Relativistic two-body Coulomb-Breit Hamiltonian in an external weak gravitational field

A construction of the Coulomb-Breit Hamiltonian for a pair of fermions, considered as a quantum two-body system, immersed in an arbitrary background gravitational field described by Einsteinʼs General Relativity is presented. Working with Fermi normal coordinates for a freely falling observer in a spacetime region where there are no background sources and ignoring the gravitational back-reaction of the system, the effective Coulomb-Breit Hamiltonian is obtained starting from the S-matrix element corresponding to the one-photon exchange between the charged fermionic currents. The contributions due to retardation are considered up to order (v/c)2 and they are subsequently written as effective operators in the relativistic quantum mechanical Hilbert space of the system. The final Hamiltonian includes effects linear in the curvature and up to order (v/c)2.