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Photons propagating in strong magnetic fields are subject to a phenomenoncalled the "vacuum birefringence" where refractive indices of two physicalmodes both deviate from unity and are different from each other. We compute thevacuum polarization tensor of a photon in a static and homogeneous magneticfield by utilizing Schwinger's proper-time method, and obtain a seriesrepresentation as a result of double integrals analytically performed withrespect to proper-time variables. The outcome is expressed in terms of aninfinite sum of known functions which is plausibly interpreted as summationover all the Landau levels of fermions. Each contribution from infinitely manyLandau levels yields a kinematical condition above which the contribution hasan imaginary part. This indicates decay of a sufficiently energetic photon intoa fermion-antifermion pair with corresponding Landau level indices. Since we donot resort to any approximation, our result is applicable to the calculation ofrefractive indices in the whole kinematical region of a photon momentum and inany magnitude of the external magnetic field.

Vacuum birefringence in strong magnetic fields: (I) Photon polarization tensor with all the Landau levels