Quantum Measures and Integrals

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.