Generalized quantum electrodynamics: one-loop correction

In this paper, we give an update on divergent problems concerning the radiative corrections of quantum electrodynamics in $(3+1)$ dimensions. In doing so, we introduce a geometric adaptation for the covariant photon propagator by including a higher derivative field. This derivation, so-called generalized quantum electrodynamics, is motivated by the stability and unitarity features. This theory provides a natural and self-consistent extension of the quantum electrodynamics by enlarging the space parameter of spinor-gauge interactions. In particular, Haag$'$s theorem undermines the perturbative characterization of the interaction picture due to its inconsistency on quantum field theory foundations. To circumvent this problem, we develop our perturbative approach in the Heisenberg picture and use it to investigate the behavior of the operator current at $1$-loop. We find the $2$- and $3$-point correlation functions are ultraviolet finite, electron self-energy and vertex corrections, respectively. On the other hand, we also explain how the vacuum polarization remains ultraviolet divergent only at $e^2$ order. Finally, we evaluate the anomalous magnetic moment, which allows us to specify a lower bound value for the Podolsky parameter.