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 typ e: an external, chronal spacetime in which two spatial regions V- at t ime 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 othe r from the past side of V+ to the future side of V-. We express the pr opagator explicitly in terms of those for the chronal spacetime and fo r the two wormholes; and from that expression we show that the propaga tor satisfies completeness and unitarity in the initial and final ''ch ronal regions'' (regions without closed timelike curves) and its propa gation from the initial region to the final region is unitary. However , within the time machine it satisfies neither completeness nor unitar ity. We also give an alternative proof of initial-region-to-final-regi on unitarity based on a conserved current and Gauss's theorem. This pr oof can be carried over without change to most any nonrelativistic tim e-machine spacetime and it is valid as long as the particle is not int eracting with itself or any other quantum particle; it can, however, i nteract with an external field (garvitational or otherwise). This resu lt is the nonrelativistic version of a theorem by Friedman, Papastamat iou, and Simon, which says that for a free scalar field quantum-mechan ical unitarity follows from the fact that the classical evolution pres erves the Klein-Gordon inner product.