Lorentz Generators for the Maxwell Field and Gauge Fixing

We propose a direct derivation of the Lorentz generators for the four-potential of electrodynamics on the basis of Wigner's theorem. The derivation relies on a study of the behaviour of polarisation vectors under $ {k}$-space differentiation. The Coulomb and Lorenz gauges are discussed in that respect, and gauge invariance under Poincaré transformations is examined. The Poincaré generators given by Bia?ynicki-Birula and Bia?ynicka-Birula are found to correspond to the Coulomb gauge case.