Bell-like examples for spin-1 hidden variable theories

Abstract

Though John Bell had claimed that his spin-1/2 example of a hidden-variable theory(HV) is an \emph{explicit} counterexample to von Neumann's proof of the non-existence of hidden variable theories empirically equivalent to quantum mechanics, such examples can be so construed only if they met all of von Neumann's requirements. In particular, that they reproduced all the observed predictions of quantum theory. To shed light on these aspects, we have, on the one hand, simplified and critically examined Bell's original example and on the other hand, constructed explicit such examples for spin-1 systems. We have clarified the relation of our example to the Kochen-Specker and Bell's powerful earlier results. Our spin-1 examples are manifestly non-contextual, yet violating the K-S constraints configuration by configuration. Nevertheless, they reproduce the correct quantum expectation values and variances for arbitrary linear combinations of the beables in one case, and for close approximants to the beables representing K-S constraints. In conformity with the K-S theorem, the variance of the K-S constraint is nonzero. The implications of this non-vanishing variance are analysed in detail. In the other example, we show how this variance can be made arbitrarily small but not zero.