Justo wrote:Hi everyone.
This is the first topic I initiate in this forum. Similar issues were already discussed in other threads, however, it is sufficiently interesting and I believe that it deserves a separate thread of its own.
In 2001 Guillaume Adenier published a refutation of Bell's theorem that is extremely interesting and in my opinion relevant to the field
https://doi.org/10.1142/9789812810809_0002Adenier's paper was largely ignored by the physical community as it usually happens with papers rejecting the Bell theorem. The reason I believe Adenier's paper is relevant is that it is based on a widespread mistake that many "Bell-believers" think is correct.
Ironically the people noticing it is a mistake assume that it is the only way of interpreting Bell's derivation and conclude that the Bell inequality is immense nonsense.
I explain how the Bell inequality can and should be interpreted, in Adenier's parlance, as a weakly objective result. The explanation can be found in section 4 of my paper
https://arxiv.org/abs/2012.10238v5I already discussed in this forum the arguments upholding the strong objective interpretation, however, I am open to here where my weakly objective interpretation fails.
There are a few reasons why your paper is mistaken,
1) Going from Equation (20) to equation (21). You make an assumption that the domain of each of the functions A(ai, λj ), B(bk, λj ) is the same as the domain of their product. The effect of this assumption is that it gives you
later when you go to equation (29). This assumption is not always true and you have no valid reason to assume that it is true for EPRB. Physically, in EPRB an angle is being changed and an outcome
is observed. But actually
is an arbitrary simplification of the real physical situation. At some angle, you expect the outcome to jump instantaneously from -1 to +1, without any transition! This doesn't normally happen in physics. If the functions are undefined for some angles where a transition happens the domains of A(ai, λj ) and B(bk, λj ) will not be the same as the domain of the product A(ai, λj )*B(bk, λj ) and
because some lambdas are unmeasurable at some angles.
2) On the issue of Freedom, you say in section 2.3 that
Justo wrote:The freedom of the experimenters to choose their settings is a fundamental hypothesis for obtaining the Bell inequality
It is not. There are only 4 expectation values that are measured in the Bell-test experiment. When Alice and Bob measure
. The settings are constant, and their average is a constant for which QM has a specific prediction of
. For a weakly objective result, each of those expectation values is obtained from a different set of particle pairs in the singlet state. Therefore there is nothing about one set that influences the other set in any way. The only requirement is that each set of pairs is in the singlet state. Therefore the freedom of Alice and Bob is irrelevant for the expectation value
and
is completely superfluous. Perhaps what you are calling freedom is not really freedom because it has nothing to do with the freedom of Alice and Bob.
Alice and Bob's freedom to do what? Certainly not in picking settings because the settings are constant as already explained.
So what do you mean by freedom? It is possible that what you mean is that Alice and Bob have the freedom to pick the same hidden variables when measuring
as when measuring
. But this is unreasonable. They have no such freedom and should not be expected to have such freedom because the hidden variables are random and hidden. Instead, you are invoking the fair sampling assumption and using the misleading term "freedom" to describe it. It has nothing to do with freedom. A fair sample of a hidden distribution is an oxymoron.
3) In your paper equation (29) is the following.
You state after the equation:
"None of the terms present in (29) are assumed to have originated from incompatible experiments, neither materialized out of counterfactual reasoning
nor pre-existed before actually measured". But this is very misleading because in writing that expression, you've already made a whole bunch of hidden assumptions. Let us examine that a bit.
You claim the expression is weakly objective, therefore you arrived at it starting from measurements on disjoint particle pairs. These measurements can be conceptualized as being represented on 2xN spreadsheets, where one column represents the value of
, and the other column represents the corresponding outcome
. For a weakly objective result on disjoint particle pairs, we have 8 such spreadsheets:
Where
is the spreadsheet of data obtained by Bob for setting
, etc. I use spreadsheets because these are ordered sets and we are multiplying corresponding rows across the spreadsheets. This is the proper starting point for the weakly objective analysis.
One hidden assumption involved in equation (29) is the assumption of fair sampling. That is, the assumption that the probability distribution of
is the same for
. This is not necessarily true but let us grant that assumption.
Secondly, in
we are doing arithmetic with functions, not plain numbers. and
we don't just have the addition of functions above, we have the addition of products of functions which we eventually factorize! This means not only do the distributions match, but the ordering of the values in our spreadsheets must match also. The
column may contain the same elements as
(cf fair-sampling assumption), but the ordering will be almost certainly different for a weakly objective result. Therefore equation (29) contains another hidden assumption that
That is the assumption that not only do the columns contain the same set of lambda values, but also that they contain the same ordering, or can be re-ordered into the same ordering. This assumption is false. Without this assumption, we cannot factorize the functions in equation (29) and the derivation does not proceed in the weakly objective scenario.
To see this, let us say you reshuffle the rows of
so that the
columns match those of
. Since column
is shared by two spreadsheets, we must apply any such permutation not just to
but also to
. However, you also have to make the lambda columns of
to match those of
and the lambda columns of
to match those of
, each time making sure to rearrange the corresponding spreadsheet from the same particle pair. This is obviously not possible. The cyclic recombination of pairs of terms means you keep undoing the order as you go around. Again it is very important to note that the fair sampling assumption (which we've granted above, only says the probability distribution of values in the lambda column is the same for all the 8 spreadsheets. It says nothing about the ordering of those values. Therefore the Fair sampling assumption does nothing to resolve the real problem. Alice and Bob do not have the freedom to select lambda columns that can be easily rearranged. Randomization makes things worse, not better.
Let us pretend that what I just demonstrated was not the case. Let us pretend that as soon as the first two permutations are done, then all the pairs of spreadsheets that need to be rearranged will automatically match and we won't have any problems and for the sake of argument, let us evaluate it for a moment. If these permutations are able to be done (despite my proof that they can't), it would mean that at the end of the day, there are
effectively only 4 2xN spreadsheets with identical lambda columns on all the spreadsheets. In other words, it would mean that there are only 4 functions
with exactly the same domain! (cf Justo's equation (29)). And it would mean, the four columns of outcomes can be combined into a single 4xN spreadsheet of outcomes! This is for all practical purposes exactly the strongly objective scenario! And therefore the derivation of an upper bound of 2 follows directly. Hopefully, now you may appreciate why Justo's equation (29) is loaded with hidden assumptions that are false.
In summary:
- The transition between (20) and (21) involves a hidden assumption about the domains of the functions, which is not necessarily true and very likely false for the physical situation of EPRB. Without this assumption,
and the derivation can't proceed.
- Even if we grant the above, (29) includes the hidden assumption that disjoint random hidden variable sets can be re-ordered. An assumption that is false.
- If we grant that the hidden variables can be reordered, the weakly objective scenario is transformed to the strongly objective one. Therefore the derivation of the inequality from the weakly objective scenario involves a hidden assumption that the weakly objective scenario is the same as the strongly objective one.