DECOHERENCE CAUSED BY TOPOLOGY IN A TIME-MACHINE SPACETIME

Nonrelativistic quantum theory of noninteracting particles in the spac etime containing a region with closed time-like curves (time-machine s pacetime) 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 e volution are classified by the number n defined as the number of times the particle returns back to its past. It corresponds also to the top ological class P-n of trajectories of a particle. The evolutions corre sponding to different values of n are noncoherent. The amplitudes corr esponding to such evolutions may not be superposed. Instead of one evo lution operator, as in the conventional (coherent) description, evolut ion of the particle is described by a family U-n of partial evolution operators. This is done in analogy with the formalism of quantum theor y of measurements, but with essential new features in the dischronal r egion (the region with closed time-like curves) of the time-machine sp acetime. Partial evolution operators U-n are equal to integrals K-n ov er the classes of paths P-n if the evolution begins and ends in the ch ronal regions. If the evolution begins or/and ends in the dischronal r egion, the integral K-n over the class P-n should be multiplied by a c ertain projector to give the partial evolution operator U-n. Thus defi ned partial evolution operators possess the property of generalized un itarity Sigma(p)U(n)(dagger)U(n) = 1 and multiplicativity U-m(t '', t' ) U-n(t', t) = U-m+n(t '', t). In the last equation however one of the numbers m or n (or both) must be equal to zero. Therefore, the part o f evolution containing repeated returning backward in dime cannot be f actorized: all backward passages of the particle have to be considered as a single act, that cannot be presented as gradually step bg step, passing through ''causal loops.'' The(generalized) multiplicativity an d unitarity take place for arbitrary time intervals including (i) prop agating in initial and final chronal regions (containing no time-like closed curves) or from the initial chronal region to the final one, an d (ii) propagating within the time machine (in the dischronal region), from the time machine to the final chronal region or from the initial chronal region to the time machine.