Consistent Quantum Reasoning

TL;DR

This paper develops a formal framework for quantum reasoning that parallels classical physics, introducing consistent quantum descriptions using frameworks and histories, and demonstrating applications like spin measurements and the quantum-classical transition.

Contribution

It introduces a new formalism for quantum reasoning based on frameworks and generalized histories, extending previous approaches without relying on density matrices or time directionality.

Findings

Formalism applies to spin measurements and diffraction experiments

Provides insights into the emergence of classicality from quantum mechanics

Defines consistency conditions for quantum histories

Abstract

Precise rules are developed in order to formalize the reasoning processes involved in standard non-relativistic quantum mechanics, with the help of analogies from classical physics. A classical or quantum description of a mechanical system involves a {\it framework}, often chosen implicitly, and a {\it statement} or assertion about the system which is either true or false within the framework with which it is associated. Quantum descriptions are no less ``objective'' than their classical counterparts, but differ from the latter in the following respects: (i) The framework employs a Hilbert space rather than a classical phase space. (ii) The rules for constructing meaningful statements require that the associated projectors commute with each other and, in the case of time-dependent quantum histories, that consistency conditions be satisfied. (iii) There are incompatible frameworks which cannot be combined, either in constructing descriptions or in making logical inferences about them, even though any one of these frameworks may be used separately for describing a particular physical system. A new type of ``generalized history'' is introduced which extends previous proposals by Omn\`es, and Gell-Mann and Hartle, and a corresponding consistency condition which does not involve density matrices or single out a direction of time. Applications which illustrate the formalism include: measurements of spin, two-slit diffraction, and the emergence of the classical world from a fully quantum description.