The Radiative Collisions of Positrons and Electrons

The general matrix elements for the collision of two electrons ($e-e$ case) or a positron and an electron ($p-e$ case) accompanied by the emission of a quantum of radiation are formulated. It is expected that the two cross sections will differ considerably both because of the existence of a dipole for the $p-e$ case, and because of the essentially different nature of the exchange phenomenon in the $p-e$ case arising there from the possibility of virtual pair creation or annihilation rather than the indistinguishability of the particles. Two cases are calculated in detail; the non-relativistic limit and the extreme relativistic limit in which the quantum energy is restricted to $\leqq E^{\frac{1}{2}}$ in the laboratory system. In the former case the cross section for the $e-e$ case is found to vanish if the momentum of the quantum is neglected, while the $p-e$ cross section is comparable to that for the "bremsstrahlung" process. In the latter case the cross sections of the two cases are of the same order of magnitude the $e-e$ cross section being greater because of the exchange terms, the exchange terms of the $p-e$ case being negligible in all of the cases considered.