Vacuum polarization in parallel homogeneous electric and magnetic fields

Abstract

The lowest-order mass operator for a real or virtual photon propagating in an arbitrary direction with respect to parallel homogeneous external electric and magnetic fields is calculated to all orders in the fields. The results are presented in a double-integral form which unfortunately does not lend itself to convenient evaluation in terms of familiar functions. However, in the weak-field approximation, $\left(\frac{e}{m^2}\right){(H^2+E^2)}^{\frac{1}{2}}\ll 1$, the birefringent properties of the vacuum may be inferred by calculating the indices of refraction and the absorption coefficient for the eigenmodes of a photon propagating in any direction in this medium. The numerical results turn out to be equivalent to the pure magnetic field calculation modified by the formal replacement $H\to {(H^2+E^2)}^{\frac{1}{2}}$. The polarization eigenmodes are a rotated version of the usual modes obtained in the pure magnetic field case.