The QED beta-Function: A First Principles Derivation

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.