Relativistic Hamiltonian with square root in the path integral formalism

Quantization of the relativistic Hamiltonian $H=\sqrt{c^2({\mathbf p}-{\mathbf A})^2+{(mc^2+V)}^2}+A_0$ is discussed. A new method is proposed for approximations of the corresponding path integral. It is shown that calculation of this integral does not give the Green's function of the Klein–Gordon equation but leads to the propagator of a scalar field in the twocomponent Feshbach–Villars formalism. The difficulties in calculating path integrals in the presence of nontrivial external fields are discussed.