Smearing of Observables and Spectral Measures on Quantum Structures

TL;DR

This paper investigates the structure of observables in quantum effect algebras, demonstrating that all observables can be viewed as smeared versions of sharp observables and establishing the existence of spectral measures for elements.

Contribution

It introduces a framework showing that all observables are smearing of sharp observables and proves the existence of spectral measures within monotone $\sigma$-complete effect algebras.

Findings

Every observable is a smearing of a sharp observable.

Spectral measures exist for all elements of the effect algebra.

The results apply to effect algebras with the Riesz Decomposition Property.

Abstract

An observable on a quantum structure is any $\sigma$-homomorphism of quantum structures from the Borel $\sigma$-algebra of the real line into the quantum structure which is in our case a monotone $\sigma$-complete effect algebras with the Riesz Decomposition Property. We show that every observable is a smearing of a sharp observable which takes values from a Boolean $\sigma$-subalgebra of the effect algebra, and we prove that for every element of the effect algebra there is its spectral measure.