Quantum-Electrodynamical Processes

Abstract

In this chapter we will gain some practical abilities in calculating various interesting quantum-electrodynamical processes that are of great importance. Thus the following chapter consists mainly of examples and problems. First, we start by applying the propagator formalism to problems related to electron-positron scattering. We shall proceed by considering more complicated processes including photons and other particles. As in the original publications of Feynman 1 we shall derive general rules for the practical calculation of transition probabilities and cross sections of any process involving electrons, positrons, and photons. These rules, although derived in a non-rigorous fashion, provide a correct and complete description of QED processes. The same set of "Feynman rules" results from a systematic treatment within the framework of quantum field theory. 3.1 Coulomb Scattering of Electrons We calculate the Rutherford scattering of an electron at a fixed Coulomb potential. The appropriate S-matrix element is given by (2.41a) and (2.42) and can be used directly. For f = i and renaming the integration variable y → x one gets S f i = −ie d 4 x ¯ ψ f (x) / A(x) Ψ i (x) (f = i). (3.1) Here e < 0 is the charge of the electron. In order to discuss (3.1) in an approximation that is solvable in practice we calculate the process in lowest order of perturbation theory. Then Ψ i (x) is approximated by the incoming plane wave ψ i (x) of an electron with momentum p i and spin s i : ψ i (x) = m 0 E i V u(p i , s i)e −ip i ·x. (3.2) V denotes the normalization volume, i.e. ψ i is normalized to probability 1 in a box with volume V. Similarly ¯ ψ f (x) is given by ¯ ψ f (x) = m 0