The exact vacuum polarization 4-tensor in the presence of a static magnetic field is calculated by a method due to Svetozarova and Tsytovich (1962). The Hermitian part of the tensor is explicitly renormalized and the anti-Hermitian part of the tensor is found to be finite and gauge invariant. An alternative method for calculating the vacuum polarization tensor is developed using the relation between the anti-Hermitian part of the tensor and the probability of emission and absorption of a photon by an electron in a static magnetic field.
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Melrose et al. (1977) studied this question.
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