  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.
