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

TL;DR

This paper derives a coordinate-space propagator for a charged scalar particle in a constant magnetic field, expressed as a sum over Landau levels, using a modified Fock-Schwinger method to factorize and analyze the series.

Contribution

It introduces a novel series representation of the propagator in coordinate space, utilizing a symmetrized factorization approach with Bessel and Laguerre functions.

Findings

Series representation of the propagator over Landau levels.

Factorization into time/z-dependent and x,y-dependent parts.

Ensures localized propagation in the plane perpendicular to the magnetic field.

Abstract

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.