Piron's approach to the foundations of quantum mechanics

W. Balzer

Gianpiero Cattaneo, Carlo Dalla Pozza, Claudio Garola +1 more

Carlo Dalla Pozza

Belal Ehsan Baaquie

Alexandru Baltag, Sonja Smets

In this paper we show how ideas coming from two areas of research in logic can reinforce each other. The first such line of inquiry concerns the “dynamic turn” in logic and especially the formalisms inspired by Propositional Dynamic Logic (PDL); while the second line concerns research into the logical foundations of Quantum Physics, and in particular the area known as Operational Quantum Logic, as developed by Jauch and Piron (Helve Phys Acta 42:842–848, 1969 ), Piron (Foundations of Quantum Physics, 1976 ). By bringing these areas together we explain the basic ingredients of Dynamic Quantum Logic, a new direction of research in the logical foundations of physics.

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.