Geometric Formulation of Nonlinear Quantum Mechanics for Density Matrices

AbstractProposals for nonlinear extenstions of quantum mechanics are dis-cussed. Two different concepts of “mixed state” for any nonlinear versionof quantum theory are introduced: (i) genuine mixture corresponds to op-erational “mixing” of different ensembles, and (ii) a mixture described bysingle density matrix without having a canonical operational possibilityto pick out its specific convex decomposition is called here an elementarymixture. Time evolution of a class of nonlinear extensions of quantummechanics is introduced. Evolution of an elementary mixture cannot begenerally given by evolutions of components of its arbitrary convex de-compositions. The theory is formulated in a “geometric form”: It canbe considered as a version of Hamiltonian mechanics on infinite dimen-sional space of density matrices. A quantum interpretation of the theoryis sketched. 1 Introduction In popular and well written book [1] on conceptual foundations of quantum me-chanics (QM) there is a subsection [Chap. 9-4, p.278] on nonlinear Schr¨odingerequation inserted into the section entitled