Geometric Formulation of Nonlinear Quantum Mechanics for Density Matrices

Proposals for nonlinear extenstions of quantum mechanics are discussed. Two different concepts of “mixed state” for any nonlinear version of quantum theory are introduced: (i) genuine mixture corresponds to operational “mixing” of different ensembles, and (ii) a mixture described by single density matrix without having a canonical operational possibility to pick out its specific convex decomposition is called here an elementary mixture. Time evolution of a class of nonlinear extensions of quantum mechanics is introduced. Evolution of an elementary mixture cannot be generally given by evolutions of components of its arbitrary convex decompositions. The theory is formulated in a “geometric form”: It can be considered as a version of Hamiltonian mechanics on infinite dimensional space of density matrices. A quantum interpretation of the theory is sketched.