Some Applications of the Mass Operator in Quantum Electrodynamics

Abstract

The one-photon mass operator is calculated explicitly up to the second order in the applied field, once for the purpose of taking matrix elements between states that satisfy the complete Dirac equation, and once for free-electron states. The former is manifestly gauge-invariant. The part linear in the applied field is also calculated for matrix elements, only one side of which satisfies the Dirac equation; it is seen, in application, to cancel precisely the non-gauge invariant part (for free electrons) of the second-order mass operator. A systematic procedure for carrying out the photon integration at the very beginning of the calculation is described and used. After the (nonperturbation) derivation of a cross-section formula in terms of the mass operator, the latter is used to rederive the integrated Klein-Nishina formula and also applied to the simple case of a constant field. The use of the mass operator technique for the calculation of inelastic cross sections is demonstrated and it is proved that, except for virtual processes induced by the radiation field (in contrast to the static field), the low-energy limit of the bremsstrahlung cross section is a multiple of the one for elastic scattering (to all orders). Finally, the lowest order-radiative corrections to Coulomb scattering are rederived.