Microcausality in curved space-time

It is well known that in Lorentz-invariant quantum field theories in flat space the commutator of spacelike separated local operators vanishes (microcausality). We provide two different arguments showing that this is a consequence of the causal structure of the classical theory, rather than of Lorentz invariance. In particular, microcausality holds in arbitrary curved space-times, where Lorentz invariance is explicitly broken by the background metric. As illustrated by an explicit calculation on a cylinder this property is rather nontrivial at the level of Feynman diagrams.