Bell inequalities and hidden variables over all possible paths in a quantum system

Abstract

Bell's theorem rests on the following fundamental condition for a local system: P(a,b|alpha, beta, lambda)= P(a|alpha, lambda)P(b|beta, lambda). Here a and b are the outcomes respectively for measurements alpha on one side, and beta on the other, of an experiment involving two entangled particles traveling in opposite directions from a source. The parameter lambda (the set of "hidden variables") represents a more complete description of the joint state of the two particles. Because of lambda, the joint probability of detection is now dependent only on lambda and the local measurement setting of alpha; similarly for the other side and the setting beta. From this equation John Bell derived a simple inequality that is violated by the predictions of quantum mechanics, which is generally taken to imply that quantum mechanics is a nonlocal theory. But, by combining Richard Feynman's formulation of quantum mechanics with a model of particle interaction described by David Deutsch, we develop a system (the "space of all paths," SP) that (1) is immediately seen to replicate the predictions of quantum mechanics, (2) has a single outcome for each quantum event (unlike MWI on which it is partly based), and (3) contains the set lambda of hidden variables consisting of all possible paths from the source to the detectors on each side of the two-particle experiment. However, the set lambda is nonmeasurable, and therefore the above equation is meaningless in SP.