Quantum Mechanics and imprecise probability

TL;DR

This paper formalizes the quantum typicality rule using imprecise probability theory, proposing a new stochastic process model that explains quantum and classical phenomena at macroscopic scales.

Contribution

It introduces a formal definition of the quantum typicality rule via imprecise probabilities and models quantum systems as imprecise stochastic processes.

Findings

The model can predict statistical experiment outcomes.

It explains the quasi-classical macroscopic evolution.

Provides a new mathematical framework for quantum probability.

Abstract

An extension of the Born rule, the {\it quantum typicality rule}, has recently been proposed [B. Galvan: Found. Phys. 37, 1540-1562 (2007)]. Roughly speaking, this rule states that if the wave function of a particle is split into non-overlapping wave packets, the particle stays approximately inside the support of one of the wave packets, without jumping to the others. In this paper a formal definition of this rule is given in terms of {\it imprecise probability}. An imprecise probability space is a measurable space $(\Omega, {\cal A})$ endowed with a {\it set} of probability measures $\cal P$. The quantum formalism and the quantum typicality rule allow us to define a set of probabilities ${\cal P}_\Psi$ on $(X^T, {\cal F})$, where $X$ is the configuration space of a quantum system, $T$ is a time interval and ${\cal F}$ is the $\sigma$-algebra generated by the cylinder sets. Thus, it is proposed that a quantum system can be represented as the {\it imprecise stochastic process} $(X^T, {\cal F}, {\cal P}_\Psi)$, which is a canonical stochastic process in which the single probability measure is replaced by a set of measures. It is argued that this mathematical model, when used to represent macroscopic systems, has sufficient predictive power to explain both the results of the statistical experiments and the quasi-classical structure of the macroscopic evolution.