Identification of Poincaré-gauge and multipolar nonrelativistic theories of QED

For over six decades, quantum electrodynamics (QED) in multipolar form has been an invaluable tool for understanding quantum-scale atomic and molecular interactions. However, its relation to the Poincare-gauge a recent topic of controversy debate. It was claimed by Rousseau Felbacq article Scientific Reports 7, 11115 (2017) that Hamiltonian QED is not same as it generally equivalent Coulomb-gauge QED. This claim subsequently refuted, but since both sides debate appear technically sound, clear reconciliation remains be given. task paramount importance due widespread use optics physics. Here, unlike other responses, we adopt method using Dirac's constrained quantisation procedure. our treatment shows are identical. We identify precise source apparent incompatibility previous results nothing more than semantic mismatch. In fact there no inconsistencies. Our firmly rigorously solidify theory.