Relativistic correction to the magnetic moment of the charged lepton

We derive a general relativistic Hamiltonian valid for both bound and scattering systems by reducing the four-component Dirac equation to a two-component Dirac-Pauli form. Unlike conventional approaches, our formulation includes first-order relativistic corrections in a compact, gauge-consistent expression applicable to arbitrary electromagnetic fields—including non-uniform and time-dependent configurations. As an application, we compute the O(α²) relativistic correction to the Landé g-factor in hydrogen-like atoms, revealing a novel mⱼ²-dependent term that generalizes the Breit result. This correction is experimentally testable in Penning trap spectroscopy. We further show that relativistic effects become comparable to QFT corrections in highly charged ions where Z ~ 1/√α