Gauge and unitary transformations in multipolar quantum optics

Multipolar quantum optics deals with the interaction of light with matter as a many-body bound system of charged particles where the coupling to electromagnetic fields is in terms of the multipolar electric polarization and magnetization. We describe two transformations applied to the conventional non-relativistic formalism, namely a gauge transformation applied directly to the fields at the Lagrangian stage and a unitary transformation applied to the old Hamiltonian. We show how such transformations lead to the same Power–Zienau–Woolley (PZW) formulation of the quantum electrodynamics (QED) of an overall electrically neutral many-body bound system of charges, including the internal motion as well as the gross dynamics of the centre of mass. Besides highlighting the utility of the multipolar formalism as a reliable and convenient platform in dealing with optical processes in atomic and molecular physics, it is shown how the analysis can also lead to the identification of the Röntgen effect arising from the gross motion of an electric dipole moment in a magnetic field and the Aharonov–Casher effect due to the motion of a magnetic dipole moment in an electric field. The importance of the two effects is pointed out in both experimental and theoretical contexts. This article is part of the theme issue ‘The quantum theory of light’.