Gauge invariance and the electric polarization field

Abstract

Physical observables associated with interactions between electromagnetic radiation and charged particles are required to be gauge invariant, i.e. independent of the gauge of the field potentials. Gauge invariance implies the involvement of the electric polarization field, P(x), in electrodynamics, but leaves undetermined the transverse component P(x)^⊥ which occurs in the interaction terms coupling charges to radiation. We put the Hamiltonian for non-relativistic electrodynamics into a completely general form displaying the arbitrary polarization field. The well-known Coulomb gauge and ‘multipolar’ Hamiltonian formalisms then arise as special cases corresponding to particular choices of P(x)^⊥. For practical calculations some choice of P(x)^⊥ has to be made, and we present a particularly useful form for neutral collection of charges, not necessarily bound, but typically appropriate to atoms and molecules. Some results concerning the independence from the arbitrary quantities of physical observables are described.