ABSTRACT
In Chapter 3 we considered continuous global symmetries, parametrized by real constants βi . In local or gauge symmetries 1 the βi are promoted to arbitrary differentiable functions βi (x) of space and time. Gauge invariance is sometimes motivated on esthetic grounds. For example, why should the phase of the electron field on the Earth be correlated with its phase on Mars? This is not entirely compelling because the standard model does involve global symmetries (though they may derive from gauge symmetries in an underlying theory). In any case, we will take the pragmatic view that gauge invariance is a powerful tool for constructing well-behaved field theories and are the unique renormalizable field theories for spin-1 particles. In particular, each generator of a gauge invariant theory must correspond to an (apparently) massless spin-1 gauge boson, which mediates an (apparently) long-range force, and the diagonal generators of an unbroken gauge theory correspond to conserved charges (Weyl, 1929). The gauge interactions are uniquely prescribed once one specifies the gauge group, the representations of the matter fields, and a gauge coupling constant g for each group factor. This approach is opposite to historical development: Maxwell’s equations of classical electrodynamics were first derived from observation and consistency, and then it was noticed that they were invariant under gauge transformations, i.e., that the vector and scalar potential involved redundant degrees of freedom. In this chapter, we outline the construction of gauge invariant Lagrangian densities. More detailed treatments include (Abers and Lee, 1973; Weinberg, 1973c,d, 1995; Peskin and Schroeder, 1995).
