Generalised gauge invariance of electromagnetism

Abstract

Classical electromagnetism in the Lorentz gauge is reviewed from the standpoint of the variational principle. The gauge condition is imposed as a constraint on the Lagrangian density of the system using a Lagrange multiplier. A similar formulation is followed for the 'complete alpha -Lorentz gauge' of Yang (1976). The uncoupled field equations in this gauge are derived and solved under simple boundary conditions. Without conforming to Maxwell's interpretation that electromagnetic radiation should propagate at speed c, the authors show that it must always do so regardless of the value of alpha . This is so because under the simple boundary conditions chosen, the electromagnetic potentials in the 'complete alpha -Lorentz gauge' are a gauge transformation of the first kind of the electromagnetic potentials in the Lorentz gauge. It is shown that electromagnetic radiation propagates at the invariant speed c under the most general of boundary conditions and under a more general type of gauge transformation. These classical results are generalised by brief reference to the Aharonov-Bohm effect. Finally, repercussions regarding advanced, as opposed to retarded, potentials and the Lorentz invariance of the formulation are considered.