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Bell's theorem - for or against Hidden Variables?
#30
What about time?  Why would it not be a hidden variable, considering that it is not a real quantity independent of spacetime?  Spacetime dependence, then, would dictate that the time parameter demands specification of measure space, and once that is done one cannot avoid including the parameter as integrated into the measure.  

The conventional way to do measurements of Bell-Aspect quantum correlations is described by Richard Gill:  "The experiment is about *counting*. There are two measurement devices which have binary settings ('1' or '2'). There is a binary outcome. ('+' or '-') At the end of the experiment you have 16 counts. So many times you saw outcome '++' and the setting was '11', so many times '+-' and the setting was '12' ... so many times '--' and the setting was '22'. Quantum mechanics predicts the probabilities of outcomes given settings e.g. Prob('-+' | '21').  The physicist's correlation is just the probability of equal outcomes minus the probability of different outcomes." *

Now if Bell-Aspect is a simple discrete counting function as Richard claims, and pairwise measures have 16 possible binary outcomes, let's try using the ultimate discrete counting device -- prime numbers.

Because all odd primes are congruent (mod 2), and unity is not considered a prime, let's take the ordered sequence of the first 8 primes on the natural line: 3, 5, 7, 11, 13, 17, 19, 23

This is the densest sequence of primes > 2 of cardinality 8, with the interesting properties that all pairs are congruent (mod 2), regardless of order, and 23 is the least prime that is not a member of a twin pair.

If we take quantum correlations as signed terms, as Richard describes, then every term has a corresponding negative value, for 16 unique counts.  That's too many, though -- we should have only 4 each (+) and ( - ) values, for 4 pairwise outcomes.  Look:

We find exactly four primes in the sequence with the Sophie Germain property
(P and 2P + 1 are both prime):  {3, 5, 11, 23}.  Let’s call this set (+).  Call ( - ) the remaining set {7, 13, 17, 19}

The one property that these combined sets have, that the conventional 4 X 4 correlation matrix does not – is reversibility. Here is why:

{+1, + 1, + 1, + 1} and {- 1, - 1, - 1, -1} with an upper bound (CHSH bound) of 2, cannot describe an evolution of particle interactions – what EPR would call hidden variables – that a wave function captures by symmetry of motion that implies reversibility of states.

We can see this clearly, by writing the delta between each term of the ordered sequences:

(+) 3 5 11 23

       + + + -
       4 8 6 4  
        - - - +

( - ) 7 13 17 19



So we generate the positive set as {3+4, 5+8, 11+6, 23-4}

The negative set is                       {7-4, 13-8, 17-6, 19+4}


So while the fundamental linear assumption of the 4 X 4 matrix is a terminating order of + + + - or - - - + (+ 3 – 1 or – 3 + 1), our continuous function is compelled to be reversible to the origin.

The delta terms all cancel, so we haven’t lost any consistency with the natural integer sequence (to which in fact, recursion is a native property). We’ve gained additional information though, of a time parameter that is sensitive to initial condition – is the origin positive or negative?

As we demonstrated, regarding the algebraic closure of the complex plane, the double-zero origin of C demands positivity parity starting condition  -- the 2 + 1 condition gives us the prime 3.  The counting function still holds, when measuring quantum correlations; however, the three delta terms are hidden variables (one term is redundant).

Karl Hess and Walter Philipp introduced the time parameter ("timelike correlated parameter") to the Bell inequality, with the same result.  “An example of outcomes for the products of equation 3.1   A(a,lambda)A(b,lambda) + A(a,lambda)A(c,lambda) – A(b,lambda)A(c,lambda) =/< + 1 that violate Bell’s inequality is + 1 for the terms with the plus sign and – 1 for the last term with the minus sign.  Because  - (-1) = + 1 the sum of the three terms is then + 3, which indeed violates the Bell inequality Eq (3.1)”  

In terms of computability, recursion native to the real line is equivalent to time reversibility native to general relativity cosmology, and dependent on initial condition.

* Gill, Paul Snively's blog, 2015
** Karl Hess, Einstein was Right p.47
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RE: Bell's theorem - for or against Hidden Variables? - by Thomas Ray - 06-16-2016, 03:18 PM

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