I don't know, I don't know. Well, I think there's a general difficulty with the phrase realism and anti-realism. I mean, Niels Bohr, in a sense, people often call him an anti-realist, but I mean, he was eminently realist about the things, the electrons, in the same way that Nancy Cartwright is. But you can be an anti-realist In saying that the actual statements themselves are not to be taken as true or false but nevertheless the things they refer to exist. Now that's I think closer to what Nancy Cartwright says but I mean the trouble is so the label anti-realist is you know has to be qualified with about 10 different ways of being an anti-realist

20:00 Well, I think it's a little bit more subtle than that, because those things are as well as pedagogics. She's anti-realist about complexity and powers and solutions. So if you think that the laws of the sky are complexity and solutions, that's why. I mean, there's an anthropological reason for that. So it's that section where she rubbishes the powers people that I think gives a clue as to what's going on beyond this. I think the book, in a sense, what one has to do is look at her arguments for anti-realism and then find out what form of anti-realism she's arguing for. This argument, I'm clear, at least from reading her book, that's what she's trying to get at. She's trying to say, look, we accept this, that quantum mechanics is inconsistent with the collapse of the wave packet. Nevertheless, there is strong abductive support for the collapse of the wave packet. Because of this, I assume anti-realism about quantum mechanics. So, I mean, really, you have to say, look, this is her argument. Question, then. What form of atheo-realism is this media committed to? And I think whatever form that is, Nancy Cartwright's committed to it. And that's what she's over to. I think that particular short section is really important to deal with. And I think that in this case, because there's very few experiments where the underlying disposition is realized in a definite, non-stop state. So when the two sets of ions split into the two beams... And then you've got a macroscopic representation of a mathematical, and it's the only representation of that.

22:30 Yes, it's a particularly nice illustration. I think this is what makes it philosophically very interesting because you cannot get out of it by the ad hoc way that people often do about saying, aha, consciousness is needed here. So I think this is what's nice. You get a total description within the Schrodinger mechanics of the total measurement. Look, can I use my option here of just saying let's move this to the end, but continue the discussion just to the end, I think otherwise we'll get the whole... Yeah, sure, sure. I just wanted to ask you if you would, at the end, say something about the quantum right, because it's not the thing that's being talked about, not deriving from... There's a particular problem about the interpretation of what she says. In what she says, how can that be why She doesn't say that there's no such thing. In fact, she does it the other way around. She uses modus tollens. In a sense, she says, look, there is a collapse of the wave packet. We know it from the practice of physicists. They need collapse of the wave packet whenever they do scattering theory. Yet, this is my view, not Nancy's. What Nancy would do is say, she would disagree with this. But would hold with that. But she doesn't take this as implying that there is no collapse of the wave packet. What she takes that as implying is that quantum mechanics is up to scale. So she uses modus tollens rather than modus prologens. In guessing problems, perhaps the wave packet 2, the x will happen. Do you need quantum mechanics? In getting from the collapse of the wave packet to the explanandum, do you need quantum mechanics, a very minor part, a very minor part of it, and that's what's, so you can in a sense argue that you can localize off the part of quantum mechanics that you use needed in order to get from here to here. She is committed at least to that part, and I think she would take it sort of a contextual attitude. Yes, that's no problem. It's interesting that where the contradiction arises can, in a sense, be localised on. I mean, this is a big problem. Philosophers always have told me that if you have an inconsistent theory, you can derive anything from it. Of course, Nancy Carter, I think, would argue, and I think she's quite right there, that in practice you can localise off contradictions from each other and use the theory despite the fact that it's actually fairly inconsistent.

25:00 And I suppose that's what development logicians generally make as well, and I think that's a point well taken. I mean, so if you can pick a game out of the four, you know, that's the thing. Look, there must be something going on, because if, in the case of B, CWP entails the negation of QP, then we can't move trying to do this and do this and that, because there's no models of CWP and QP. So if you want to, if you have an argument that this is resolved by, you know, reinterpreting all of it, then it can't be just. There is a very next question about if you accept that this is right, if you accept that there is no, to cross out this A and put in what Nancy thinks, there is no viable alternative explanation. If you accept this, then it's not at all clear in what sense this is an argument for anti-realism, but I haven't chosen to address that here. I'm just saying in a sense, parenthetically, because somebody asked me, what does Nancy Cartwright do about this? Well, she would put in no here instead of a, and then use this as part of her platform for anti-realism. I think it's an interesting question, whether she's entitled to make the step from here and here to anti-realism, or of what sort. That's the actual bit where she picks on a local anti-realism for the case, yes, I think that's part of the problem in defining terms like realism and anti-realism, you have to say at least make them contextual to a particular area of physics, not, so to speak, talk about it writ large, so I think that's a separate talk and I think an interesting one, but not the one I'd like to get on with now. So what I'd like to do now is to give you what I think This is the pillar to the Cartwright argument and to try and show that there is a viable alternative explanation of the collapse of the Cartwright correlation without appealing to the collapse of the wave packet. So we don't need the collapse of the wave packet in quantum theory. It's useful, but given this here we ought to get rid of it, particularly given that there is really no need for it, there is an alternative explanation. So that's what I want to get onto this day.

27:30 Consider an extended Stern-Berlet experiment. So we have the astellite ions going through our magnetic poles here, M1, and that's brought immediately behind the first magnet, the second magnet, M2, a very strong one, here, and the electrons now go through the fields sequentially, and we make it such that... And this magnet here captures all the electrons, irrespective of whether they're displaced this way or that way, which come out of the first magnet. It's a very strong field here which captures all the electrons coming out, and such that, irrespective of their initial z displacements, irrespective of whether they're pushed this way or that way by the first field, Ions in Fy plus at T dash are given a large positive Z displacement by M2 and initial Z displacement ions in Fy minus at T dash are given a large negative Z displacement by M. This M2 interaction constitutes, since required by Cc, an immediate repeat conservative measurement of Sy. And hence we can apply CC to tell us the following. It will tell us that all ions, this is CC talking, CC tells us that all ions emerging from M1, which have positive Z displacement, are deflected further positively, Z positively, by M2. Okay, that's just the condition of the immediate repeat measurement. The first measurement consists of, let's suppose the first measurement consists of registering y, sy, a half, plus a half, that is to say positive z displacement, then this cc tells us that that must be repeated in the second measurement.

30:00 So, or more formally, doing it in statistical terms, what this tells us is this, the probability, which is the formal restatement in statistical terms of this, Z plus 1 is equal to 1, where Z plus 2 stands for positive Z displacement by M2, and Z plus 1, positive Z displacement by M1. So this is just, this is just, C-C tells us this. Now, C-W-P, the collapse of the wave pattern, clearly explains this, the same step as going from here to here. But how else can we explain it? That's the problem. How can we explain this C-C without involving the collapse of the wave pattern? Well, I'm going to suggest that we can do it by invoking the following two axioms. Here we go a little bit more formal than I have so far. I'm going to, in particular, assume that people can understand density operating law. For those of you who can't, perhaps you can follow the conversation anyway. The first axiom is a condition for a quantity to have a determinant value. The quantity Q for a system S is determinant in value. If and only if the density operator for S of T, sorry, the terminating value of T, if and only if W of S of T, which is the density operator for S of T, is diagonal in the eigenvectors of Q, the operator representing Q,

32:30 So the idea is that with any particular system, you have a set of Hilbert spaces. For example, for our ions here, you have a spin space and you have a configuration space. Both are Hilbert spaces. And the same quantity, the spin, They're represented by an operator in the spin space, but also, of course, in the spin space across the configuration space. There's another operator that represents the spin in that direct product space. And what I'm saying is we pick the smallest Hilbert space on which a particular physical quantity is represented. And if the density operator is diagonal in the eigenvectors of that quantity, of that operator which represents that quantity, on the smallest part of the space, that's what, those are the conditions for, say, the Q as a determinant of matter. Now that is, putting it in slightly more formal terms, that is if and only if... WST, to use a form which will be familiar to those of you who have done measurement theory, and I mean WST, is equal to sigma PI, projection operator, FI, for some PI, that sigma PI is equal to 1, PI is greater than or equal to 0, where the FI, eigenvectors, Q, should be degenerate, degenerate. So that's the axiom. There's a certain naturalness once you've got the density operator theory. There's a certain naturalness about this axiom. It's what you would expect. It's essentially saying when interference effects vanish, and this is just the condition that interference effects vanish between these different eigenstates, when interference effects vanish, then you have a determinant value and only then. With this little extra bit about, it's being on the smallest little space on which physical quantity is represented by an operator. So it's a very natural condition that's got this idea, knowing the theorem's effects, therefore the tournament value.

35:00 It's an extension of that to density operator formulas. The second axiom, a small extension of that, and a very natural one again, Exhaustive and mutually exclusive in the statistical sense, say a probability sum to 1 and the probability of the conjunction of any two of them is 0. Mutually exclusive in the statistical sense, if and only if just this condition here holds. Ws2 is equal to sigma Pi, Pfi, the sum. So these are the two axioms that I'm going to use and introduce here, natural axioms, part of the Long-Learn approach. And so on and so forth, to solve the, to explain C-C, the half-wave correlation without assuming the collapse of the wave. So how is this done? W-S-T, the density operator for the system S at time t, on, we're now considering a particular ion, a particular silver ion going through these magnets. WST' can easily be calculated using the von Neumann rule for density operators, and it's just this, it's in fact, sorry, the von Neumann rule tells us, this is on the spin space, and the density operator on the spin space for each ion at T' is in fact trace. Where's the dash going? It's on the T. It's the density operator of an ion at the time of T' when it comes out of the first magnet. On the spin space. So the spin space is in fact the trace taken over the configuration space of phi. And that's very easily calculated. So just look online for all of them. So here we have very easily this nice diagonal form.

37:30 And notice it has this nice diagonal form when you take the trace, even though phi, which I've now unfortunately rubbed out, had this long diagonal form. All of these terms can be used to define the We're also going to assume what I will call the conditional statistical passivity of measurement, which says this, for any r, any number r, for probability, The first condition of the measured value being Q. Q has a determinate value. Sorry, the value R determinately. So it just says, if there's a determinate value there, then the probability of measurement showing that determinate value is 1. That's a very natural passivity of measurement. So that's the second condition I'm assuming. What this tells us, applying this condition here to the second magnet, it tells us that the probability Z minus 2, the probability of getting a negative reflection on the second magnet, given that Sy possesses the value determinately of minus 1.5, that makes it equal to 1.

40:00 In other words, this here is matched by that response, a negative response on the second magnet. Moreover, and this is the third set of assumptions... However, we can assume that Z plus 2 and Z minus 2 are mutually exclusive and exhaustive, that's just in virtue of the fact... In fact, this second measurement here is an ideal measurement. It must give one response, it must either give a response out of the board or into the board, so therefore we can assume that these two possible responses, into the board and out of the board, are mutually exclusive and exhaustive in the statistical sense. Now, with that machinery, which might seem a lot of machinery to crack this nut, and is, but I mean can be independently justified in other contexts as well, I know I can use these answers to explain a whole range of funny phenomena in quantum mechanics, but I'm just applying them here to this one case, we can now go ahead and very quickly derive what we want to prove. So let's call, can I get even David's name, for, yes this is for, I've got a 3 there somewhere here. Now, from 4, it's very easy to show, from this one here, it's very easy to show that since Sy positive and Sy negative are mutually exclusive and exhaustive in the statistical sense, and since the same holds for the Z plus 2 and Z minus 2, it's a trivial theorem of probability, to show that from this here, from 4, we can derive the following. And I can go over that proof at the end if you want, but it's just a simple, really simple two-step proof, and it essentially works by, this is a conditional probability, you let this be the probability of Z-2 and Sy-, you must be equal to the probability of Z-2, you then expand out, you use the addition rule for probabilities using this particular pair, and then you use this particular pair, and then you end up very quickly with that result here.

42:30 So this is an absolutely trivial derivation. Let's call this 4-dash. It's exactly what you would expect given that. You get this given these two. But from 3 and 4-dash, now putting this together with this, you can see by standard probability theory, the proof is a little bit more difficult but I'm happy to go over it in the end. From 3 and 4-dash we can get the following. Let's call this 5. Given z plus 1 is equal to 1. You can see intuitively why that works. Here we've got the probability of Z plus 2 given Z plus 1 is 1. Here we've got Sy plus given Z plus 2 is 1. And we would expect that we can substitute the Z plus 2 here with the Z plus 1 given that we've got this correlation between Z plus 2 and Z plus 1. So it's what you'd expect and as I say it drops out. It's slightly more complicated than it might seem because this probability here is conditional on M2 occurring. And this probability here is conditional on M1 and M2 both occurring. But even so, the proof goes through. It's a little bit more complicated than you might think, because these are slightly different conditional probabilities. But anyway, you derive this, and now we can similarly derive, of course, the following, substituting pluses and minuses throughout the proofs. And now we're at our exponents. This is what we're going to use to explain the Cartesian correlation. By reversing this argument from 5 and 5 dash, by reversing this argument and assuming passivity, 4, we can derive 3, and 3 of course is our explanandum. This is the Cartwright correlation. So in other words, what we have here is an interesting pattern, very much like the Scriven examples in the Minnesota article, where you have an explanandum justifying the explanands, and then we use the explanands to...

45:00 In a circular fashion, apparently circular fashion, to argue back to the explanandum and to provide an explanation for it. So let me write down what's happening here. So from here we can derive our explanands. Thus, let me now put down intuitively what happens because it's a bit hard to see just from the, when you put it up like this in logical form. Thus, the prior correlation, the prior correlations 5 and 5' these two here, this is correlating spin y being positive. So this is an initial correlation at the time t-dash when you come out of the magnet, when you come out of the first magnet. So this is an initial correlation.