Nonrelativistic quantum theory of noninteracting 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 [Formula: see text] of trajectories of a particle. The evolutions corresponding to different values of n are noncoherent. 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 Un 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 Un are equal to integrals Kn over the classes of paths [Formula: see text] if the evolution begins and ends in the chronal regions. If the evolution begins or/and ends in the dischronal region, the integral Kn over the class [Formula: see text] should be multiplied by a certain projector to give the partial evolution operator Un. Thus defined partial evolution operators possess the property of generalized unitarity ∑nUn†Un=1 and multiplicativity Um(t″, t′)Un(t′, t)=Um+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 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.” The (generalized) multiplicativity and unitarity take place for arbitrary time intervals including (i) propagating in initial and final chronal regions (containing no time-like closed curves) or from the initial chronal region to the final one, and (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.
An Algebraic Approach to the Quantum Theory of Measurements
