Electromagnetic response in strong magnetic fields. General formalism and vacuum polarization for parallel propagation

A unified formalism appropriate for calculating the electromagnetic response of both an electron gas and the vacuum in an arbitrarily strong magnetic field is developed. The particular case of propagation parallel to the magnetic field in vacuum is discussed in detail in this paper and the full expression for the corresponding ${\Pi }_{\mu \nu }(\overrightarrow{\mathrm{k}}, \omega )$ is obtained. The pole structures of the dynamical polarizabilities related to ${\Pi }_{\mu \nu }(\overrightarrow{0}, \omega )$ are examined and the existence of a singlet longitudinal massive photon inferred. The static, $k$-dependent, electric and magnetic polarizabilities of the vacuum are evaluated from ${\Pi }_{\mu \nu }(\overrightarrow{\mathrm{k}}, 0)$.