Jauch–Piron states in W*-algebraic quantum mechanics

Abstract

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.