Non-Local EPR Correlations using Quaternion Spin

Comment: I have extensively rewritten the paper and changed the title in response to discussions on forums.

Comment: According to the abstract of this paper, "a statistical simulation is presented which reproduces the correlation obtained from EPR coincidence experiments without non-local connectivity". This statement is false. At the end of the paper, formula (32) is presented, which states that the correlation obtained is a mixture of correlations obtained in two "sub-models". But both of these two submodels define correlations which satisfy the Bell-CHSH inequalities, and hence so does their probabilistic mixture.

The author seems to be rather confused about Bell's theorem. On the first page of this paper, he writes "Although this work is not about Bell’s theorem, we note that there are no hidden variables (HV). The only local variable is the angle that orients a spin on the Bloch sphere, first identified in the 1920’s”. His “local variable” is the classical variable which he has introduced in order to explain the correlations predicted by quantum mechanics. That is exactly the definition of a "hidden variable" in this context. Everybody in the whole world working in the foundations of quantum mechanics will call it a “hidden variable”. Bell, Bohr and Einstein would all call it a hidden variable. LHV theories all start by “finding” such a hidden variable. Sanctuary says that he has found the hidden variable and says it is even a variable which people talked about in the 20’s. Numerous local hidden variables theories have been proposed in the past (none of them successful) with the same hidden variable (a direction of spin or of polarization).

Sanctuary's model is a local hidden variables model in Bell’s sense because he introduces functions A and B, and a variable lambda having some probability distribution rho which does not depend on the experimenters settings a and b, such that the experimentally observed outcomes are x = A(a, lambda), y = B(b, lambda). Such a model is always called a *local* hidden variables model because the function A is a function of lambda and of setting a, but not of setting b; the function B is a function of lambda and of setting b, not of setting a. Whether or not the variable “lambda” is associated with some location, some part of the experiment, is irrelevant. The adjective “local” in the expression “local hidden variables model” refers to the functions A and B, not to the variable lambda. Bell said this explicitly many times.

More precisely, Sanctuary's model would be a local hidden variables model in Bell’s sense if the observed outcomes x, y were elements of the set {-1, +1}. The standard quantum mechanical predictions for the EPR-B model give not only mean values and correlations but also the possible values of the outcomes. The experimental correlation of his formula (32) is a probability mixture of two correlation functions both of which satisfy Bell-CHSH inequalities since he has a local hidden variables theory for “p” detections and another local hidden variables theory for “c” detections. The paper does not contain a graph of E_exp(a, b). It is not equal to the famous EPR-B correlation.

If one would imagine each particle pair having two attributes, namely a polarisation angle lambda together with the identity “p - type” or “c - type”, and the detectors not distinguishing which type the incoming particles are but just registering the +/-1 outcome of the relevant one of his two “sub-models", then formula (32) would represent the experimentally observed correlation function. As said before, it does not violate CHSH inequalities and it is not equal to the EPR-B correlation, so this model does not “explain” QM predictions and it does not explain actual experimental results obtained to date.