A pedestrian approach to von Neumann’s hidden variables proof
Leonardo Andreta de Castro, Carlos Alexandre Brasil
Abstract Hidden variables are hypothetical parameters that, if known, would deterministically specify quantum measurement outcomes, aiming to restore determinism and causality typical of classical mechanics. This work provides a modern and pedagogical exposition of John von Neumann’s 1932 proof on the incompatibility of hidden variables with quantum mechanics. We detail his two main conclusions: (1) dispersion-free ensembles (as defined within the framework of hidden variable theories) are incompatible with quantum mechanics; and (2) the quantum state provides a complete description of physical reality. We critically analyze the key assumptions in von Neumann’s proof, highlighting later challenges—particularly John Bell’s critique and the development of contextual hidden-variable theories by Kochen and Specker. This discussion clarifies von Neumann’s historical role as a pioneer in this debate, while showing how modern hidden-variable theories, such as Bohmian mechanics, remain compatible with quantum predictions. Conversion tables are included between this article’s notation and von Neumann’s original book.
European Journal of Physics•Nov 13, 2025•1 citation