On a Jordan-algebraic formulation of quantum mechanics: Hilbert space construction

TL;DR

This paper explores a Jordan-algebraic approach to quantum mechanics, constructing a real Hilbert space and deriving the Schrödinger equation within this framework, offering a novel algebraic perspective.

Contribution

It introduces a Jordan-algebraic Hilbert space construction inspired by the GNS-construction, providing a new algebraic foundation for quantum mechanics.

Findings

Constructed a real Hilbert space from Jordan algebras

Derived a Schrödinger equation in the Jordan-algebraic framework

Established a Jordan-representation of observables on the Hilbert space

Abstract

In this note I discuss some aspects of a formulation of quantum mechanics based entirely on the Jordan algebra of observables. After reviewing some facts of the formulation in the \CS -approach I present a Jordan-algebraic Hilbert space construction (inspired by the usual GNS-construction), thereby obtaining a real Hilbert space and a (Jordan-) representation of the algebra of observables on this space. Taking the usual case as a guideline I subsequently derive a Schr\"odinger equation on this Hilbert space.