Scattering theory, cross sections, and the lattice of propositions (Q1065335)

The paper starts with a discussion of states defined as probability measures on a complete orthocomplemented lattice \({\mathcal L}\). In particular, if states \(\mu_ 1\) and \(\mu_ 2\) assign probabilities \(P[p| \mu_ 1]\) and \(P[p| \mu_ 2]\) to a proposition p of \({\mathcal L}^ a \)metric d(.,.) on the convex set \({\mathcal M}({\mathcal L})\) of states on \({\mathcal L}\) is defined by \[ d(\mu_ 1,\mu_ 2)=\sup_{p\in {\mathcal L}}| P[p| \mu_ 1)-P[p| \mu_ 2]|. \] The paper then specialises to \({\mathcal L}({\mathcal H})\), the lattice of subspaces of a Hilbert space \({\mathcal H}\), appropriate for quantum mechanics. It is shown, for example, that any state may be approximated in the above metric by a finite convex linear combination of pure states defined by vectors in a dense subset of \({\mathcal H}\), and the philosophical significance of this is commented upon. It is also shown that an isometric operator W in \({\mathcal H}\) induces a corresponding mapping in \({\mathcal M}({\mathcal L}({\mathcal H}))\), which is isometric on \({\mathcal M}\) if A is unitary on \({\mathcal H}\). The standard asymptotic results of the quantum theory of potential scattering are expressed in terms of states of \({\mathcal M}\), and in particular an isometric scattering operator on \({\mathcal M}\) is constructed. This is a concrete exemplification of the work of \textit{N. S. Kronfli} [ibid. 2, 345-349 (1969)]. The remainder of the paper deals with cross-sections for potential scattering. The number of particles emerging in a cone \({\mathcal C}\) per unit time per unit average incident flux, averaged over all displacements of the incident beam perpendicular to the scattering centre, is denoted by \(\sigma\) (\({\mathcal C})\). Two calculations, one using classical mechanics, the other using quantum mchanics, then lead to expressions for \(\sigma\) (\({\mathcal C})\) which are the same if the differential cross- section \(d\sigma\) /d\(\omega\) in the classical expression is identified with \(| f|^ 2\) in the quantum one, where f is the scattering amplitude.