Phonon splitting in the magnetised vacuum

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Abstract
An explicit result for the quadratic vacuum polarisation tensor in a static uniform magnetic field is derived using the Gehenian representation of the electron propagator. The formalism of relativistic-quantum plasma physics is used to calculate the probability for a photon to split into two photons in the magnetised vacuum, without approximation in the magnetic field strength or in the photon frequency or wavenumber. The exact effect of photon dispersion on photon splitting is included. It is shown that the probability for photon splitting in both the weak-field and low-frequency limits is greatest when the energy of the initial photon is divided equally between the two final photons. Errors are indicated in earlier results for the box-diagram contribution to photon splitting in the magnetised vacuum.