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We consider the Dirac equation coupled to an external electromagnetic fieldin curved four-dimensional spacetime with a given timelike worldline $\gamma$representing a classical clock. We use generalised Fermi normal coordinates ina tubular neighbourhood of $\gamma$ and expand the Dirac equation up to, andincluding, the second order in the dimensionless parameter given by the ratioof the geodesic distance to the radii defined by spacetime curvature, linearacceleration of $\gamma$, and angular velocity of rotation of the employedspatial reference frame along $\gamma$. With respect to the time measured bythe clock $\gamma$, we compute the Dirac Hamiltonian to that order. On top ofthis `weak-gravity' expansion we then perform a post-Newtonian expansion up to,and including, the second order of $1/c$, corresponding to a `slow-velocity'expansion with respect to $\gamma$. As a result of these combined expansions wegive the weak-gravity post-Newtonian expression for the Pauli Hamiltonian of aspin-half particle in an external electromagnetic field. This extends andpartially corrects recent results from the literature, which we discuss andcompare in some detail.

Geometric post-Newtonian description of massive spin-half particles in curved spacetime