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 the particle and the antiparticle. The analyses involve time-dependent 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 Schrödinger equation and to avoid problems with vacuum instabilities. The severe ordering ambiguities we found are essentially the same ones as the well known non-relativistic case.