  Non-relativistic quantum theory of non-interacting particles in the spacetime
containing a region with closed time-like curves (time-machine spacetime) is
considered with the help of the path-integral technique. It is argued that, in
certain conditions, a sort of superselection may exist for evolution of a
particle in such a spacetime. All types of evolution are classified by the
number $n$ defined as the number of times the particle returns back to its
past. It corresponds also to the topological class of trajectories of the
particle. The evolutions corresponding to different values of $n$ are
non-coherent. The amplitudes corresponding to such evolutions may not be
superposed. Instead of one evolution operator, as in the conventional
(coherent) description, evolution of the particle is described by a family
$U_n$ of partial evolution operators. This is done in analogy with the
formalism of quantum theory of measurements, but with essential new features in
the dischronal region (the region with closed time-like curves) of the
time-machine spacetime. Partial evolution operators $U_n$ are equal to
integrals over the topological classes of paths if the evolution begins and
ends in the chronal regions. If the evolution begins or/and ends in the
dischronal region, the integral over the topological class should be multiplied
by a certain projector to give the partial evolution operator $U_n$. Thus
defined partial evolution operators possess the property of generalized
unitarity and multiplicativity. The part of evolution containing repeated
returning backward in time cannot be factorized: all backward passages of the
particle have to be considered as a single act, that cannot be presented as
gradually, step by step, passing through `causal loops'.
