ON NONLINEAR QUANTUM MECHANICS

Pavel B\'ona

Abstract. There is a ``natural" formal extension of quantum
mechanics (QM) containing, besides the traditional (linear) QM of
systems with finite number of degrees of freedom, its nonlinear
generalization, as well as classical mechanics (CM), and also
some other (``quasiclassical") theories. A formal expression of
this extension has the form of a Poisson system on an
infinite-dimensional manifold. The essentially statistical nature
of interpretation of this theory requires introduction of
description of physical quantities (``observables") by projection
measure valued functions on the state space and their
interpretation. A brief review of parts of this theory is given.
Some open problems are mentioned.

Keywords. Projective Hilbert space, K\"ahler structure, Poisson bracket,
Fr\`echet differential, spectral measure, symmetry, coadjoint action,
momentum map.

MS classification. 53C55, 58B20, 81B99, 81C25.