Completeness of Decoherence Functionals

TL;DR

This paper investigates the conditions under which decoherence functionals in the consistent histories approach to quantum theory can be constructed to match given probability distributions, focusing on the completeness of these functionals.

Contribution

It demonstrates that for any partition of the unit operator and any probability distribution, there exists a decoherence functional that makes the set consistent and reproduces that distribution.

Findings

Existence of decoherence functionals matching arbitrary probability distributions.

Completeness of decoherence functionals in separable Hilbert spaces.

Connection between partitions of unity and decoherence functional construction.

Abstract

The basic ingredients of the `consistent histories' approach to a generalized quantum theory are `histories'and decoherence functionals. The main aim of this program is to find and to study the behaviour of consistent sets associated with a particular decoherence functional $d$. In its recent formulation by Isham it is natural to identify the space $\UP$ of propositions about histories with an orthoalgebra or lattice. When $\UP$ is given by the lattice of projectors $\PV$ in some Hilbert space $\V$, consistent sets correspond to certain partitions of the unit operator in $\V$ into mutually orthogonal projectors $\{\a_1,\a_2,\ldots\}$, such that the function $d(\a,\a)$ is a probability distribution on the boolean algebra generated by $\{\a_1,\a_2,\ldots\}$. Using the classification theorem for decoherence functionals, proven previously, we show that in the case where $\V$ is some separable Hilbert space there exists for each partition of the unit operator into a set of mutually orthogonal projectors, and for any probability distribution $p(\a)$ on the corresponding boolean algebra, decoherence functionals $d$ with respect to which this set is consistent and which are such that for the probability functions $d(\a,\a)=p(\a)$ holds.