When can hidden variables be excluded in quantum mechanics?

Abstract

The problem of hidden variables is examined in the axiomatic formulation of quantum mechanics based on the algebra of observables. After a brief introductory survey of the earlier investigations, we first investigate the structure ofC*-algebras which allow dispersion-free positive linear functionals. The result obtained is a direct generalization of the well-known result of von Neumann concerning the hidden variables. In the next Section, we assume, as before, that the observables form the Hermitian elements of aC*-algebra. But we now relax the requirement on «states» and allow the so-called monotone-positive functionals (which are not necessarily linear) to represent states. It is then shown that even when such generalized states are allowed, a system admits hidden variables only if its algebra of observables is Abelian;i.e., only if all observables are mutually compatible. In another Section, we investigate the question of hidden variables under the assumption that the observables, instead of forming aC*-algebra, have a certain more general algebraic structure.