Quantum measures and the coevent interpretation

TL;DR

This paper reviews quantum measure theory, introduces a new quantum integral representation, and explores the relationship between quantum measures and classical measures on an extended logical space, focusing on finite systems.

Contribution

It presents a novel representation of the quantum integral and establishes a method to transfer quantum measures to classical measures on an extended logic.

Findings

Pure quantum measures are strictly contained in extremal quantum measures.

Quantum measures on finite spaces can be transferred to classical measures on an extended logic.

The new integral representation simplifies computations of quantum Lebesgue integrals.

Abstract

This paper first reviews quantum measure and integration theory. A new representation of the quantum integral is presented. This representation is illustrated by computing some quantum (Lebesgue)${}^2$ integrals. The rest of the paper only considers finite spaces. Anhomomorphic logics are discussed and the classical domain of a coevent is studied. Pure quantum measures and coevents are considered and it is shown that pure quantum measures are strictly contained in the extremal elements for the set of quantum measures bounded above by one. Moreover, we prove that any quantum measure on a finite event space $\ascript$ can be transferred to an ordinary measure on an anhomomorphic logic $\ascript ^*$. In this way, the quantum dynamics on $\ascript$ can be described by a classical dynamics on the larger space $\ascript ^*$.