  Generalized Fourier transformation between the position and the momentum
representation of a quantum state is constructed in a coordinate independent
way. The only ingredient of this construction is the symplectic (canonical)
geometry of the phase-space: no linear structure is necessary. It is shown that
the "fractional Fourier transform" provides a simple example of this
construction. As an application of this techniques we show that for any linear
Hamiltonian system, its quantum dynamics can be obtained exactly as the lift of
the corresponding classical dynamics by means of the above transformation.
Moreover, it can be deduced from the free quantum evolution. This way new,
unknown symmetries of the Schr\"odinger equation can be constructed. It is also
argued that the above construction defines in a natural way a connection in the
bundle of quantum states, with the base space describing all their possible
representations. The non-flatness of this connection would be responsible for
the non-existence of a quantum representation of the complete algebra of
classical observables.
