Klein-Gordon and square-root operator equations for two-spinors and   scalars: perturbation calculations for hydrogen-like systems
  There exists a Klein-Gordon-like equation for a spin-1/2 particle in an electromagnetic field with 2-spinors as wave functions that is a direct consequence of the corresponding Dirac equation. Thus, it reproduces the same binding energies for an electron in a hydrogen atom as the Dirac equation. There is also a square-root equation for 2-spinors which can give the same binding energies up to the forth order of the fine structure constant, inclusively, what will be shown by means of a comparison of the non-relativistic limit plus a first relativistic correction of both equations for 2-spinors within the framework of a perturbation analysis. A parallel will also be drawn to the spin-0 case. 
Tobias Gleim