The multipolar Hamiltonian in radiation theory

The multipolar Hamiltonian is widely used in applications of quantum electrodynamics to quantum optics and theoretical chemistry. In this paper, it is shown how this form of the Hamiltonian, normally derived within the Coulomb gauge, may be obtained from a Lagrangian in an arbitrary gauge. The method involves the construction of the Routhian functional to eliminate ignorable coordinates from which the Hamil­tonian is obtained. Further, the electrostatic interactions arise from a constraint and are independent of the initial choice of the scalar potential. The contributions from Röntgen currents, arising from dielectric motion, are allowed for by including nuclear motions in the theory.