From: naturman@uctvax.uct.ac.za
Newsgroups: sci.physics
Subject: Value Rule is sometimes false
Date: 5 Oct 92 15:44:55 +0200
Organization: University of Cape Town

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

Am I correct in saying that the Value Rule (VR) is used in the derivation of
the Bell Inequality (BI). The VR in interpretations of QM formalism says that
if the probability of an observable Q having a value q in state Psi is zero,
then Q cannot have value q in Psi.

It is certainly agreed that VR is false for observables corresponding to
operators with an uncountable spectrum of eigenvalues, since any such
eigenvalue has prob. zero in a non-eigenstate.

However, it is often assumed that VR holds if the operator has a finite
spectrum. But this is false if the operator is actually degenerate. To see 
this let X denote the position of a particle along some fixed axis,
corresponding to the self-adjoint operator X^. Let E = {-1,0,1} and let
s:R-->E (where R is the real numbers) be the signum function given by

                    ( -1 if r<0
              s(r) ={  0 if r=0
                    (  1 if r>0

Let Q^ be the operator s(X^). Then Q^ is self-adjoint. Assuming all
self-adjoint operators correspond to observables there is an observable Q
corresponding to Q^. Then Spectrum(Q^) = E which is finite. Now consider a 
state Psi where prob(r>0) = prob(r<0), i.e a probability density amplitude
symmetric about r=0. Then if p:E-->[0,1] is the probability function for Q 
having particular values in E in state Psi. Then p(-1) = p(1) = 0.5 while
p(0) = 0. If we accept VR then we must conclude that Q cannot have value 0 in
state Psi. This is false since X can have value 0, and consequently Q has value
q = s(0) = 0. (We only need Restricted Functional Correspondence here since
X^ is non-degenerate.)

Now when spin was discovered the mathematical formalism was expanded to produce 
spin operators Sx^, Sy^, Sz^ etc etc, BUT IS WAS BLINDLY ASSUMED THAT THE
SPIN EIGENVECTORS ARE NON-DEGENERATE. If we assume that they are degenerate due
to the dependence of the spin operator on some "hidden" operator in the manner
discussed above with X^ and Q^, then we must reject VR.

If BI needs VR then as far as I can see, an interpretation which is realist,
deterministic, denies general Functional Correspondence and which also denies
VR, then avoids all forms of Nonlocality, avoids BI and avoids the
Kochen-Specker paradox.

Any comments?!! 

Please only reply if you are really familiar with the subject. Don't reply if
you have free-will hangups or a fetish for retaining Nonlocality at all costs.
 
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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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