  We give from first principles the non-relativistic limit of scalar and Dirac
fields in curved spacetime. We aim to find general relativistic corrections to
the quantum theory of particles affected by Newtonian gravity, a regime
nowadays experimentally accessible. We believe that the ever-improving
measurement accuracy and the theoretical interest in finding general
relativistic effects in quantum systems require the introduction of corrections
to the Schr\"{o}dinger-Newtonian theory. We rigorously determine these
corrections by the non-relativistic limit of fully relativistic quantum
theories in curved spacetime. For curved static spacetimes, we show how a
non-inertial observer (equivalently, an observer in the presence of a
gravitational field) can distinguish a scalar field from a Dirac field by
particle-gravity interaction. We study the Rindler spacetime and discuss the
difference between the resulting non-relativistic Hamiltonians. We find that
for sufficiently large acceleration, the gravity-spin coupling dominates over
the corrections for scalar fields, promoting Dirac particles as the best
candidates for observing non-Newtonian gravity in quantum particle
phenomenology.
