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

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.