I have seen work with introducing relativistic corrections to the momentum term of the Hamiltonian by taking higher order terms in the Taylor series, but do people work with $H = \sqrt{P^2 c^2 - m^2 c^4} - mc^2$ where $P$ is the ordinary spatial momentum operator? If not, why isn't this an interesting/useful case?

Note: of course this is just the free particle case. This is the simplest case, which is why I'm asking about it.