  Photons propagating in strong magnetic fields are subject to a phenomenon
called the "vacuum birefringence" where refractive indices of two physical
modes both deviate from unity and are different from each other. We compute the
vacuum polarization tensor of a photon in a static and homogeneous magnetic
field by utilizing Schwinger's proper-time method, and obtain a series
representation as a result of double integrals analytically performed with
respect to proper-time variables. The outcome is expressed in terms of an
infinite sum of known functions which is plausibly interpreted as summation
over all the Landau levels of fermions. Each contribution from infinitely many
Landau levels yields a kinematical condition above which the contribution has
an imaginary part. This indicates decay of a sufficiently energetic photon into
a fermion-antifermion pair with corresponding Landau level indices. Since we do
not resort to any approximation, our result is applicable to the calculation of
refractive indices in the whole kinematical region of a photon momentum and in
any magnitude of the external magnetic field.
