Comment on “ Exclusion of time in the theorem of Bell ”

– Bell’s inequalities are a variant of Boole’s legendary consistency “conditions of possible experience.” Although they do not specifically refer to spatially separated subsystems, they apply to spatially separated particles as well. Such an interpretation appears to be immune to arguments involving time dependencies put forward recently by Hess and Philipp [1]. In the middle of the 19th century the English mathematician George Boole formulated a theory of “conditions of possible experience” (COPE) [2–4]. These conditions subsume the consistency requirements satisfied by relative frequencies or probabilities of classical events. They are expressed by certain equations or inequalities. More recently, COPE for a particular setup relevant in the quantum mechanical context have been studied by Bell and others. To name a specific example, the Clauser, Horne, Shimony, and Holt (CHSH) inequalities are just COPE for a particular physical setup studied by CHSH. Pitowsky, by pointing out this fact, has given a geometrical interpretation of COPE in terms of correlation polytopes [5]. Thereby, the rows of the truth tables of events and their joints are interpreted as vectors of Euclidean space. A correlation polytope is defined by taking all such vectors and interpreting them as the extreme points of the polytope. The physical interpretation of the inequalities representing the boundaries of the Pitowsky correlation polytope is this: Any face of the polytope has an “inside” and an “outside,” and corresponds to a Boole-Bell type inequality. It can be viewed as a sort of demarcation line, a maximal border, between the classically allowed probabilities and the ones (outside of the polytope) which are inconsistent with a classical description of observables as a Boolean algebra. Quantum probabilities exceed the borders of the classical correlation polytopes: they ”lie outside.” But also quantum probabilities are subject to certain constraints and do not violate the inequalities maximally. The elementary algebraic derivation of Boole’s COPE and hence also of Bell-type inequalities do not refer to any spatial or temporal dependence whatsoever. They just state consistency requirements for classical probabilities and correlations of a multiparticle system. Thereby, the particles could, but need not be spatially separated. However, note that these arguments apply also to systems which are spatially separated and thus also to the EinsteinPodolsky-Rosen (EPR) setup.