by minkwe » Wed Aug 25, 2021 12:39 pm
At this point, it is also fitting to close this rabbit trail and tie it back to what Joy was saying at the beginning to Justo concerning the additivity of expectation values.
Bell derived
 - E(a_1,b_2) + E(a_2, b_1) + E(a_2, b_2) \le 2)
In order to complete this derivation, he made some implicit assumptions about the data from a Bell-test experiment. Those assumptions can be interpreted two ways:
1 - The terms in the inequality represent measurements on separate distinct series of particles, and the four 2xN lists of outcomes can be reduced to a single 4xN table of outcomes.
2 - The terms in the inequality represent counterfactual measurements on the same set of particles.
Justo claims it is (1) that Bell used, despite Bell clearly appearing to use (2) in his original paper. Looks like wishful thinking to me but note that if (1) is true, then there is no practical difference between the two interpretations because to derive the inequality, everything needs to go through a 4xN spreadsheet (ie, four functions) anyway. But we know now that (1) is False. Therefore whether Bell liked it or not, knew it or not, he effectively used (2) to derive his inequality
Bell then proceeded to substitute values from QM into the expression to generate what he called a "violation".
 - E(a_1,b_2) + E(a_2, b_1) + E(a_2, b_2) = 2 \sqrt{2})
The problem is that
is not a meaningful quantity in QM if the terms in that expression are supposed to mean the same thing as the terms in
. This is because interpreted in the same way as implied by Bell's own derivation, the terms represent incompatible measurements that do not commute. Therefore, Bell made a mistake in using the linear combination of those expectations to calculate
.
In other words, Bell did not properly calculate the correct
for the terms he had derived in his inequality. He naively assumed, just like von Neuman, that he could just add them up and it will be fine. In his words about von Neumann:
Bell wrote:It was not the objective measurable predictions of quantum mechanics which ruled out hidden variables. It was the arbitrary assumption of a particular (and impossible) relation between the results of incompatible measurements either of which might be made on a given occasion but only one of which can in fact be made.
Bell wrote:Yet the von Neuman proof, if you actually come to grips with it, falls apart in your hands! There is nothing to it. It’s not just flawed, it’s silly. If you look at the assumptions it made, it does not hold up for a moment. It’s the work of a mathematician, and he makes assumptions that have a mathematical symmetry to them. When you translate them into terms of physical disposition, they’re nonsense. You may quote me on that: the proof of von Neuman is not merely false but foolish.
Looking at my Scenario 4, we have a 4xN spreadsheet of numbers, all real experimental data, with no counterfactuals. We also have particle pairs generated in the singlet state. Yet, for scenario 4, we have
 - E(a_1,b_2) + E(a_2, b_1) + E(a_2, b_2) = \sqrt{2})
We can contrast this with my Scenario 2, where we have four 2xN spreadsheets of numbers, all real experimental data, no counterfactuals, with singlet state particle pairs, resulting in
 - E(a_1,b_2) + E(a_2, b_1) + E(a_2, b_2) = 2\sqrt{2})
Now we see very clearly the problem. We have two expressions, both correct, representing two different scenarios, both from QM, that disagree with each other. Bell used
to compare with
. This was the mistake. Doing so is equivalent to assuming that you can just pick individual terms from a different scenario and add them up and have the correct result. Not so, as is now evident from comparing Scenarios 2 and 4.
All of this could have been avoided had Bell been taught proper mathematical notation as a student. 60 years of physics wasted for a very naive mistake, and I'm not joking. I know what you may be thinking -- "...,
Bell could not have been so stupid, he must have been doing something smarter than it looks. What about all those smart physicists who have been thinking about this for years, etc, etc ...".
My answer to this would be, don't be so sure. If you understand the issue and it makes sense then truth is not democratic. If the argument makes sense and you reject it because "other smart people" don't get it yet, then you are not a free thinker.
Oh, and BTW, for the remaining Bell believers, what type of non-locality is it that you believe in, that QM is unable to "violate" for scenario 4? There does seem to be a very careful orchestration of this so-called nonlocality requiring a lot of mathematical and notational trickery to achieve. Some non-locality that.
At this point, it is also fitting to close this rabbit trail and tie it back to what Joy was saying at the beginning to Justo concerning the additivity of expectation values.
Bell derived
[tex]{\cal S}_{Bell} = E(a_1,b_1) - E(a_1,b_2) + E(a_2, b_1) + E(a_2, b_2) \le 2[/tex]
In order to complete this derivation, he made some implicit assumptions about the data from a Bell-test experiment. Those assumptions can be interpreted two ways:
1 - The terms in the inequality represent measurements on separate distinct series of particles, and the four 2xN lists of outcomes can be reduced to a single 4xN table of outcomes.
2 - The terms in the inequality represent counterfactual measurements on the same set of particles.
Justo claims it is (1) that Bell used, despite Bell clearly appearing to use (2) in his original paper. Looks like wishful thinking to me but note that if (1) is true, then there is no practical difference between the two interpretations because to derive the inequality, everything needs to go through a 4xN spreadsheet (ie, four functions) anyway. But we know now that (1) is False. Therefore whether Bell liked it or not, knew it or not, he effectively used (2) to derive his inequality
Bell then proceeded to substitute values from QM into the expression to generate what he called a "violation".
[tex]{\cal S}_{QM} = E(a_1,b_1) - E(a_1,b_2) + E(a_2, b_1) + E(a_2, b_2) = 2 \sqrt{2}[/tex]
The problem is that [tex]{\cal S}_{QM}[/tex] is not a meaningful quantity in QM if the terms in that expression are supposed to mean the same thing as the terms in [tex]{\cal S}_{Bell}[/tex]. This is because interpreted in the same way as implied by Bell's own derivation, the terms represent incompatible measurements that do not commute. Therefore, Bell made a mistake in using the linear combination of those expectations to calculate [tex]{\cal S}_{QM}[/tex].
In other words, Bell did not properly calculate the correct [tex]{\cal S}_{QM}[/tex] for the terms he had derived in his inequality. He naively assumed, just like von Neuman, that he could just add them up and it will be fine. In his words about von Neumann:
[quote="Bell"]It was not the objective measurable predictions of quantum mechanics which ruled out hidden variables. It was the arbitrary assumption of a particular (and impossible) relation between the results of incompatible measurements either of which might be made on a given occasion but only one of which can in fact be made.
[/quote]
[quote="Bell"]Yet the von Neuman proof, if you actually come to grips with it, falls apart in your hands! There is nothing to it. It’s not just flawed, it’s silly. If you look at the assumptions it made, it does not hold up for a moment. It’s the work of a mathematician, and he makes assumptions that have a mathematical symmetry to them. When you translate them into terms of physical disposition, they’re nonsense. You may quote me on that: the proof of von Neuman is not merely false but foolish.[/quote]
Looking at my Scenario 4, we have a 4xN spreadsheet of numbers, all real experimental data, with no counterfactuals. We also have particle pairs generated in the singlet state. Yet, for scenario 4, we have
[tex]{\cal S}_{QM_4} = E(a_1,b_1) - E(a_1,b_2) + E(a_2, b_1) + E(a_2, b_2) = \sqrt{2}[/tex]
We can contrast this with my Scenario 2, where we have four 2xN spreadsheets of numbers, all real experimental data, no counterfactuals, with singlet state particle pairs, resulting in
[tex]{\cal S}_{QM_2} = E(a_1,b_1) - E(a_1,b_2) + E(a_2, b_1) + E(a_2, b_2) = 2\sqrt{2}[/tex]
Now we see very clearly the problem. We have two expressions, both correct, representing two different scenarios, both from QM, that disagree with each other. Bell used [tex]{\cal S}_{QM_2}[/tex] to compare with [tex]{\cal S}_{Bell}[/tex]. This was the mistake. Doing so is equivalent to assuming that you can just pick individual terms from a different scenario and add them up and have the correct result. Not so, as is now evident from comparing Scenarios 2 and 4.
All of this could have been avoided had Bell been taught proper mathematical notation as a student. 60 years of physics wasted for a very naive mistake, and I'm not joking. I know what you may be thinking -- "..., [i]Bell could not have been so stupid, he must have been doing something smarter than it looks. What about all those smart physicists who have been thinking about this for years, etc, etc ...[/i]".
My answer to this would be, don't be so sure. If you understand the issue and it makes sense then truth is not democratic. If the argument makes sense and you reject it because "other smart people" don't get it yet, then you are not a free thinker.
Oh, and BTW, for the remaining Bell believers, what type of non-locality is it that you believe in, that QM is unable to "violate" for scenario 4? There does seem to be a very careful orchestration of this so-called nonlocality requiring a lot of mathematical and notational trickery to achieve. Some non-locality that.
