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Bell's inequality



In 1935 von Neumann  published a proof that no more complete theory could reproduce the predictions of quantum mechanics[34]. This proof was accepted for nearly three decades even though Bohm  published an example of a more complete theory in 1952[6]. Bohm thought at the time that his theory at some level must disagree with standard and thus was not in conflict with von Neumann's proof.

Bell suspected there was something wrong with von Neumann's proof and, in the early sixties, he wrote a refutation[4]. At the end of this paper (which was not published for several years) he suggests that the broad result claimed by von Neumann was not possible. Influenced by Bohm's work he wondered if one might show that no local realistic theory 

could be consistent with . The term realistic does not have an unambiguous meaning in physics but Bell defined what he meant. In 1964 he published a paper[3] that described an inequality (now referred to as Bell's inequality) that any local theory must obey. The equality relates two experimentally controllable parameters to two detections. Figure gif on page gif

illustrates the setup. In this figure the experimental parameters are referred to as polarizers. The inequality relates the angles between the polarizer to the probability that there will be a joint detection of two events. In a local theory in which the measurements are determined by an objective state (local realistic theory as defined by Bell) there are various inequalities that constraint this correlation function. Quantum mechanics predicts these constraints do not hold. The inequality is only predicted to hold if a change in the probability of joint detections occurs in less time that it takes light to travel from either polarizer to the more distant detector. Quantum mechanics predicts this change can happen in an arbitrarily short time.

In the 1970's Eberhard  derived Bell's result without reference to `realistic' theories[12,13]. It applies to all local theories. Eberhard also showed that the nonlocal effects that predicts cannot be used for superluminal communication.

How does violate locality? Two principles of physics are involved: the singlet state  and and the act of measurement  . In the properties of pairs of particles in a singlet state remain connected even if the particles become separated by a great distance. They are still part of a single wave function. The wave function  is not something that exists in physical space. It is defined only in configuration space  where there are a separate set of spatial coordinates for every particle. The wave function evolves locally in configuration space. To make predictions in physical space we must project the configuration wave function model onto physical space to compute the probability that an event will be observed at a detector. The combination of wave function evolution in configuration space and this projection operation is irreducibly nonlocal.

Because claims that probabilities are irreducible it is not possible to send a superluminal signal  . The only way to model what is happening mathematically is for information to be transferred superluminally from one of the polarizers to the detector more distant from it. However because of quantum uncertainty there is no way to know in what direction this information transfer occurs. You can only prove there must have been the superluminal transfer of information  by comparing the results form the distant detectors. Superluminal transfer of information without superluminal signals is only possible in a theory that claims probabilities are irreducible.

Physicists often claim that there is no superluminal transfer of information predicted. They claim that something is happening that does not fall within the domain of our classical mathematics. Perhaps this is true but I prefer to write as if classical mathematics 

holds even for . There is not a shred or experimental evidence that suggests it does not. There have been experimental tests but none of them are conclusive. The experimental verdict is still out on locality in nature.

Bell's result has, to a degree, converted the metaphysical measurement problem to an experimental question. If locality holds then there is a space time structure to the changes in the wave function associated with an observation and through tests of Bell's inequality we will be able to experimentally observe this structure which is outside of any accepted physical theory.

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