  Non-local boundary conditions, such as the Atiyah-Patodi-Singer (APS)
conditions, for Dirac operators on Riemannian manifolds are well understood
while not much is known for such operators on spacetimes with timelike
boundary. We define a class of Lorentzian boundary conditions that are local in
time and non-local in the spatial directions and show that they lead to a
well-posed Cauchy problem for the Dirac operator. This applies in particular to
the APS conditions imposed on each level set of a given Cauchy temporal
function.
