Resuming the themes proposed by EPR paradox, J. S. Bell obtained in 1964 the important result of find a property of the local hidden variables theories not compatible with the Quantum Mechanics. Using this property it turned possible to organize crucial experiments, whose outcome would have allowed to take the debate on reality and causality away from the subjectiveness of the philosophical options, making it to reenter in the methodology commonly accepted by the physicists in the resolution of their theoretical divergences.
In order to illustrate the result of Bell, we propose a simplified version of it, suggested by J. J. Sakurai and found in Wikipedia.
Let consider, for example, the decay of a pion π0 in a pair of electrons e+, e-. Let say a e b the two electrons.
Since the pion has spin 0, the electrons must have opposite spin.
If an observer, intercepting the electron a, finds that its spin component along a given direction s is positive, he can assert, without to execute any measure on b, that the spin component of b along the same direction s is negative.
If we accept the Einstein reality criterion, the spin of b is an element of reality , an intrinsic, local, constant and perfectly determined property of the electron b. Because of the symmetry, we can say the same about the electron a.
For the mainstream interpretation of Quantum Mechanics the things are different: the two electrons form a whole system, described by a single state function ψ. Using this function we can only assert that the electron b, like the electron a, has 50% of probability to have positive spin and 50% of of probability to have negative spin: to talk about the reality of the value of its spin before carrying out the measure on a is a physical nonsense.
Therefore, if we accept the Einstein criterion, we must admit that the information given from the state function ψ aren't complete; they only supply statistic means on a great number of measures executed on many pairs of electrons. Instead a complete theory must be able to define what happens to every single electron.
Let suppose therefore that for every electron the spin component along a given direction preexists the measurement and is determined by some currently inaccessible parameter.
If we choose three directions r, s, t, each angled toward the other, (none of they opposite), the signs of the spin components along the three directions will have the following combinations with respective frequencies fi
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If we measure the signs of the spin component of the two electrons a and b along two different directions, for example, r and s, the frequency of the pair (r+, s+) is given from the sum of the frequencies of the two initial combinations which can lead to this result:
If the directions are r and t, in the same way we obtain
Finally, if the directions are t and s , we obtain
Obviously the fi aren't negative, therefore
that is
and, since the frequencies fi, after a greatest number of measurements, tend to probabilities pi
This relation is a possible version of the Bell's inequality and must be satisfied by all the theories that admit the existence of local hidden variables.
The prevision of the results of these electronic spin measurements calculated on the basis of Quantum Mechanics can violate this inequality, that is the probability in the first term exceeds the sum of the probabilities in the second term.
Therefore, if a realization of experiments of this type gives results which violate the Bell's inequality, we must conclude that the local hidden variables theories are without scientific foundation.
Starting from the publication of the Bell paper, many physicists have made many variants of this experiment: we remember only someone among those which more rose the attention of the scientific community:
All these experiments have verified that the Bell's inequality violation is much greater than standard deviation, corroborating the orthodox interpretation of Quantum Mechanics.
At the present time, therefore, the majority of the physicists who have been interested to the argument believes that the trials of solution of the EPR paradox based on local hidden variables theories seems without future.
However, the debate does not appear still ended: the theoretical bases of the Bell's inequality and the experimental tests based on it or on its derivations are subjected to continuous critical reviews like those cited in Wikipedia - Bell's theorem, in particular those of C. H. Thompson.
In last years some researchers are engaged in the formulation of new hidden variables theories based on various approaches, in particular inserting the quantum phenomena in the context of the recent developments of General Relativity: among these there are Gerard 'T Hooft, (Nobel for Physics 1999), and Mark J. Hadley.
An account of these jobs can be seen in Was Einstein Right?, by G. Musser in Scientific American - September 2004.
The fact itself that the problem, in the course of seventy years, has stimulated so many intelligences at theoretical level and absorbed so many resources in terms of job and instruments, is a demonstration that an underground inquietude has continued to wind in a century, the 1900's, which, after the relativity and quantum revolutions has been a century, to use Kuhn's terminology, of 'normal science', in which the physical knowledges have had an enormous quantitative expansion and most numerous technological applications, but did not produce important cultural changes with respect to the standard set up in the thirties.
Maybe it's too much to expect too many revolutions in a single century and very probably these physical and epistemological problems will remain for long.