A real space split operator method for the Klein–Gordon equation

The Klein–Gordon equation is a Lorentz invariant equation of motion for spinless particles. We propose a real space split operator method for the solution of the time-dependent Klein–Gordon equation with arbitrary electromagnetic fields. Split operator methods for the Schrödinger equation and the Dirac equation typically operate alternately in real space and momentum space and, therefore, require the computation of a Fourier transform in each time step. However, the fact that the kinetic energy operator K^ in the two-component representation of the Klein–Gordon equation is a nilpotent operator, that is K^2=0, allows us to implement the split operator method for the Klein–Gordon equation entirely in real space. Consequently, the split operator method for the Klein–Gordon equation does not require the computation of a Fourier transform and may be parallelized efficiently by domain decomposition.