n a magnetic field, calculated by Shabad[@shabad1; @shabad2] in an external constant magnetic field $\Pi_{\mu\nu}(x,x^{\prime\prime}\vert A^{ext})$ by starting from the electron-positron Green function in the Furry picture, and by using the Schwinger proper time method. The expression obtained was used by Shabad [@shabad2] to investigate the photon dispersion equation in vacuum in presence of an external magnetic field. It was found a strong deviation from the light cone curve near the energy thresholds for pair creation, which suggests that the photon propagation behavior in the external classical magnetic field is strongly influenced by the virtual electron-positron pairs of vacuum near these thresholds, showing a behavior similar to that of a massive particle. These phenomena become especially significant near the critical field $B_c=m_0^2/e
\sim 4,41 \cdot 10^{13}$ Gauss, where $m_0, e$ are respectively the electron mass and charge.

The photon magnetic moment might have astrophysical and cosmological consequences. For instance, photons passing by a strongly magnetized star, would experience an additional shift to the usual gravitational one produced by the star mass.

In presence of an external field the current vector is non vanishing $j(x)_{\mu}=ie Tr\gamma_{\mu} G(x,x|A^{ext}) \neq 0$, where $G(x,x^{\prime}|A^{ext})$ is the electron-positron Green’s function in the external field. By calling the total electromagnetic field by $A^t_{\mu}= A^{ext}_{\mu}+A_{\mu}$, the QED Schwinger-Dyson equation for the photon field $A_\mu(x)$, propagating in the external field $A_\mu(x)^{ext}$ is $$\left[\square
\eta_{\mu\nu}-\partial_\mu\partial_\nu\right]
A^\nu(x)+\int\Pi_{\mu\nu}(x,x^\prime\vert A^{ext})
A^\nu(x^\prime) d^4x^\prime=0,\label{sdpmBF}$$ where $\mu,\nu=1,2,3,4$. The expression (\[sdpmBF\]) is actually the set of Maxwell equations in a neutral polarized vacuum, where the second term corresponds to the approximation of the four-current linear in $A_{\mu}$, where the coefficient is the polarization operator 