🚀 go-pugleaf

Only reply to this if you are deeply familiar with Realism and Locality
paradoxes in Quantum Mechanics.

In the derivation of the Bell Inequallity, is the Value Rule used. The Value
Rule (VR) states that if the probability of an observable Q having the value
q in state \Psi is 0 then Q cannot have the value q in state \Psi.

Physicists realize that VR fails for observables whose corresponding
self-adjoint operator has a continuous spectrum of eigenvalues, since any such
eigenvalue has prob. = 0.

However physicists do not seem to realize that VR can also fail for a discrete
spectrum. Consider the set X = {-1,0,1} and define a probability measure 
p:X-->[0,1] by p(-1) = p(1) = 0.5 and p(0) = 0. Now suppose that Q is an
observable whose operator has eigenvalues X. By VR, Q cannot have the value 0.

To see that this is false let S:R-->X (where R is the set of real numbers) be
the signum function given by 
                                    ( -1 if r<0
                            S(r) =  {  0 if r=0
                                    (  1 if r>0

Let Z denote the position of a particle along some axis corresponding to
operator Z^. Let Q^ = S(Z^). Then Q^ is a self-adjoint operator and PROVIDED
WE ASSUME THAT EVERY SELF-ADJOINT OPERATOR DOES CORRESPOND TO AN OBSERVABLE
there is an observable Q corresponding to Q^. Then the spectrum of Q is X. Let
/Psi be a state where the probability of Z having value z<0 is the same as that
of it having z>0. Then the probability distribution of eigenvalues for Q is
simply p defined above. But if Z has value 0 then so will Q, violating VR.
(PROVIDING we assume that values of Q are related to those of Z by q = S(z).
This assumption does not lead to a Kochen-Specker paradox since Z^ is
non-degenerate.)

If the Bell Inequallity relies on VR for its derivation then if we take a view
of QM which is realist, deterministic, denies VR and denies strong functionality
then this view avoids all locality paradoxes (including the Bell Inequallity),
and avoids the Kochen-Specker paradox.

In fact, since as I've shown above, VR is false I would say that it should
always be denied.

(People with free-will hangups and a morbid desire to preserve non-locality need
not reply to this.)
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* Baby Doc Naturman                                                         *
* USS Enterprise                           "It's life Jim,                  *
* Final Frontier                                  but not as we know it."   *
* naturman@uctvax.uct.ac.za                                                 *
*                                                       Spock of Vulcan     *
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