The Loss of Unitarity in the Vicinity of a Time Machine

We construct the propagator of a non-relativistic non-interacting particle in a flat spacetime in which two regions have been identified. This corresponds to the simplest "time machine". We show that while completeness is lost in the vicinity of the time machine it holds before the time machine appears and it is recovered afterwards. Unitarity, however, is not satisfied anywhere. We discuss the implications of these results and their relationship to the loss of unitarity in black hole evaporation.