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We present a second-quantized field theory of massive spin one-half particlesor antiparticles in the presence of a weak gravitational field treated as aspin two external field in a flat Minkowski background. We solve thedifficulties which arise from the derivative coupling and we are able tointroduce an interaction picture. We derive expressions for the scatteringamplitude and for the outgoing spinor to first-order. In several appendices,the link with the canonical approach in General Relativity is established and ageneralized stationary phase method is used to calculate the outgoing spinor.We show how our expressions can be used to calculate and discuss phase shiftsin the context of matter-wave interferometry (especially atom or antiatominterferometry). In this way, many effects are introduced in a unifiedrelativistic framework, including spin-gravitation terms: gravitational redshift, Thomas precession, Sagnac effect, spin-rotation effect, orbital and spinLense-Thirring effects, de Sitter geodetic precession and finally the effect ofgravitational waves. A new analogy with the electromagnetic interaction ispointed out.Comment: 32 pages, LaTex, needs rotating and lscape usepackages, to appear in  C. L\"ammerzahl, C.W.F. Everitt, F.W. Hehl (eds.): ``Gyros, Clocks, and  Interferometers: Testing Relativistic Gravity in Space'', Springer-Verlag,  Berlin 200

Relativistic Phase Shifts for Dirac Particles Interacting with Weak Gravitational Fields in Matter—Wave Interferometers