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

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.