Quantum propagator for a nonrelativistic particle in the vicinity of a time machine

We study the propagator of a nonrelativistic, noninteracting particle in any nonrelativistic ''time-machine'' spacetime of the following type: an external, chronal spacetime in which two spatial regions V- at time t- and V+ at time t+ are connected by two temporal wormholes, one leading from the past side of V- to the future side of V+ and the other from the past side of V+ to the future side of V-. We express the propagator explicitly in terms of those for the chronal spacetime and for the two wormholes; and from that expression we show that the propagator satisfies completeness and unitarity in the initial and final ''chronal regions'' (regions without closed timelike curves) and its propagation from the initial region to the final region is unitary. However, within the time machine it satisfies neither completeness nor unitarity. We also give an alternative proof of initial-region-to-final-region unitarity based on a conserved current and Gauss's theorem. This proof can be carried over without change to most any nonrelativistic time-machine spacetime and it is valid as long as the particle is not interacting with itself or any other quantum particle; it can, however, interact with an external field (gravitational or otherwise). This result is the nonrelativistic version of a theorem by Friedman, Papastamatiou, and Simon, which says that for a free scalar field quantum-mechanical unitarity follows from the fact that the classical evolution preserves the Klein-Gordon inner product.