  We study the non-relativistic limit of quantum fields for an inertial and a
non-inertial observer. We show that non-relativistic particle states appear as
a superposition of relativistic and non-relativistic particles in different
frames. Hence, the non-relativistic limit is frame-dependent. We detail this
result when the non-inertial observer has uniform constant acceleration. Only
for low accelerations, the accelerated observer agrees with the inertial frame
about the non-relativistic nature of particles locally. In such a
quasi-inertial regime, both observers agree about the number of particles
describing quantum field states. The same does not occur when the acceleration
is arbitrarily large (e.g., the Unruh effect). We furthermore prove that wave
functions of particles in the inertial and the quasi-inertial frame are
identical up to the coordinate transformation relating the two frames.
