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Coordinate-space representation of a charged scalar particle propagator in a constant magnetic field expanded as a sum over the Landau levels

A coordinate-space representation for a charged scalar particle propagator in a constant magnetic field was obtained as a series over the Landau levels. Using the recently developed modified Fock-Schwinger method, an intermediate expression was constructed and symmetrized, thus, allowing for factorization of the series terms into two factors. The first one, a sum of Bessel functions, depends on time and $z$-coordinate, where the $z$-axis is chosen to be a direction of the magnetic field, and has a structure similar to the propagator of a free field. The second one, a product of a Laguerre polynomial and a damping exponential, depends on $x,y$-coordinates, which form a plane perpendicular to the direction of the magnetic field, and ensures the localized propagation in the $x,y$-plane.