Why Do the Quantum Observables Form a Jordan Operator Algebra?

TL;DR

This paper presents a new axiomatic approach to quantum mechanics, deriving the Jordan algebra structure of observables from statistical postulates related to conditional probabilities, connecting it to the Hilbert space formalism.

Contribution

It shows how the Jordan algebra structure of quantum observables can be derived from statistical properties, providing a novel axiomatic foundation for quantum mechanics.

Findings

Derivation of Jordan algebra structure from conditional probability properties

Connection between Jordan algebras and C*-algebras established

Extension of quantum mechanics to include types II and III von Neumann algebras

Abstract

The Jordan algebra structure of the bounded real quantum observables was recognized already in the early days of quantum mechanics. While there are plausible reasons for most parts of this structure, the existence of the distributive nonassociative multiplication operation is hard to justify from a physical or statistical point of view. Considering the non-Boolean extension of classical probabilities, presented in a recent paper, it is shown in this paper that such a multiplication operation can be derived from certain properties of the conditional probabilities and the observables, i.e., from postulates with a clear statistical interpretation. The well-known close relation between Jordan operator algebras and C*-algebras then provides the connection to the quantum-mechanical Hilbert space formalism, thus resulting in a novel axiomatic approach to general quantum mechanics that includes the types II and III von Neumann algebras.