Impossibility of the strong magnetic fields generation in an electron-positron plasma

We examine the issue of whether a magnetic field can be amplified in a background matter consisting of electrons and positrons self-interacting within the Fermi model. For this purpose, we compute the antisymmetric contribution to the photon polarization tensor in this matter having nonzero temperature and chemical potential. It is shown that this contribution is vanishing in the static limit. Then we study a particular case of a degenerate relativistic electron gas present in a magnetar. We demonstrate that a seed magnetic field is attenuated in this case. Thus, contrary to the recent claim, we show that there is no magnetic field instability in such a system, which can lead to the magnetic field growth. Therefore, the recently proposed mechanism cannot be used for the explanation of strong magnetic fields of magnetars.