  The paper consists of two parts. In the first part Schroedinger's equation
for a charged quantum particle in a Galilei-Newton curved space-time is derived
in a fully geometrical way. Gravitational and electromagnetic fields are coded
into space metric and space-time connection. The fundamental geometrical object
is a quantum connection in a Hermitian line bundle over the 7-dimensional jet
space of 3-velocities. The secondary object is the bundle of Hilbert spaces
over absolute time. Time appears as a superselection quantity while Shroedinger
equation is interpreted as parallel transport in this bundle. In the second
part the problem of measurement in quantum theory is discussed as a part of a
more general problem of coupling between quantum and classical systems. The
standard framework of quantum theory is extended so as to allow for dynamical
central observables within dissipative dynamics. It is shown that within this
approach one obtains not only Liouville equation that describes statistical
ensembles, but also a piecewise-deterministic random process describing
sequences of "events" that can be monitored by a continuous observation of the
single, coupled classical system. It also describes "quantum jumps" or "wave
packet reductions" that accompany these events. Two example are worked out in
some details. The last one deals with the problem oof "how to determine the
wave function ?".
