  We present a second-quantized field theory of massive spin one-half particles
or antiparticles in the presence of a weak gravitational field treated as a
spin two external field in a flat Minkowski background. We solve the
difficulties which arise from the derivative coupling and we are able to
introduce an interaction picture. We derive expressions for the scattering
amplitude and for the outgoing spinor to first-order. In several appendices,
the link with the canonical approach in General Relativity is established and a
generalized stationary phase method is used to calculate the outgoing spinor.
We show how our expressions can be used to calculate and discuss phase shifts
in the context of matter-wave interferometry (especially atom or antiatom
interferometry). In this way, many effects are introduced in a unified
relativistic framework, including spin-gravitation terms: gravitational red
shift, Thomas precession, Sagnac effect, spin-rotation effect, orbital and spin
Lense-Thirring effects, de Sitter geodetic precession and finally the effect of
gravitational waves. A new analogy with the electromagnetic interaction is
pointed out.
