On the Hermitian projective line as a home for the geometry of Quantum Theory

TL;DR

This paper explores the use of Hermitian projective lines as a geometric framework for Quantum Theory, emphasizing their connection with associative involutive algebras and related algebraic structures.

Contribution

It refines previous proposals by analyzing structural features linking quantum physics to Hermitian projective geometries and introduces their algebraic foundations.

Findings

Hermitian projective lines relate to associative involutive algebras

Connections established with Jordan-Lie and Lie-Jordan algebras

Framework supports geometric formulation of Quantum Theory

Abstract

In the paper "Is there a Jordan geometry underlying quantum physics?" (Int. J. Theor. Phys., to appear; arXiv:0801.3069 [math-ph]), generalized projective geometries have been proposed as a framework for a geometric formulation of Quantum Theory. In the present note, we refine this proposition by discussing further structural features of Quantum Theory: the link with associative involutive algebras, and with Jordan-Lie and Lie-Jordan algebas. The associated geometries are (Hermitian) projective lines over an associative algebra; their axiomatic definition and theory will be given in subsequent work with M. Kinyon.