The magnetized vacuum has dispersive properties similar to a material medium. Its response may be described by a hierarchy of response tensors, which are functions of B∕Bc. The linear response tensor, referred to as the vacuum polarization tensor, has a relatively simple form, first derived in the 1930s, that applies at frequencies, ω≪2m, well below the pair-creation threshold. In this limit the linear and nonlinear response tensors may be derived from the Heisenberg-Euler Lagrangian, which includes a static electric field as well as a static magnetic field. When the low-frequency approximation is not made, the linear response tensor may be calculated from the Feynman amplitude for the bubble diagram. As in the unmagnetized case, this amplitude diverges, and it needs to be regularized. The magnetic field introduces no new divergences, so that the difference between the tensors for B∕Bc≠0 and their limits for B∕Bc→0 are necessarily divergence-free. The quadratic nonlinear response tensor of the magnetized vacuum is nonzero for B∕Bc≠0, and it allows a three-wave interaction, referred to as photon splitting.
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