Convex and Sequential Effect Algebras

TL;DR

This paper develops a mathematical framework for quantum mechanics using convex effect algebras, characterizing classical and quantum cases, and introduces a sequential product to analyze conditional probabilities.

Contribution

It introduces convex and sequential effect algebras, providing a new formalism that captures quantum and classical structures with a focus on states, effects, and their sequential composition.

Findings

Characterization of classical and quantum convex effect algebras

Representation of quantum effect algebras on complex Hilbert spaces

Introduction of a sequential product for effects to study conditional probabilities

Abstract

We present a mathematical framework for quantum mechanics in which the basic entities and operations have physical significance. In this framework the primitive concepts are states and effects and the resulting mathematical structure is a convex effect algebra. We characterize the convex effect algebras that are classical and those that are quantum mechanical. The quantum mechanical ones are those that can be represented on a complex Hilbert space. We next introduce the sequential product of effects to form a convex sequential effect algebra. This product makes it possible to study conditional probabilities and expectations.