Quantum measures and integrals

TL;DR

This paper develops a framework connecting quantum measures and integrals within Hilbert spaces, introducing operators, properties, and a quantum integral with applications to quantum theory.

Contribution

It introduces a novel approach to quantum measures and integrals using Hilbert space operators, establishing their properties and quantization of random variables.

Findings

Operators D(A,B) and μ(A) have positivity and additivity properties.

Quantum integral defined via trace with a density operator.

Example illustrating the theoretical framework.

Abstract

We show that quantum measures and integrals appear naturally in any $L_2$-Hilbert space $H$. We begin by defining a decoherence operator $D(A,B)$ and it's associated $q$-measure operator $\mu (A)=D(A,A)$ on $H$. We show that these operators have certain positivity, additivity and continuity properties. If $\rho$ is a state on $H$, then $D_\rho (A,B)=\rmtr\sqbrac{\rho D(A,B)}$ and $\mu_\rho (A)=D_\rho (A,A)$ have the usual properties of a decoherence functional and $q$-measure, respectively. The quantization of a random variable $f$ is defined to be a certain self-adjoint operator $\fhat$ on $H$. Continuity and additivity properties of the map $f\mapsto\fhat$ are discussed. It is shown that if $f$ is nonnegative, then $\fhat$ is a positive operator. A quantum integral is defined by $\int fd\mu_\rho =\rmtr (\rho\fhat\,)$. A tail-sum formula is proved for the quantum integral. The paper closes with an example that illustrates some of the theory.