We have discovered an oversight in John S. Bell's hidden variable formulation of his inequality.
Everyone
assumes Bell's result holds for all possible hidden variable theories
but that assumption is false. Nonrecurrent hidden variables do not
abide by Bell's inequality. The probability that two instances of
a nonrecurrent variable are the same is zero, exactly zero (see W Feller
Vol II page 4). Each instance of a nonreccurrent hidden variable is
unique. All sequences are unique. All samples are unique.
All contexts are unique. It is assumed that the
source of the hidden variables is not a function of the parameters
of any measurement device. However, the uniqueness of
nonreccurrent hidden variables says when a measurement
parameter changes the hidden variables change (because they are
recorded at a different time). That induces a form of context
sensitivity to measurements. That sensitivity means that Bell's
formulation can not hold for nonrecurrent hidden variables, for when the
parameters for Alice and Bob are (a,b) the hidden variables are
different from those when the parameters are (a, b').. Bell
assumed that in situation 1 (a,b) the hidden variables can be made
common with situation 2 (a,b') but that is false for nonrecurrent hidden
variables. If the variables are discrete it would be possible to
satisfy Bell's contraint. Here is the difference between recurrent
and nonrecurrent variables
Recurrent <A
1A
2> = <1> = 1
Nonrecurrent <A
1A
2> = <A
1><A
2> = 0
The latter follows from the fact that different samples are independent for nonrecurrent hidden variables.
Using
independence and following Bell's steps of his derivation we arrive at
a different inequality. We show that inequality places no
constraints on correlations.
The same argument holds for the CHSH inequality and the GHZ expression.
Nonrecurrent hidden variables are a powerful lever that pries open quantum issues.
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