Feynman Propagator for a Spinless Relativistic Particle in Feshbach–Villars Representation

In this paper we used the Feynman path integral technique to calculate propagators for the Feshbach–Villars equation and the generalized Feshbach–Villars equation. We calculate these propagators for three cases: for a free particle, for a particle in a constant magnetic field and in an inhomogeneous magnetic field. Some special relations between Pauli matrices, which appear in the Feshbach–Villars equation, allow us to express the Feynman propagator for the free particle in a compact form, and in the one-dimensional case we can calculate this propagator in explicit form in terms of Bessel functions. For the second case, i.e. for a particle in a constant magnetic field, using a relation between the Feshbach–Villars Hamiltonian and a certain associated Schrödinger Hamiltonian, which appears in the Feshbach–Villars representation, we are able to express the path integral propagator for Feshbach–Villars equation in explicit form by means of Landau levels for the associated Schrödinger Hamiltonian. In the third case, i.e. for inhomogeneous magnetic field, in the same way as in the second case, we can express the path integral propagator for Feshbach–Villars equation in the explicit form in terms of eigenfunctions of problem.