In the previous chapter, we investigated elementary processes in QED and calculated cross-sections to lowest order in perturbation theory. Taking higher orders into account, one will obtain corrections of the order of the Sommerfeld fine-structure constant α to the lowest-order results. However, performing such calculations, one will encounter divergent integrals. Nevertheless, we will try to remedy this situation in three steps. First, we want to regularize the theory, which means that we modify the theory in order to keep it finite and well-defined to all orders in perturbation theory. This step is called regularization for which there are several methods. Specific methods of regularization include: Pauli–Villars regularization, dimensional regularization, lattice regularization, Riemann's ξ-function regularization, etc. Here, we will mainly investigate Pauli–Villars regularization and dimensional regularization. Second, we recognize that the non-interacting particles (i.e. the free asymptotic fields) are not identical to the real physical particles that interact. Thus, the interactions modify the properties of the particles such as the masses and the charges. Of course, all the relevant predictions of the theory must be expressed in terms of the properties of the physical particles and not the noninteracting (or bare) particles. This step is called renormalization. Third, we have to revert from the regularized theory back to QED, and thus, the infinities of QED will appear in the relations between the bare and physical particles. Of course, these relations as well as the bare particles are completely unobservable.