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Reviews in Mathematical Physics

JAUCH–PIRON STATES AND σ-ADDITIVITY

L. J. Bunce, Jan Hamhalter

Published Jun 1, 2000• 5 citations
Publisher

Abstract

We study σ-additivity of the physically plausible Jauch–Piron states on a von Neumann algebra M. Amongst other consequences we extend our earlier results [5] by showing that geometric conditions much weaker than pureness imply σ-additivity for a Jauch–Piron state and, further, that if M is properly infinite and the continuum hypothesis is assumed to be true then all Jauch–Piron factor states are σ-additive.

DOI
10.1142/s0129055x00000307

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We study σ-additivity of the physically plausible Jauch–Piron states on a von Neumann algebra M. Amongst other consequences we extend our earlier results [5] by showing that geometric conditions much weaker than pureness imply σ-additivity for a Jauch–Piron state and, further, that if M is properly infinite and the continuum hypothesis is assumed to be true then all Jauch–Piron factor states are σ-additive.
A state φ on a W*-algebra ℳ is said to fulfill the Jauch–Piron condition if φ(p)=φ(q)=1 for projections p,q∈ℳ implies φ(p∧q)=1. Here p∧q denotes the infimum of p and q in the projection lattice of ℳ. The Jauch–Piron condition is a compatibility condition between the algebraic and the lattice-theoretic approach for the description of physical systems. Normal (i.e., σ-weakly continuous) states always fulfill the Jauch–Piron condition. It is argued that states not fulfilling this condition should be regarded as unphysical. It is shown that a state φ on a σ-finite factor ℳ is singular if and only if projections e, f∈ ℳ exist such that φ(e)=φ( f )=1 and e∧f=0. In particular, any pure state φ on ℳ fulfilling the Jauch–Piron condition is normal, which implies that the underlying factor ℳ is of type I. Furthermore, the following result is proved: Let φ be a pure Jauch–Piron state on W*-algebra ℳ with separable predual and without any commutative summand. Then φ is normal and a central projection z0∈ ℳ exists such that φ(z0)=1 and z0 ℳz0 is a factor of type I. Thus, cum grano salis, pure Jauch–Piron states exist only on commutative W*-algebras and type I factors. The former case corresponds to classical theories, the latter to Hilbert-space quantum mechanics. The implications of these results on the interpretation of quantum mechanics are discussed.
We show in this note that if [Formula: see text] is a Boolean subalgebra of the lattice quantum logic [Formula: see text], then each state on [Formula: see text] can be extended over [Formula: see text] as a Jauch–Piron state provided [Formula: see text] is Jauch–Piron unital with respect to [Formula: see text] (i.e. for each nonzero [Formula: see text], there is a Jauch–Piron state [Formula: see text] on [Formula: see text] such that [Formula: see text]). We then discuss this result for the case of [Formula: see text] being the Hilbert space logic [Formula: see text] and [Formula: see text] being a set-representable logic.