Is there a Jordan geometry underlying quantum physics?

TL;DR

This paper explores the mathematical structure of Jordan geometry as a potential underlying framework for quantum physics, highlighting its relation to Jordan algebras and foundational quantum issues.

Contribution

It provides a detailed mathematical description of Jordan geometry and discusses its possible connections to the foundations of quantum theory.

Findings

Jordan geometry relates to the algebra of quantum observables

Potential links between Jordan geometry and quantum foundations are proposed

Mathematical framework may offer new insights into quantum structure

Abstract

There have been several propositions for a geometric and essentially non-linear formulation of quantum mechanics. From a purely mathematical point of view, the point of view of Jordan algebra theory might give new strength to such approaches: there is a ``Jordan geometry'' belonging to the Jordan part of the algebra of observables, in the same way as Lie groups belong to the Lie part. Both the Lie geometry and the Jordan geometry are well-adapted to describe certain features of quantum theory. We concentrate here on the mathematical description of the Jordan geometry and raise some questions concerning possible relations with foundational issues of quantum theory.