Gauge invariance classical electrodynamics

A simple example in classical electrodynamics of what is now known as gauge invariance was introduced by Heaviside [3,4], who reduced the original electrodynamical equations of Maxwell to their present form. Therefore, these equations are more properly known as the Maxwell-Heaviside equations and, in the terminology of contemporary gauge field theory, are identifiable as U(l) Yang-Mills equations [15]. The subj ect of this chapter is 0(3) Yang-Mills gauge theory applied to electrodynamics and electroweak theory. 

QUANTUM ELECTRODYNAMICS POTENTIALS, GAUGE INVARIANCE, AND ANALOGY TO CLASSICAL ELECTRODYNAMICS... 

In classical electrodynamics, the field equations for the Maxwell field A/( depend only on the antisymmetric tensor which is invariant under a gauge transformation A/l A/l + ticduxix), where x is an arbitrary scalar field in space-time. Thus the vector field A/( is not completely determined by the theory. It is customary to impose an auxiliary gauge condition, such as 9/x/Fx = 0, in order to simplify the field equations. In the presence of an externally determined electric current density 4-vector j11, the Maxwell Lagrangian density is... 

In theories that are gauge invariant we are free either to try to work in a manifestly gauge-invariant fashion, or, since it cannot affect the final physical results, to choose a convenient gauge in which to work. In classical electrodynamics one often uses the Lorentz gauge in which... 

There has been an unusual amount of debate concerning the development of 0(3) electrodynamics, over a period of 7 years. When the 2 (3) field was first proposed [48], it was not realized that it was part of an 0(3) electrodynamics homomorphic with Barrett s SU(2) invariant electrodynamics [50] and therefore had a solid basis in gauge theory. The first debate published [70,79] was between Barron and Evans. The former proposed that B,3> violates C and CPT symmetry. This incorrect assertion was adequately answered by Evans at the time, but it is now clear that if B<3) violated C and CPT, so would classical gauge theory, a reduction to absurdity. For example, Barrett s SU(2) invariant theory [50] would violate C... 

---