Jauch-Piron States

Pavel Pták

Frederik Jozef Belinfante

Nicolas Hadjisavvas, F. Thieffine, Mioara Mugur-Schächter

Josef Tkadlec

A Boolean orthoposet is the orthoposet P fulfilling the following condition: if a ^ b = 0 then a ? b. Boolean orthoposets enjoy some properties of Boolean algebras but, in some sense, they are far away of them (3, 4, 8). An important notion in quantum logic theory is the notion of Jauch-Piron state (2, 5, 6, 7). In this paper, we show connections between Boolean orthoposets and orthoposets with a proper number of Jauch-Piron states: We demonstrate that an orthoposet with enough two-valued Jauch-Piron states is Boolean and, on the other hand, we give several examples of Boolean orthoposets without any two- valued Jauch-Piron state.

L. J. Bunce, Jan Hamhalter

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.

Anton Amann

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.