  Calculations of reaction rates for the third-order QED process of photon
splitting in strong magnetic fields traditionally have employed either the
effective Lagrangian method or variants of Schwinger's proper-time technique.
Recently, Mentzel, Berg and Wunner (1994) presented an alternative derivation
via an S-matrix formulation in the Landau representation. Advantages of such a
formulation include the ability to compute rates near pair resonances above
pair threshold. This paper presents new developments of the Landau
representation formalism as applied to photon splitting, providing significant
advances beyond the work of Mentzel et al. by summing over the spin quantum
numbers of the electron propagators, and analytically integrating over the
component of momentum of the intermediate states that is parallel to field. The
ensuing tractable expressions for the scattering amplitudes are satisfyingly
compact, and of an appearance familiar to S-matrix theory applications. Such
developments can facilitate numerical computations of splitting considerably
both below and above pair threshold. Specializations to two regimes of interest
are obtained, namely the limit of highly supercritical fields and the domain
where photon energies are far inferior to that for the threshold of
single-photon pair creation. In particular, for the first time the
low-frequency amplitudes are simply expressed in terms of the Gamma function,
its integral and its derivatives. In addition, the equivalence of the
asymptotic forms in these two domains to extant results from effective
Lagrangian/proper-time formulations is demonstrated.
