Department of Physics,

Institute for Theoretical Physics,

Technical University of Clausthal,

Leibnizstr. 10,

38678 Clausthal-Zellerfeld,

GERMANY

E-mail: asi@pt.tu-clausthal.de

**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.