ABSTRACT
Gauge invariance and Lorentz covariance are the two fundamental physical principles underlying both classical and quantum electrodynamics. While Lorentz covariance embodies the global invariance of electrodynamics under the space-time rotations of special relativity in hyperbolic space-time, gauge invariance is a local symmetry reflecting the fact that the four-potential can be modified according to
(4.1)
without changing the electromagnetic field; here, is an arbitrary function of the four-position. The term
local symmetry
can be explained as follows: while the local character of the gauge transform is obvious, the fact that it is a symmetry may seem surprising at first glance because it looks quite different from usual symmetries, such as a 2
π
/
n
rotation leaving an
n
-polygon unchanged. However, by definition, a symmetry is an operation that transforms an object into itself; for the gauge symmetry, that object is
F
µν. This idea is illustrated schematically in Figure 4.1. Moreover, as prescribed by Noether’s theorem, there is an invariant quantity associated with this local symmetry: the four-current satisfies the conservation equation
(4.2)
In this chapter we explore the concept of electrodynamical gauge invariance and its connection with charge conservation. We review the approach first taken by Weyl, who introduced pure infinitesimal geometry to describe gauge invariance, and we introduce a few fundamental ideas, including Noether’s theorem and the relativistic field Lagrangian, which prove extremely useful when extended to QED and modern quantum field theories based on gauge invariance.
