Quantum Mechanics and Imprecise Probability

Abstract

An extension of the Born rule, the quantum typicality rule, has recently been proposed [B. Galvan in 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 imprecise probability. An imprecise probability space is a measurable space $(\varOmega,{\mathcal{A}})$ endowed with a set of probability measures ℘. The quantum formalism and the quantum typicality rule allow us to define a set of probabilities ${\mathcal{P}}_{\varPsi}$ on (X ^T,ℱ), where X is the configuration space of a quantum system, T is a time interval and ℱ is the σ-algebra generated by the cylinder sets. Thus, it is proposed that a quantum system can be represented as the imprecise stochastic process $(X^{T},{\mathcal{F}},{\mathcal {P}}_{\varPsi})$ , 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.