  We discuss the non-relativistic limit of quantum field theory in an inertial
frame, in the Rindler frame and in the presence of a weak gravitational field,
highlighting and clarifying several subtleties. We study the following topics:
(a) While the action for a relativistic free particle is invariant under the
Lorentz transformation, the corresponding action for a non-relativistic free
particle is not invariant under the Galilean transformation, but picks up extra
contributions at the end points. This leads to an extra phase in the
non-relativistic wave function under a Galilean transformation, which can be
related to the rest energy of the particle even in the non-relativistic limit.
(b) We show how the solution to the generally covariant Klein-Gordon equation
in a non-inertial frame, which has a time-dependent acceleration, reduces to
the quantum mechanical wave function in the presence of an appropriate
(time-dependent) gravitational field, in the non-relativistic limit. The extra
phase acquired by the non-relativistic wave function in an accelerated frame
actually arises from the gravitational time dilation and survives in the
non-relativistic limit. (c) We provide a detailed description of the
non-relativistic limit of the Feynman propagator in a weak gravitational field,
and discuss related issues. [Abridged Abstract]
