A Is Bell's 1964 Proof Sound?

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Summary
Did Bell have make three errors proving in his famous theorem?
I have some questions about J. S. Bell’s famous theorem as presented in his1964 paper.1 These are about his theoretical assumptions and reasoning, not about experimental observations such as Aspect-type experiments. While some questions relate to the experiments, others do not because Aspect’s work2 used photons from a calcium radiative cascade, not the spin-1/2 particles discussed by Bell.

As in Bell's paper, Alice and Bob are distant observers of spin.


1. My first question is practical. Bell considers “a pair of spin one-half particles formed somehow in the singlet spin state and moving freely in opposite directions.” While Bell’s description seems innocuous, is it even possible to prepare such a state? Suppose, for example, that we allow a spin-0 boson to decay into two spin-1/2 fermions (say πo e- + e+). Since the initial state is spherically symmetric, its decay products should radiate in spherical waves, like ripples on a pond, not as particles moving in opposition directions. As singlet states are spherically symmetric, this will be true in general. (We can separate electrons from positrons electromagnetically, but is it possible to do so without transferring angular momentum?)

Of course, cloud, bubble and spark chamber observations show well-defined tracks, but such observations involve von Neumann’s process 1 (the physics of observation) and not process 2 (the physics of unobserved time development). Only process 2 physics is relevant until the quanta strike Alice’s and Bob’s detectors -- and it entails spherical symmetry.

So, Bell has one particle arriving at Alice’s detector, and another at Bob’s detector. Yet if, as in my example, the quanta have equal mass, both will arrive simultaneously at both detectors. This is not a mere conceptual difference, but has physical implications. If the electron and positron waves are co-extensive, their four-currents will cancel, and no EM field will be produced. If they are separate, each will produce an EM field and, collectively, a dipole field. This physical difference requires a different mathematical model of detection. (My example also seems to imply charge-exchange (particle-antiparticle) entanglement.)


2. My central question deals with detector independence. Bell states: “Measurements can be made, say by Stern-Gerlach magnets, on selected components of the Spins σl and σ2, … we make the hypothesis, and it seems one at least worth considering, that if the two measurements are made at places remote from one another the orientation of one magnet does not influence the result obtained with the other.”

This is crucial in questioning local realism, yet it seems to contradict accepted physics. While not proven from more fundamental considerations, the antisymmetry of multi-fermion wavefunctions under interchange of coordinates is an accepted principle of quantum physics. To be fully covariant, the interchange must include time as well as space coordinates. In Dirac's many-time formulation3, this constrains the multi-electron wave function, ψ(x1, t1, x2, t2, ...), by transtemporal equations of the form

ψ(…, xi, ti, …, xj, tj, …) = -ψ(…, xj, tj, . …, xi, ti, …).

If we take values of xi, ti in Alice’s detector at the time of her observations and of xj, tj in Bob’s detector at the time of his observations, we can see that the wavefunctions of detector electrons are not independent as Bell assumes, but highly cross-constrained.

Alice and Bob can select the orientation of their detectors, but they can’t violate the antisymmetry constraining their electrons. As those electrons interact with incident quanta to produce detection events, observing a detection result could inform us about correlative remote detection results, at least statistically. This doesn’t predict entanglement, but it seems to refute Bell’s independence assumption.


3. Finally, Bell further writes: “Since we can predict in advance the result of measuring any chosen component of σ2, by previously measuring the same component of σl, it follows that the result of any such measurement must actually be predetermined. Since the initial quantum mechanical wave function does not determine the result of an individual measurement, this predetermination implies the possibility of a more complete specification of the state.”

This conclusion ignores the role of the detector state in process 1. Let Alice’s and Bob’s Stern-Gerlach magnets be orthogonally aligned. Then, no matter what the direction of the measured spins, the magnitude of their vector sum will be 21/2/2. Since the initial angular momentum was zero, the measured valued cannot represent the state prior to measurement. Conservation requires another source of angular momentum, which can only be the detection system.

Since the detection system can contribute to the measured result, predetermination need not imply the possibility of a more complete specification of the state. It may only imply that a full description of process 1 must include the state of the detection system. This is the point made by Heisenberg in his 1927 uncertainty paper4, i.e. that measurements invariably combine detector state information with system state information.


Notes:

1. J. S. Bell, "On the Einstein Podolsky Rosen Paradox," Physics 1:3 (1964), 195-290. https://cds.cern.ch/record/111654/files/vol1p195-200_001.pdf

2. Alain Aspect, Philippe Grangier, Gérard Roger, "Experimental Tests of Realistic Local Theories via Bell's Theorem," Phys. Rev. Lett. 47 :7, (1981), 460–3. Alain Aspect, Jean Dalibard, Gérard Roger. "Experimental Test of Bell's Inequalities Using Time-Varying Analyzers," Phys. Rev. Lett. 49:25 (1982), 1804-7.

3. Paul A. M. Dirac, “Relativistic Quantum Mechanics,” Proc. R. Soc. Lond. A 136 (1932), 453-64. doi:10.1098/rspa.1932.0094

4. Werner Heisenberg, “Ueber den anschaulichen Inhalt der quantentheoretischen Kinematik and Mechanik,” Z. für Physik, 43 (1927), 172-98.
 

Nugatory

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Bell’s theorem states that if a theory conforms to certain assumptions then it must obey Bell’s inequality. That reasoning is not invalidated by any challenge to the assumptions; the truth of A is irrelevant to the truth of a statement of the form “if A then B”.

To find a flaw in Bell’s theorem, you would have to show that a theory that obeys Bell’s assumptions (so is not quantum mechanics, which explicitly violates detector independence) could violate Bell’s inequality. Failing that, Bell’s conclusion that no theory conforming to his assumptions can reproduce all the predictions of quantum mechanics (in particular, the predicted violation of the inequality) still stands.
 
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Bell’s theorem states that if a theory conforms to certain assumptions then it must obey Bell’s inequality. That reasoning is not invalidated by any challenge to the assumptions; the truth of A is irrelevant to the truth of a statement of the form “if A then B”.

To find a flaw in Bell’s theorem, you would have to show that a theory that obeys Bell’s assumptions (so is not quantum mechanics, which explicitly violates detector independence) could violate Bell’s inequality. Failing that, Bell’s conclusion that no theory conforming to his assumptions can reproduce all the predictions of quantum mechanics (in particular, the predicted violation of the inequality) still stands.
You seem not to have read my post. It discusses the soundness of the supposed proof of Bell's theorem. The fact that the theorem is of the form: A =>B, does not mean that I'm questioning A. Rather I'm questioning the assumptions Bell made, which do not depend on whether A is true or not. One of his assumptions, the one I am chiefly questioning, is that Alice's and Bob's detectors operate independently. He says of it:

The vital assumption [2] is that the result B for particle 2 does not depend on the setting a, of the magnet for particle 1, nor A on b.
So, the assumption of detector independence is critical to Bell's proof, as it is reflected in his mathematical model of the problem. His conclusion depends on the form of that model, which assumes that Alice's and Bob's detectors operate independently.

i can only have missed the point if I failed to show that the detectors are interdependent.
 
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The fact that the theorem is of the form: A =>B, does not mean that I'm questioning A. Rather I'm questioning the assumptions Bell made, which do not depend on whether A is true or not.
A means all of the assumptions that go into the form A => B. If you are questioning any of the assumptions, you are questioning A, not questioning A => B.
 
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Since the initial state is spherically symmetric, its decay products should radiate in spherical waves, like ripples on a pond, not as particles moving in opposition directions.
You're confusing the spin state with the momentum state. They're not the same. Bell only said the particles were prepared in the singlet spin state. He did not say they were prepared in a spherical wave of momentum.
 

Nugatory

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So, the assumption of detector independence is critical to Bell's proof, as it is reflected in his mathematical model of the problem. His conclusion depends on the form of that model, which assumes that Alice's and Bob's detectors operate independently.
Right, detector independence is a key assumption. Bell’s theorem is “If detector independence holds [and the other assumptions] then the inequality must hold”.

Detector independence is one of the A’s in the “if A then B” formulation, where B is “the inequality holds”.
i can only have missed the point if I failed to show that the detectors are interdependent.
There is no need to show that they are independent. All Bell is doing is considering the implications if they are independent.
 
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My central question deals with detector independence.
And questioning that means questioning the assumptions, not questioning whether the theorem is valid given that the assumptions are true. In other words, questioning A, not A => B. If the detectors are not in fact independent, then A is false; but as has already been pointed out, that has no bearing whatever on whether the hypothetical A => B, which is Bell's theorem, is true.
 
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Since the initial angular momentum was zero, the measured valued cannot represent the state prior to measurement.
Of course not; the measurement will change the state, and, as you say, this change of state is caused by the interaction of the measured system with the detector. But none of this contradicts anything Bell said.

First, note that what you quote here from Bell specifically talks about the case where you measure the same component of spin on both particles; in that case, knowing the measurement result on one particle lets you predict with certainty the result on the other. But you are talking about a different case, where you measure different, orthogonal components of spin on the two particles. Knowing one measurement result in your case does not allow you to predict with certainty the result of the measurement on the other particle.

Second, Bell never says that the measured values represent the state prior to measurement. He only says that the fact that, if you measure the same component of spin on both particles, knowing one result lets you predict with certainty the other result, means, when combined with his other assumptions, that the results for all possible measurements on both particles must be predetermined. Whether or not the predetermination involves specifying more precisely the effects of interaction with the detectors is irrelevant; Bell makes no claim about that one way or the other.
 
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Right, detector independence is a key assumption. Bell’s theorem is “If detector independence holds [and the other assumptions] then the inequality must hold.” ... All Bell is doing is considering the implications if they are independent.
This is not what Bell said he is doing in his 1964 paper. I quote his conclusion:

In a theory in which parameters are added to quantum mechanics to determine the results of individual measurements, without changing the statistical predictions, there must be a mechanism whereby the setting of one measuring device can influence the reading of another instrument, however remote. Moreover, the signal involved must propagate instantaneously, so that such a theory could not be Lorentz invariant.
I agree that there is a mechanism linking Alice's and Bob's detector states, namely the transtemporal exchange principle. (1) This need involve no long range "influence," only local adherence to the exchange principle. As we lack a fundamental derivation of the exchange principle, we don't know if it is a consequence of local symmetry considerations or if it involves "spooky action at a distance." We also know, by considering EPR experiments in in different frames of reference, that either observation could be temporally prior to the other, so neither can has a unique claim to being the cause or the effect of such "influence." (2) We can formulate the exchange principle in a covariant way, so deterministic measurements require neither violation of covariance nor hidden variables.
 
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I quote his conclusion
What you are quoting from his conclusion is not talking about how his theorem is proved, which is what @Nugatory was talking about. In your quote, Bell is talking about the implications of the fact that the predictions of QM violate his theorem, and therefore any theory that can reproduce the predictions of QM must violate one or more of the premises of his theorem. So what you quoted from Bell is irrelevant to what you quoted from @Nugatory.
 
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And questioning that means questioning the assumptions, not questioning whether the theorem is valid given that the assumptions are true. In other words, questioning A, not A => B. If the detectors are not in fact independent, then A is false; but as has already been pointed out, that has no bearing whatever on whether the hypothetical A => B, which is Bell's theorem, is true.
As announced in the thread title, I'm discussing whether Bell's reasoning is sound, not if it is valid. For reasoning to be valid, it need only conform to valid logical forms. For reasoning to be sound, it must be valid and be based on true premises. While unsound reasoning can accidentally arrive at true conclusion, knowing that a theorem is true (as you claim Bell's is) requires sound reasoning.

While "Bell's Theorem" is used to name many related things, I have explicitly confined myself to what Bell claimed to have proved in in his 1964 paper, which draws the conclusion I quoted above. I am not the only one to see this as "Bell's theorem." Consider this formulation.
No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics.
-- C.B. Parker. McGraw-Hill Encyclopaedia of Physics (2nd ed., 1994)). McGraw-Hill. p. 542.

This is not a conditional form, as you claim the theorem is.

Bell never says that the measured values represent the state prior to measurement. He only says that the fact that, if you measure the same component of spin on both particles, knowing one result lets you predict with certainty the other result, means, when combined with his other assumptions, that the results for all possible measurements on both particles must be predetermined. Whether or not the predetermination involves specifying more precisely the effects of interaction with the detectors is irrelevant; Bell makes no claim about that one way or the other.
Yes, Bell does not say that "the measured values represent the state prior to measurement." He says that they are predetermined -- and, because of his assumed detector independence, the predetermination of Bob's observation is independent of what Alice observers. But, since the detectors are linked by the exchange principle, Alice's actual observation gives her information about the state of Bob's detector that need not be true absent Alice's actual observation. Why? Because if Alice set her detector to a different orientation, the electrons in Bob's detector would be subject to a different set of exchange constraints.

The claim Bell does make is that predetermination justifies the assumption that the probability distribution for each detector is independent of the other. The exchange constraints give us every reason to expect them to be correlated.

the transtemporal exchange principle

What is this?
The exchange principle as I have given it -- using different times for each electron, as suggested by Dirac's many time formulation, as given in the paper I cited. As the exchange principle is usually formulated, it applies only to a single time slice and so is not covariant. As I have given it, it links the wave function at all times, hence "transtemporal." This feature resolves the question of direction of influence. Symmetry is not an "influence," but it can transcend a particular time.
 
We can formulate the exchange principle in a covariant way, so deterministic measurements require neither violation of covariance nor hidden variables.
What can it mean for a measurement to be deterministic, other than there is a (hidden) variable that encodes the outcome prior to the measurement event?
 
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What you are quoting from his conclusion is not talking about how his theorem is proved, which is what @Nugatory was talking about. In your quote, Bell is talking about the implications of the fact that the predictions of QM violate his theorem, and therefore any theory that can reproduce the predictions of QM must violate one or more of the premises of his theorem. So what you quoted from Bell is irrelevant to what you quoted from @Nugatory.
The question is: what did Bell claim to have proved? I quoted it. Yes, there is an inequality, but it is a means to the conclusion I quoted, not the end of the line of argumentation. As I noted, "Bell's Theorem" is not well defined and is even applied to go-nogo theorems Bell did not author. The verbal claim is what C.B. Parker calls "Bell's Theorem" in McGraw-Hill Encyclopaedia of Physics. Still, we need not argue over a term I did not even mention it in the thread title. The physics question is can we have a deterministic measurement theory without spooky action at a distance?
 
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What can it mean for a measurement to be deterministic, other than there is a (hidden) variable that encodes the outcome prior to the measurement event?
It can mean that the measured result is determined by manifest, not hidden variables, viz. jointly by the the state of the measured system and the state of the detector. Other than mathematical complexity, there is no reason not to calculate the interactions between the incident quanta and the atomic electrons of the detector, subject to the myriad of constraints imposed by the exchange principle. Why should process 1 events be exempt from accepted physics?
 
The question is: what did Bell claim to have proved? I quoted it. Yes, there is an inequality, but it is a means to the conclusion I quoted, not the end of the line of argumentation. As I noted, "Bell's Theorem" is not well defined and is even applied to go-nogo theorems Bell did not author. The verbal claim is what C.B. Parker calls "Bell's Theorem" in McGraw-Hill Encyclopaedia of Physics. Still, we need not argue over a term I did not even mention it in the thread title. The physics question is can we have a deterministic measurement theory without spooky action at a distance?
See 3.2.2 here: https://plato.stanford.edu/entries/bell-theorem/

I suppose you're right that in 1964 specifically this was not totally clear, but was worked out in discussions later, and certainly today the point still isn't always phrased properly. But the fact is you can have a local, realist, deterministic theory that violates Bell ineqs IF the hidden variables are conspiratorial (which include the retrocausal and superdeterminsitic variations).

As to your other point, which I think is that fermionic antisymmetrization suggests a violation of detector independence, I have heard this suggestion before, though don't see how it would be a strong enough constraint. But when this is invoked, it is done as a proof in principle for non-local hidden variables/spooky action at a distance.

Put another way, what you are talking about is called the violation of "parameter independence" which is equivalent to choosing non-locality to satisfy Bell's theorem. The alternative is violating "source independence" which produces the local, conspiratorial HV approaches.
 
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I'm discussing whether Bell's reasoning is sound, not if it is valid. For reasoning to be valid, it need only conform to valid logical forms. For reasoning to be sound, it must be valid and be based on true premises.
You are misunderstanding the form of Bell's reasoning, and that of mathematical theorems more generally. Every mathematical theorem is a hypothetical: it makes some assumptions and then draws some interesting logical conclusion from those assumptions. In other words, it is a valid argument, in your terminology, and that is all that matters. Mathematics, as such, does not care whether or not the premises of a theorem are "true".

In physics, mathematical theorems can be used for a number of purposes. One of them is building a theory--a way of predicting the results of experiments. To be used for this purpose, the premises of the theorem must of course be satisfied by the physical system involved in the experiment; in other words, the argument of the theorem must be sound, in your terminology.

However, Bell is not using his theorem for that purpose. He is using it, in fact, precisely in order to investigate the consequences of the fact that the predictions of QM violate his theorem. In other words, he took the trouble to work out and publish his theorem precisely because it is not sound--because, in our actual universe, the conclusion of the theorem is not satisfied. And therefore, since the theorem itself is a valid mathematical argument, at least one of its premises must be false in our actual universe. He published the theorem in order to focus attention precisely on the question: which of the premises of the theorem are in fact false?

Note that, so far, I am using "Bell's Theorem" in a way different from the way you are apparently using it. See further comments below.

This is not a conditional form, as you claim the theorem is.
Ok, so you want to use "Bell's Theorem" to refer to something different than I was using it to refer to above. Fine. Then let's drop the term "Bell's Theorem", since it is only causing confusion, and focus attention on the actual logical arguments. There are two of them, which I will call Argument A and Argument B.

Argument A: If certain premises are satisfied, the correlations between measurements on two separated systems must obey a certain inequality.

Argument B: The actual predictions of quantum mechanics violate the inequality referred to in Argument A. Therefore, no theory that satisfies the premises of Argument A can possibly reproduce all of the predictions of quantum mechanics.

Argument A is the argument I was talking about in my previous posts, and in the first part of this post. It is obviously a hypothetical (or "conditional form"), of the form "A => B" that was referred to in previous posts. And, as I said above, it is a valid argument.

However, Argument A is, as I said above, is not a sound argument. Why? Because Argument B is a sound argument (not just a valid argument). And the conclusion of Argument B is precisely that Argument A is not sound! (Technically, we would need an additional premise that the predictions of quantum mechanics match the results of actual experiments; but Bell took that for granted even when he published his paper, and experiments in the decades since have only strengthened that belief.) You are correct that Argument B is not a conditional; only Argument A is.

So you are wasting your time trying to investigate whether "Bell's Theorem" is sound. We already know that Argument A (which is what I, and others, were previously using the term "Bell's Theorem" to refer to) is not sound; and we already know that Argument B (which is what you now say you are using the term "Bell's Theorem" to refer to) is sound. So there is nothing that needs to be investigated on that score. What needs to be investigated is, which of the premises of Argument A are not true in our actual universe?
 
It can mean that the measured result is determined by manifest, not hidden variables, viz. jointly by the the state of the measured system and the state of the detector. Other than mathematical complexity, there is no reason not to calculate the interactions between the incident quanta and the atomic electrons of the detector, subject to the myriad of constraints imposed by the exchange principle. Why should process 1 events be exempt from accepted physics?
The theorem doesn't turn on whether the variables are hidden or manifest.
 

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