Abelian Gauge Theories

Abstract

This chapter focuses on abelian gauge theory, whose physical realization is Quantum Electrodynamics (QED). The chapter is organized as follows. It begins with elementary considerations about the massive vector field in perturbation theory. It shows that coupling to matter field leads to field theories that are renormalizable in four dimensions only if the vector field is coupled to a conserved current. In the latter case the massless vector limit can be defined. The corresponding field theories are gauge invariant. It then discusses the specific properties of gauge invariant theories and mentions the IR problem of physical observables. It quantizes gauge theories starting directly from first principles. The formal equivalence between different gauges is established. Regularization methods are presented which allow overcoming the new diffculties one encounters in gauge theories. The abelian gauge symmetry, broken by gauge fixing terms, then leads to a set of WT identities which are used to prove the renormalizability of the theory. The gauge dependence of correlation functions in a set of covariant gauges is determined. Renormalization group equations follow and the RG β-function is calculated at leading order. As an introduction to the next chapter, the abelian Higgs mechanism is analyzed. Finally, the chapter comments about stochastic quantization of gauge theories.