Vacuum polarization tensor in inhomogeneous magnetic fields
  We develop worldline numerical methods, which combine string-inspired with Monte-Carlo techniques, for the computation of the vacuum polarization tensor in inhomogeneous background fields for scalar QED. The algorithm satisfies the Ward identity exactly and operates on the level of renormalized quantities. We use the algorithm to study for the first time light propagation in a spatially varying magnetic field. Whereas a local derivative expansion applies to the limit of small variations compared to the Compton wavelength, the case of a strongly varying field can be approximated by a derivative expansion for the averaged field. For rapidly varying fields, the vacuum-magnetic refractive indices can exhibit a non-monotonic dependence on the local field strength. This behavior can provide a natural limit on the self-focussing property of the quantum vacuum. 
Holger Gies and Lars Roessler
hep-ph hep-th