Quantisations, Representations of Lie Algebras related to Diffeomorphism Groups and Nonlinear Transformations

Abstract:
Quatum Theory is intrinsically linear and very successful. However, nonlinear extensions are of physical and mathematical interest. A method - BOREL quantisation - is presented which is based (non relativistic case) on the classification of linear representations of the kinematical algebra through self adjoint operators and a generic introduction of time. The result is a family of nonlinear Schroedinger equations (DG equations) for pure one-particle states. We explain the backgound of this nonlinearity: Unitary inequivalent rpresentations of the kinematical algebra are transformed into each other through nonlinear transformations. This transformation can be used to nonlinear version not only for the Hamiltonian but for all quantum mechanical observables.