#### Document Type

Article

#### Publication Date

2011

#### Keywords

Quantum measures, Quantum integrals, Decoherence functionals

#### Abstract

We show that quantum measures and integrals appear naturally in any L2-Hilbert space H. We begin by defining a decoherence operator D(A, B) and it’s associated q-measure operator µ(A) = D(A, A) on H. We show that these operators have certain positivity, additivity and continuity properties. If ρ is a state on H, then Dρ(A, B) = tr [ρD(A, B)] and µρ(A) = Dρ(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 fb on H. Continuity and additivity properties of the map f 7→ fb are discussed. It is shown that if f is nonnegative, then fb is a positive operator. A quantum integral is de- fined by R f dµρ = tr(ρfb). A tail-sum formula is proved for the quantum integral. The paper closes with an example that illustrates some of the theory.

#### Recommended Citation

Gudder, S. (2011). Quantum measures and integrals. Mathematics Preprint Series. Retrieved from https://digitalcommons.du.edu/math_preprints/39

## Comments

The final version of this article published in Reports on Mathematical Physics is available online at: https://doi-org.du.idm.oclc.org/10.1016/S0034-4877(12)60019-6