Pascual Jordan's Ansatz revisited
There have been several propositions for a geometric and essentially
non-linear formulation of quantum mechanics (most of them following the fundamental paper "Geometrization of
Quantum Mechanics" by T.W.B. Kibble (1979)).
From a purely mathematical
point of view, the point of view of Jordan algebra theory
might turn out to 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 will concentrate on the mathematical description of the Jordan
geometry and raise some questions concerning possible relations
with foundational issues of quantum theory.