Green’s functions under magnetic effects in a nontrivial topology

We compute the propagators of Bose and Dirac fields in four-dimensional Euclidean space in the presence of an external magnetic field, using a hybrid formulation that combines the Ritus and Schwinger methods. This approach yields explicit expressions for both propagators in coordinate and momentum space. By performing a suitable gauge transformation, we eliminate the components of the propagators that break translation invariance. We further derive the Matsubara frequencies of the fields within a toroidal topology. The resulting framework provides a unified mathematical structure that incorporates thermal and finite-volume effects in Green’s functions under various boundary conditions, while simultaneously offering an analytically tractable representation of all Landau levels.