XXVII Workshop on Geometric Methods in Physics 29.06-05.07.2008

Wolfgang Bertram

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.