  This article reviews the microlocal construction of Feynman propagators for
normally hyperbolic operators acting on vector bundles over globally hyperbolic
spacetimes and its consequences. It is shown that for normally hyperbolic
operators that are selfadjoint with respect to a hermitian bundle metric, the
Feynman propagators can be constructed to satisfy a positivity property that
reflects the existence of Hadamard states in quantum field theory on curved
spacetimes. We also give a more direct construction of the Feynman propagator
for the Dirac operator on a globally hyperbolic spacetime. Even though the
natural bundle metric on spinors is not positive-definite, in this case we can
give a direct microlocal construction of a Feynman propagator that satisfies
positivity.
