No-Go Theorem for Modeling with Von Neumann Observables

Abstract

How far can one proceed with quantum formalism in modeling psychological effects? It is clear that each theory has its own domain of applicability. By borrowing the formalism of quantum mechanics (QM) and applying it in psychology, cognitive, social, and political sciences, one has to be aware that once he might approach the boundary of applicability of this formalism. The special attention must be paid to quantum-like merging of different psychological effects. To illustrate the problem of effects’ merging, consider distinguishing of classical statistical physics and QM via the Bell type experiments. For each pair of experimental settings, the experimental data can be described by classical probability theory. But, the combination of the experimental situations corresponding to selections of a few pairs of setting leads to a deviation from classical probability (in the form of the violation of the Bell inequalities). This example is of only illustrative value. Here we speak about impossibility to combine a few experimental contexts. A psychological effect is a class of experimental contexts. So, the question is about possibility to merge a few special classes of contexts. A few years ago it was found (by Khrennikov, Basieva, Dzhafarov, and Busemeyer) that one of the basic psychological effects, namely, the question order effect,   cannot be integrated with the response replicability effect in the framework of von Neumann measurement theory, i.e., representation of observables (questions, tasks) by Hermitian operators and state updates by projections. (Refined said, we consider quantum instruments of the von Neumann-Lüders class.) In this chapter, we finally present this result which has already been so often discussed in the previous chapters.