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This article gives global microlocalisation constructions for normallyhyperbolic operators on globally hyperbolic spacetimes on vector bundles ingeometric terms. As an application, this is used to generalise theDuistermaat-H\"ormander construction of Feynman propagators, thereforeincorporating the most important non-scalar geometric operators. It is shownthat for normally hyperbolic operators that are selfadjoint with respect to ahermitian bundle metric, the Feynman propagators can be constructed to satisfya positivity property that reflects the existence of Hadamard states in quantumfield theory on curved spacetimes. We also give a more direct construction ofthe Feynman propagators for Dirac-type operators on a globally hyperbolicspacetime. Even though the natural bundle metric on spinors is notpositive-definite, in this case, we can give a direct microlocal constructionof a Feynman propagator that satisfies positivity.

On microlocalization and the construction of Feynman propagators for normally hyperbolic operators