  We perform the canonical quantization of a relativistic spinless particle
moving in a curved and static spacetime. We show that the classical theory
already describes at the same time both particle and antiparticle. The analyses
involves time-depending constraints and we are able to construct the
two-particle Hilbert space. The requirement of a static spacetime is necessary
in order to have a well defined Schr\"odinger equation and to avoid problems
with vacuum instabilities. The severe ordering ambiguities we found are in
essence the same ones of the well known non-relativistic case.
