Abstract
Quantum electrodynamics (QED) is a quantum-relativistic theory for electromagnetism.
It is among the most successful quantum field theories, which collectively
describe the dynamics of elementary particles at high energies. Unfortunately, the
dynamical expressions for interacting theories do not have analytical solutions. Approximate
solutions are obtained through perturbation theory, although a significant
obstacle is encountered when some of the terms in the expansion diverge. These divergences
have historically caused much debate around the legitimacy of QED, and
it was not until the modern theory of renormalization and the invention of the renormalization
group that QED gained a wider acceptance as a legitimate calculational
tool. The renormalization group yielded surprising physical consequences, such as
running coupling constants and anomalous dimensions. Of primary interest to this
presentation is the beta-function of QED, a result of the renormalization group that describes
the dependence of the electromagnetic coupling on particle momentum. The
beta-function is of both theoretical and experimental importance, as its derivation yields
information about the legitimacy of perturbation theory, the presence of a Landau
ghost, and the electric charge polarization of the vacuum; the latter of which is a
contribution to the famous Lamb shift. Here the beta-function is derived from first
principles. Starting with the formulation of quantum fields in Fock space we will
discuss the formalism around free and interacting quantum field theories, ultimately
discussing renormalization, QED, the QED beta-function and its implications.