Algebraic constraints on hidden variables

Mikl�s R�dei

Mićo Đurđević

A general algebraic formalism for studying contextual hidden variable theories is developed. The contextuality is understood as a manifestation of an inadequacy in the algebra of quantum observables for the complete description of the system. It is shown that it is always possible to 'improve' the algebra of quantum observables, explicitly paying attention to contexts, such that this improved algebra becomes a base for a hidden variable theory.

F. Pollini

Sofia Wechsler

A recent article of Colbeck and Renner tackled the problem whether entanglements may be explained by combined models of local and non-local hidden variables. To the difference from previous works they considered models in which each pair of entangled particles behaves in the same way, and the particles in the pair are equivalent, i.e. each of them produces its response to a measurement according to both local and non-local hidden variables. Their article aimed at proving that the local hidden variable component in such models has no effect on the measurement results, i.e. only the non-local variables are relevant. However, their proof deals with a very restrictive case and assumes questionable constraints on the hidden variables. The present text studies the Colbeck and Renner class of models on a less restrictive case and under no constraints on the hidden variables. It is shown again that the local component cannot have any influence on the results. However, the Colbeck and Renner class of models is not the only one possible. A different class is described, and it admits local hidden variables by the side of the non-local influence. This class presents a couple of advantages.

Miklós Rédei

Miklós Rédei

Based partly on proving that algebraic relativistic quantum field theory (ARQFT) is a stochastic Einstein local (SEL) theory in the sense of SEL which was introduced by Hellman (1982b) and which is adapted in this paper to ARQFT, the recently proved maximal and typical violation of Bell's inequalities in ARQFT (Summers and Werner 1987a–c) is interpreted in this paper as showing that Bell's inequalities are, in a sense, irrelevant for the problem of Einstein local stochastic hidden variables, especially if this problem is raised in connection with ARQFT. This leads to the question of how to formulate the problem of local hidden variables in ARQFT. By giving a precise definition of hidden-variable theory within the operator algebraic framework of quantum mechanics, it will be argued that the aim of hidden-variable investigations is to determine those classes of quantum theories whose elements represent a statistical content that cannot be reduced in a given way. In some particular way to be stated, a proposition will be stated which distinguishes quantum field theories whose statistical content cannot be reduced without violating some relativistic locality principle.