Bound States and the Ladder Approximation in Positron Annihilation

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Abstract
A discussion is given of the consistency requirements which are relevant to the calculation of the properties of a positron before annihilating in an electron gas, and, in particular, of the sense in which one can meaningfully speak of positronium-like bound states. It is shown that such states can be explicitly introduced by making an ansatz for them. The annihilation rate computed from this ansatz is, however, somewhat lower than the experimental values. The properties of the ladder approximation are also analyzed, and it is shown that the frequently stated opinion that the divergence of this approximation at sufficiently low densities corresponds to the formation of positronium-like states is incorrect. It is shown that this divergence corresponds to an unphysical instability of the whole system. Some cancellation mechanisms which help to explain the success of the ladder approximation for high and intermediary electron densities are demonstrated. Whenever possible, comparisons are given with the important and similar case of a heavy impurity.