Kochen-Specker theorem for von Neumann algebras

Abstract

The Kochen-Specker theorem has been discussed intensely ever since its original proof in 1967. It is one of the central no-go theorems of quantum theory, showing the non-existence of a certain kind of hidden states models. In this paper, we first offer a new, non-combinatorial proof for quantum systems with a type $I_n$ factor as algebra of observables, including $I_{\infty }$. Afterwards, we give a proof of the Kochen-Specker theorem for an arbitrary von Neumann algebra $\mathcal{R}$ without summands of types $I_1$ and $I_2$, using a known result on two-valued measures on the projection lattice $\mathcal{P}(\mathcal{R})$. Some connections with presheaf formulations as proposed by Isham and Butterfield are made.

22 pages, no figures

References in corpus (2)

• All quantum observables in a hidden-variables model must commute simultaneously
• A Topos Perspective on the Kochen-Specker Theorem: IV. Interval Valuations

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