# A simple and quantum-mechanically motivated characterization of the formally real Jordan algebras

## TL;DR

This paper explores the mathematical structure of quantum mechanics by characterizing formally real Jordan algebras through simple, quantum-mechanically motivated postulates, aiming to clarify their role in the foundations of quantum theory.

## Contribution

It provides a new characterization of formally real Jordan algebras based on minimal postulates inspired by quantum mechanics, addressing gaps in the reconstruction of quantum theory.

## Key findings

- Nearly reconstructs finite-dimensional quantum theory from four postulates
- Identifies limitations in current reconstruction excluding 2D Hilbert space and exceptional Jordan algebras
- Discusses potential physical generalizations of the quantum logic framework

## Abstract

Quantum theory's Hilbert space apparatus in its finite-dimensional version is nearly reconstructed from four simple and quantum-mechanically motivated postulates for a quantum logic. The reconstruction process is not complete, since it excludes the two-dimensional Hilbert space and still includes the exceptional Jordan algebras, which are not part of the Hilbert space apparatus. Options for physically meaningful potential generalizations of the apparatus are discussed.