Local Gauge Invariance

Abstract

Quantum electrodynamics stands out as a fundamental theory of electromagnetism that incorporates the principles of quantum mechanics and special relativity in a consistent way. A conspicuous ingredient of the theory, whose classical formulation was first given by Maxwell (1864), is its invariance under local gauge transformations. Under these transformations wave functions (or fields) associated with charged particles change their phase by an amount that may vary from one space-time point to another. It is possible to generalize these gauge transformations to more complicated groups of transformations, leading to a large class of so-called gauge theories. Experimentally, it is now a well-established fact that these theories underly the strong, weak and electro-magnetic interactions between elementary particles, as we shall discuss in due course. We should point out here that also the gravitational force is described by a gauge theory, as the theory of general relativity is invariant under con-tinuous reparametrizations of space-time, which are called general coordinate transformations. However, general coordinate transformations are of a differ-ent nature than the generalized local phase transformations and fall outside the framework of this book. In this chapter we will start by examining the immediate consequences of local gauge invariance, and discuss the principal concepts that are involved.