Mathematical Constraints on Gauge in Maxwellian Electrodynamics

The structure of classical electrodynamics based on the variational principle together with causality and spacetime homogeneity is analyzed. It is proved that in this case the 4-potentials are defined uniquely. On the other hand, the approach where Maxwell equations and the Lorentz law of force are regarded as cornerstones of the theory allows gauge transformations. For this reason, the two theories are not equivalent. A simple example substantiates this conclusion. Quantum physics is linked to the variational principle and it is proved that the same result holds for it. The compatibility of this conclusion with gauge invariance of the Lagrangian density is explained. Several alternative possibilities that may follow this work are pointed out.