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I will split this post so to make it easier to reply, if needed.
I. Background
=============
The EPR Paradox (1935) left the scientific community in a divided state. Some saw QM, even then a successful theory, as the likely victor in the debate over realism. But many remained unconvinced, and felt that hidden variables existed which could restore determinism in a classical fashion. But then Bell (1964) stunned the theoretical community with his paper on EPR. After, it became clear to the majority of physicists that QM could not be reconciled with the classical form of determinism, herein called local realism (LR). Bell's Theorem demonstrates that the following 3 things cannot all be true:
i) The experimental predictions of quantum mechanics (QM) are correct in all particulars
ii) Hidden variables exist (particle attributes really exist independently of observation)
iii) Locality holds (a measurement at one place does not affect a measurement result at another)
QM predicts that certain LR scenarios, if they existed, would have negative likelihood of occurance (in defiance of common sense). Any LR theory - in which ii) and iii) above are assumed to be true - will make predictions for values of these scenarios which is significantly different than the QM predicted values. QM does not acknowledge the existence of these scenarios, often called hidden variables (HV or LHV), so it does not have a problem with this consequence of Bell's Theorem.
Why would Bell mean the end of LR? The reason is that by 1964, QM had scored a long series of victories. Its formalism explained things that were unimaginable, and in ever increasing accuracy. It made prediction after prediction in advance of experiment. When the experimental technology eventually caught up, QM would be shown to be correct yet again. It had endured no significant predictive failures. So Bell seemed like a sure bet.
Eventually, experimental technology caught up yet again, and Aspect soundly confirmed Bell in favor of QM. LR was ruled out by 9 standard deviations by 1982, and 30 by 1998. No surprise in that.
During recent years, a chorus of tenacious opponents has attempted to argue that Aspect is NOT, after all, evidence in favor of QM and against LR. They soundly reject Aspect with clains of "accidentals" and failure to achieve "fair sampling". Maybe. Let's take a look under the hood.
II. After Bell, before Aspect
=============================
Let's roll things back to before Aspect's confirming experiments. As noted above, QM was already held in such high esteem (and why not?) that few expected to see its predictions rejected in favor of LR. After all, Bell showed that the assumption of independent reality of hidden variables led to the prediction of negative probabilities for the results of certain outcomes in spin tests of photons in a correlated singlet state. This was a sure sign to most that these cases could not exist. So we already had the following:
a. QM was a star theory that could do no wrong (perhaps a slight exaggeration :)
b. QM was at odds with LR, and the two could not peacefully coexist.
c. The QM predictions matched classical optics with its cos^2 function.
d. There were no specific/consistent alternative predictions made by local realists.
I could arguably stop here, and there is already plenty of evidence for QM and against LR.
III. The Setup to be Discussed
==============================
See elsewhere for more information on Bell tests. It is enough to state that we will refer to the same setup used in the thread "Bell's Theorem and Negative Probabilities".
In the Realistic view, we could imagine that the spin answers to polarizer settings A, B and C all exist at the same time - even if we could only measure 2 at a time. Therefore, there are 8 possible outcomes probabilities, cases [1..8] below, that must total to 100% (probability=1). This is "common sense" and is a requirement of LR (but not QM). The permutations are:
[1] A+ B+ C+ (and the likelihood of this is >=0)
[2] A+ B+ C- (and the likelihood of this is >=0)
[3] A+ B- C+ (and the likelihood of this is >=0)
[4] A+ B- C- (and the likelihood of this is >=0)
[5] A- B+ C+ (and the likelihood of this is >=0)
[6] A- B+ C- (and the likelihood of this is >=0)
[7] A- B- C+ (and the likelihood of this is >=0)
[8] A- B- C- (and the likelihood of this is >=0)
With "relative" settings for A, B and C of 0, 45 and 67.5 degrees respectively, we solve for 2 special cases [SC] - essentially pointed out by Bell:
SC = [3] + {6]
= (X + Y - Z) / 2
where
X = correlations between measurements at A and C, a relative difference of 67.5 degrees
Y = non-correlations between measurements at A and B, a relative difference of 45 degrees
Z = correlations between measurements at B and C, a relative difference of 22.5 degrees
and leading to (where "QM." is prefixed to the Quantum Mechanical prediction, and "LR." is prefixed to the Local Realistic prediction):
[QM.SC] = -.1036 (prediction per QM, if the cases existed)
[LR.SC] >= 0 (prediction per LR, simply by assuming the cases existed)
There ended up being significant discrepancies between the predicted values of X, Y and Z that led to [QM.SC] and [LR.SC].
QM predicts QM.X = .1464 while LR predicts* LR.X > .2500 (big difference)
QM predicts QM.Y = .5000 while LR predicts* LR.X =.5000 (no difference)
QM predicts QM.Z = .8536 while LR predicts* LR.Z < .7500 (big difference)
* Though not specifically required, there are the assumptions used here that LR.X + LR.Z = 1 and QM.Y = LR.Y but strictly any values of LR.X, LR.Y and LR.Z are acceptable IF LR.SC = LR.X + LR.Y - LR.Z >= 0. Let's assume these values, coming the closest to the QM predictions and therefore having the smallest discrepancies, are the predictions of LR.
I note also that Caroline Thompson's LR model, the "Chaotic Ball" makes reference to a roughly linear function which matches the above values closely (see her page 3, below formula 1 and her figure 14).
IV. Tests of X, Y and Z
=======================
So all we need to do is measure X, Y and Z and we will solve the puzzle. Or do we? The QM predictions match a known optics formula: cos^2(angle). LR matches no otherwise known spin statistics. In other words, QM is working from a mathematical formalism while LR has no specific values to offer us. QM is an otherwise successful theory; LR has long been abandoned as a working tool. Why bother with any tests at all?
Scientists, being a thorough lot, tested away - of course. Through the years, Aspect and others performed tests of X, Y and Z in various forms and combinations. Always the results were substantial and ever-increasing deviations from the LR expecation values that had the smallest variance with QM.
I could arguably stop here, and there is already plenty of evidence for QM and against LR.
I. Background
=============
The EPR Paradox (1935) left the scientific community in a divided state. Some saw QM, even then a successful theory, as the likely victor in the debate over realism. But many remained unconvinced, and felt that hidden variables existed which could restore determinism in a classical fashion. But then Bell (1964) stunned the theoretical community with his paper on EPR. After, it became clear to the majority of physicists that QM could not be reconciled with the classical form of determinism, herein called local realism (LR). Bell's Theorem demonstrates that the following 3 things cannot all be true:
i) The experimental predictions of quantum mechanics (QM) are correct in all particulars
ii) Hidden variables exist (particle attributes really exist independently of observation)
iii) Locality holds (a measurement at one place does not affect a measurement result at another)
QM predicts that certain LR scenarios, if they existed, would have negative likelihood of occurance (in defiance of common sense). Any LR theory - in which ii) and iii) above are assumed to be true - will make predictions for values of these scenarios which is significantly different than the QM predicted values. QM does not acknowledge the existence of these scenarios, often called hidden variables (HV or LHV), so it does not have a problem with this consequence of Bell's Theorem.
Why would Bell mean the end of LR? The reason is that by 1964, QM had scored a long series of victories. Its formalism explained things that were unimaginable, and in ever increasing accuracy. It made prediction after prediction in advance of experiment. When the experimental technology eventually caught up, QM would be shown to be correct yet again. It had endured no significant predictive failures. So Bell seemed like a sure bet.
Eventually, experimental technology caught up yet again, and Aspect soundly confirmed Bell in favor of QM. LR was ruled out by 9 standard deviations by 1982, and 30 by 1998. No surprise in that.
During recent years, a chorus of tenacious opponents has attempted to argue that Aspect is NOT, after all, evidence in favor of QM and against LR. They soundly reject Aspect with clains of "accidentals" and failure to achieve "fair sampling". Maybe. Let's take a look under the hood.
II. After Bell, before Aspect
=============================
Let's roll things back to before Aspect's confirming experiments. As noted above, QM was already held in such high esteem (and why not?) that few expected to see its predictions rejected in favor of LR. After all, Bell showed that the assumption of independent reality of hidden variables led to the prediction of negative probabilities for the results of certain outcomes in spin tests of photons in a correlated singlet state. This was a sure sign to most that these cases could not exist. So we already had the following:
a. QM was a star theory that could do no wrong (perhaps a slight exaggeration :)
b. QM was at odds with LR, and the two could not peacefully coexist.
c. The QM predictions matched classical optics with its cos^2 function.
d. There were no specific/consistent alternative predictions made by local realists.
I could arguably stop here, and there is already plenty of evidence for QM and against LR.
III. The Setup to be Discussed
==============================
See elsewhere for more information on Bell tests. It is enough to state that we will refer to the same setup used in the thread "Bell's Theorem and Negative Probabilities".
In the Realistic view, we could imagine that the spin answers to polarizer settings A, B and C all exist at the same time - even if we could only measure 2 at a time. Therefore, there are 8 possible outcomes probabilities, cases [1..8] below, that must total to 100% (probability=1). This is "common sense" and is a requirement of LR (but not QM). The permutations are:
[1] A+ B+ C+ (and the likelihood of this is >=0)
[2] A+ B+ C- (and the likelihood of this is >=0)
[3] A+ B- C+ (and the likelihood of this is >=0)
[4] A+ B- C- (and the likelihood of this is >=0)
[5] A- B+ C+ (and the likelihood of this is >=0)
[6] A- B+ C- (and the likelihood of this is >=0)
[7] A- B- C+ (and the likelihood of this is >=0)
[8] A- B- C- (and the likelihood of this is >=0)
With "relative" settings for A, B and C of 0, 45 and 67.5 degrees respectively, we solve for 2 special cases [SC] - essentially pointed out by Bell:
SC = [3] + {6]
= (X + Y - Z) / 2
where
X = correlations between measurements at A and C, a relative difference of 67.5 degrees
Y = non-correlations between measurements at A and B, a relative difference of 45 degrees
Z = correlations between measurements at B and C, a relative difference of 22.5 degrees
and leading to (where "QM." is prefixed to the Quantum Mechanical prediction, and "LR." is prefixed to the Local Realistic prediction):
[QM.SC] = -.1036 (prediction per QM, if the cases existed)
[LR.SC] >= 0 (prediction per LR, simply by assuming the cases existed)
There ended up being significant discrepancies between the predicted values of X, Y and Z that led to [QM.SC] and [LR.SC].
QM predicts QM.X = .1464 while LR predicts* LR.X > .2500 (big difference)
QM predicts QM.Y = .5000 while LR predicts* LR.X =.5000 (no difference)
QM predicts QM.Z = .8536 while LR predicts* LR.Z < .7500 (big difference)
* Though not specifically required, there are the assumptions used here that LR.X + LR.Z = 1 and QM.Y = LR.Y but strictly any values of LR.X, LR.Y and LR.Z are acceptable IF LR.SC = LR.X + LR.Y - LR.Z >= 0. Let's assume these values, coming the closest to the QM predictions and therefore having the smallest discrepancies, are the predictions of LR.
I note also that Caroline Thompson's LR model, the "Chaotic Ball" makes reference to a roughly linear function which matches the above values closely (see her page 3, below formula 1 and her figure 14).
IV. Tests of X, Y and Z
=======================
So all we need to do is measure X, Y and Z and we will solve the puzzle. Or do we? The QM predictions match a known optics formula: cos^2(angle). LR matches no otherwise known spin statistics. In other words, QM is working from a mathematical formalism while LR has no specific values to offer us. QM is an otherwise successful theory; LR has long been abandoned as a working tool. Why bother with any tests at all?
Scientists, being a thorough lot, tested away - of course. Through the years, Aspect and others performed tests of X, Y and Z in various forms and combinations. Always the results were substantial and ever-increasing deviations from the LR expecation values that had the smallest variance with QM.
I could arguably stop here, and there is already plenty of evidence for QM and against LR.