A five-dimensional perspective on the Klein-Gordon equation

We discuss the Klein-Gordon (KG) equation using a 5D space-time approach. We explicitly show that the KG equation in flat space-time admits a consistent probabilistic interpretation with positively defined probability density. However, the probabilistic interpretation is not covariant. In the non-relativistic limit, the formalism reduces naturally to that of the Schr\"odinger equation. We further discuss other interpretations of the KG equation (and their non-relativistic limits) resulting from the 5D space-time picture. Finally, we apply our results to the problem of hydrogenic spectra and calculate the canonical sum of the hydrogenic atom.