  In a recent work the first named author, Levitin and Vassiliev have
constructed the wave propagator on a closed Riemannian manifold $M$ as a single
oscillatory integral global both in space and in time with a distinguished
complex-valued phase function. In this paper, first we give a natural
reinterpretation of the underlying algorithmic construction in the language of
ultrastatic Lorentzian manifolds. Subsequently we show that the construction
carries over to the case of static backgrounds thanks to a suitable reduction
to the ultrastatic scenario. Finally we prove that the overall procedure can be
generalised to any globally hyperbolic spacetime with compact Cauchy surfaces.
As an application, we discuss how, from our procedure, one can recover the
local Hadamard expansion which plays a key role in all applications in quantum
field theory on curved backgrounds.
