  Classically, the dynamics in a non-globally hyperbolic spacetime is ill
posed. Previously, a prescription was given for defining dynamics in static
spacetimes in terms of a second order operator acting on a Hilbert space
defined on static slices. The present work extends this result by giving a
similar prescription for defining dynamics in stationary spacetimes obeying
certain mild assumptions. The prescription is defined in terms of a first order
operator acting on a different Hilbert space from the one used in the static
prescription. It preserves the important properties of the earlier one: the
formal solution agrees with the Cauchy evolution within the domain of
dependence, and smooth data of compact support always give rise to smooth
solutions. In the static case, the first order formalism agrees with second
order formalism (using specifically the Friedrichs extension). Applications to
field quantization are also discussed.
