Decoherence Caused by Topology in a Time-Machine Spacetime

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'.